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Sample records for fredholm integral equations

  1. Nonlinear Fredholm Integral Equation of the Second Kind with Quadrature Methods

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    M. Jafari Emamzadeh

    2010-06-01

    Full Text Available In this paper, a numerical method for solving the nonlinear Fredholm integral equation is presented. We intend to approximate the solution of this equation by quadrature methods and by doing so, we solve the nonlinear Fredholm integral equation more accurately. Several examples are given at the end of this paper

  2. Numerical method for solving linear Fredholm fuzzy integral equations of the second kind

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    Abbasbandy, S. [Department of Mathematics, Imam Khomeini International University, P.O. Box 288, Ghazvin 34194 (Iran, Islamic Republic of)]. E-mail: saeid@abbasbandy.com; Babolian, E. [Faculty of Mathematical Sciences and Computer Engineering, Teacher Training University, Tehran 15618 (Iran, Islamic Republic of); Alavi, M. [Department of Mathematics, Arak Branch, Islamic Azad University, Arak 38135 (Iran, Islamic Republic of)

    2007-01-15

    In this paper we use parametric form of fuzzy number and convert a linear fuzzy Fredholm integral equation to two linear system of integral equation of the second kind in crisp case. We can use one of the numerical method such as Nystrom and find the approximation solution of the system and hence obtain an approximation for fuzzy solution of the linear fuzzy Fredholm integral equations of the second kind. The proposed method is illustrated by solving some numerical examples.

  3. On Fredholm-Stieltjes quadratic integral equation with supremum

    International Nuclear Information System (INIS)

    Darwish, M.A.

    2007-08-01

    We prove an existence theorem of monotonic solutions for a quadratic integral equation of Fredholm-Stieltjes type in C[0,1]. The concept of measure of non-compactness and a fixed point theorem due to Darbo are the main tools in carrying out our proof. (author)

  4. Optimal Homotopy Asymptotic Method for Solving the Linear Fredholm Integral Equations of the First Kind

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    Mohammad Almousa

    2013-01-01

    Full Text Available The aim of this study is to present the use of a semi analytical method called the optimal homotopy asymptotic method (OHAM for solving the linear Fredholm integral equations of the first kind. Three examples are discussed to show the ability of the method to solve the linear Fredholm integral equations of the first kind. The results indicated that the method is very effective and simple.

  5. Numerical Treatment of Fixed Point Applied to the Nonlinear Fredholm Integral Equation

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    Berenguer MI

    2009-01-01

    Full Text Available The authors present a method of numerical approximation of the fixed point of an operator, specifically the integral one associated with a nonlinear Fredholm integral equation, that uses strongly the properties of a classical Schauder basis in the Banach space .

  6. On randomized algorithms for numerical solution of applied Fredholm integral equations of the second kind

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    Voytishek, Anton V.; Shipilov, Nikolay M.

    2017-11-01

    In this paper, the systematization of numerical (implemented on a computer) randomized functional algorithms for approximation of a solution of Fredholm integral equation of the second kind is carried out. Wherein, three types of such algorithms are distinguished: the projection, the mesh and the projection-mesh methods. The possibilities for usage of these algorithms for solution of practically important problems is investigated in detail. The disadvantages of the mesh algorithms, related to the necessity of calculation values of the kernels of integral equations in fixed points, are identified. On practice, these kernels have integrated singularities, and calculation of their values is impossible. Thus, for applied problems, related to solving Fredholm integral equation of the second kind, it is expedient to use not mesh, but the projection and the projection-mesh randomized algorithms.

  7. A Special Variant of the Moment Method for Fredholm Integral Equations of the Second Kind

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    S. A. Solov’eva

    2015-01-01

    Full Text Available We consider the linear Fredholm integral equation of the second kind, where the kernel and the free term are smooth functions. We find the unknown function in this class as well.Exact and approximate methods for the solution of linear Fredholm integral equations of the second kind are well developed. However, classical methods do not take into account the structural properties of the kernel and the free term of equation.In this paper we develop and justify a special variant of the moment method to solve this equation, which takes into account the differential properties of initial data. The proposed paper furthers studies of N.S Gabbasov, I.P. Kasakina, and S.A Solov’eva. We use approximation theory, version of the general theory of approximate methods of analysis that Gabdulkhayev B.G suggested, and methods of functional analysis to prove theorems. In addition, we use N.S. Gabbasov’s ideas and methods in papers that are devoted to the Fredholm equations of the first kind, as well as N.S. Gabbasov and S.A Solov’eva’s investigations on the Fredholm equations of the third kind in the space of distributions.The first part of the paper provides a description of the basic function space and elements of the theory of approximation in it.In the second part we propose and theoretically justify a generalized moment method. We have demonstrated that the improvement of differential properties of the initial data improves the approximation accuracy. Since, in practice, the approximate equations are solved, as a rule, only approximately, we prove the stability and causality of the proposed method. The resulting estimate of the paper is in good agreement with the estimate for the ordinary moment method for equations of the second kind in the space of continuous functions.In the final section we have shown that a developed method is optimal in order of accuracy among all polynomial projection methods to solve Fredholm integral equations of the second

  8. Optimal Homotopy Asymptotic Method for Solving System of Fredholm Integral Equations

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    Bahman Ghazanfari

    2013-08-01

    Full Text Available In this paper, optimal homotopy asymptotic method (OHAM is applied to solve system of Fredholm integral equations. The effectiveness of optimal homotopy asymptotic method is presented. This method provides easy tools to control the convergence region of approximating solution series wherever necessary. The results of OHAM are compared with homotopy perturbation method (HPM and Taylor series expansion method (TSEM.

  9. Handbook of integral equations

    CERN Document Server

    Polyanin, Andrei D

    2008-01-01

    This handbook contains over 2,500 integral equations with solutions as well as analytical and numerical methods for solving linear and nonlinear equations. It explores Volterra, Fredholm, WienerHopf, Hammerstein, Uryson, and other equations that arise in mathematics, physics, engineering, the sciences, and economics. This second edition includes new chapters on mixed multidimensional equations and methods of integral equations for ODEs and PDEs, along with over 400 new equations with exact solutions. With many examples added for illustrative purposes, it presents new material on Volterra, Fredholm, singular, hypersingular, dual, and nonlinear integral equations, integral transforms, and special functions.

  10. Some new retarded nonlinear Volterra-Fredholm type integral inequalities with maxima in two variables and their applications.

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    Xu, Run; Ma, Xiangting

    2017-01-01

    In this paper, we establish some new retarded nonlinear Volterra-Fredholm type integral inequalities with maxima in two independent variables, and we present the applications to research the boundedness of solutions to retarded nonlinear Volterra-Fredholm type integral equations.

  11. ADM For Solving Linear Second-Order Fredholm Integro-Differential Equations

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    Karim, Mohd F.; Mohamad, Mahathir; Saifullah Rusiman, Mohd; Che-Him, Norziha; Roslan, Rozaini; Khalid, Kamil

    2018-04-01

    In this paper, we apply Adomian Decomposition Method (ADM) as numerically analyse linear second-order Fredholm Integro-differential Equations. The approximate solutions of the problems are calculated by Maple package. Some numerical examples have been considered to illustrate the ADM for solving this equation. The results are compared with the existing exact solution. Thus, the Adomian decomposition method can be the best alternative method for solving linear second-order Fredholm Integro-Differential equation. It converges to the exact solution quickly and in the same time reduces computational work for solving the equation. The result obtained by ADM shows the ability and efficiency for solving these equations.

  12. Stochastic Analysis of Gaussian Processes via Fredholm Representation

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    Tommi Sottinen

    2016-01-01

    Full Text Available We show that every separable Gaussian process with integrable variance function admits a Fredholm representation with respect to a Brownian motion. We extend the Fredholm representation to a transfer principle and develop stochastic analysis by using it. We show the convenience of the Fredholm representation by giving applications to equivalence in law, bridges, series expansions, stochastic differential equations, and maximum likelihood estimations.

  13. Shifted Legendre method with residual error estimation for delay linear Fredholm integro-differential equations

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    Şuayip Yüzbaşı

    2017-03-01

    Full Text Available In this paper, we suggest a matrix method for obtaining the approximate solutions of the delay linear Fredholm integro-differential equations with constant coefficients using the shifted Legendre polynomials. The problem is considered with mixed conditions. Using the required matrix operations, the delay linear Fredholm integro-differential equation is transformed into a matrix equation. Additionally, error analysis for the method is presented using the residual function. Illustrative examples are given to demonstrate the efficiency of the method. The results obtained in this study are compared with the known results.

  14. Numerical treatments for solving nonlinear mixed integral equation

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    M.A. Abdou

    2016-12-01

    Full Text Available We consider a mixed type of nonlinear integral equation (MNLIE of the second kind in the space C[0,T]×L2(Ω,T<1. The Volterra integral terms (VITs are considered in time with continuous kernels, while the Fredholm integral term (FIT is considered in position with singular general kernel. Using the quadratic method and separation of variables method, we obtain a nonlinear system of Fredholm integral equations (NLSFIEs with singular kernel. A Toeplitz matrix method, in each case, is then used to obtain a nonlinear algebraic system. Numerical results are calculated when the kernels take a logarithmic form or Carleman function. Moreover, the error estimates, in each case, are then computed.

  15. Introducing Differential Equations Students to the Fredholm Alternative--In Staggered Doses

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    Savoye, Philippe

    2011-01-01

    The development, in an introductory differential equations course, of boundary value problems in parallel with initial value problems and the Fredholm Alternative. Examples are provided of pairs of homogeneous and nonhomogeneous boundary value problems for which existence and uniqueness issues are considered jointly. How this heightens students'…

  16. Global existence for Volterra-Fredholm type neutral impulsive functional integrodifferential equations

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    V. Vijayakumar

    2012-09-01

    Full Text Available n this paper, we study the global existence of solutions for the initial value problems for Volterra-Fredholm type neutral impulsive functional integrodifferential equations. Using the Leray-Schauder's Alternative theorem, we derive conditions under which a solution exists globally. An application is provided to illustrate the theory.

  17. Exponential Convergence for Numerical Solution of Integral Equations Using Radial Basis Functions

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    Zakieh Avazzadeh

    2014-01-01

    Full Text Available We solve some different type of Urysohn integral equations by using the radial basis functions. These types include the linear and nonlinear Fredholm, Volterra, and mixed Volterra-Fredholm integral equations. Our main aim is to investigate the rate of convergence to solve these equations using the radial basis functions which have normic structure that utilize approximation in higher dimensions. Of course, the use of this method often leads to ill-posed systems. Thus we propose an algorithm to improve the results. Numerical results show that this method leads to the exponential convergence for solving integral equations as it was already confirmed for partial and ordinary differential equations.

  18. Nonlinear Ritz approximation for Fredholm functionals

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    Mudhir A. Abdul Hussain

    2015-11-01

    Full Text Available In this article we use the modify Lyapunov-Schmidt reduction to find nonlinear Ritz approximation for a Fredholm functional. This functional corresponds to a nonlinear Fredholm operator defined by a nonlinear fourth-order differential equation.

  19. Projected regression method for solving Fredholm integral equations arising in the analytic continuation problem of quantum physics

    International Nuclear Information System (INIS)

    Arsenault, Louis-François; Millis, Andrew J; Neuberg, Richard; Hannah, Lauren A

    2017-01-01

    We present a supervised machine learning approach to the inversion of Fredholm integrals of the first kind as they arise, for example, in the analytic continuation problem of quantum many-body physics. The approach provides a natural regularization for the ill-conditioned inverse of the Fredholm kernel, as well as an efficient and stable treatment of constraints. The key observation is that the stability of the forward problem permits the construction of a large database of outputs for physically meaningful inputs. Applying machine learning to this database generates a regression function of controlled complexity, which returns approximate solutions for previously unseen inputs; the approximate solutions are then projected onto the subspace of functions satisfying relevant constraints. Under standard error metrics the method performs as well or better than the Maximum Entropy method for low input noise and is substantially more robust to increased input noise. We suggest that the methodology will be similarly effective for other problems involving a formally ill-conditioned inversion of an integral operator, provided that the forward problem can be efficiently solved. (paper)

  20. Application of Reproducing Kernel Method for Solving Nonlinear Fredholm-Volterra Integrodifferential Equations

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    Omar Abu Arqub

    2012-01-01

    Full Text Available This paper investigates the numerical solution of nonlinear Fredholm-Volterra integro-differential equations using reproducing kernel Hilbert space method. The solution ( is represented in the form of series in the reproducing kernel space. In the mean time, the n-term approximate solution ( is obtained and it is proved to converge to the exact solution (. Furthermore, the proposed method has an advantage that it is possible to pick any point in the interval of integration and as well the approximate solution and its derivative will be applicable. Numerical examples are included to demonstrate the accuracy and applicability of the presented technique. The results reveal that the method is very effective and simple.

  1. Simulating propagation of coherent light in random media using the Fredholm type integral equation

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    Kraszewski, Maciej; Pluciński, Jerzy

    2017-06-01

    Studying propagation of light in random scattering materials is important for both basic and applied research. Such studies often require usage of numerical method for simulating behavior of light beams in random media. However, if such simulations require consideration of coherence properties of light, they may become a complex numerical problems. There are well established methods for simulating multiple scattering of light (e.g. Radiative Transfer Theory and Monte Carlo methods) but they do not treat coherence properties of light directly. Some variations of these methods allows to predict behavior of coherent light but only for an averaged realization of the scattering medium. This limits their application in studying many physical phenomena connected to a specific distribution of scattering particles (e.g. laser speckle). In general, numerical simulation of coherent light propagation in a specific realization of random medium is a time- and memory-consuming problem. The goal of the presented research was to develop new efficient method for solving this problem. The method, presented in our earlier works, is based on solving the Fredholm type integral equation, which describes multiple light scattering process. This equation can be discretized and solved numerically using various algorithms e.g. by direct solving the corresponding linear equations system, as well as by using iterative or Monte Carlo solvers. Here we present recent development of this method including its comparison with well-known analytical results and a finite-difference type simulations. We also present extension of the method for problems of multiple scattering of a polarized light on large spherical particles that joins presented mathematical formalism with Mie theory.

  2. Integral equation for Coulomb problem

    International Nuclear Information System (INIS)

    Sasakawa, T.

    1986-01-01

    For short range potentials an inhomogeneous (homogeneous) Lippmann-Schwinger integral equation of the Fredholm type yields the wave function of scattering (bound) state. For the Coulomb potential, this statement is no more valid. It has been felt difficult to express the Coulomb wave function in a form of an integral equation with the Coulomb potential as the perturbation. In the present paper, the author shows that an inhomogeneous integral equation of a Volterra type with the Coulomb potential as the perturbation can be constructed both for the scattering and the bound states. The equation yielding the binding energy is given in an integral form. The present treatment is easily extended to the coupled Coulomb problems

  3. Integral equations of the first kind, inverse problems and regularization: a crash course

    International Nuclear Information System (INIS)

    Groetsch, C W

    2007-01-01

    This paper is an expository survey of the basic theory of regularization for Fredholm integral equations of the first kind and related background material on inverse problems. We begin with an historical introduction to the field of integral equations of the first kind, with special emphasis on model inverse problems that lead to such equations. The basic theory of linear Fredholm equations of the first kind, paying particular attention to E. Schmidt's singular function analysis, Picard's existence criterion, and the Moore-Penrose theory of generalized inverses is outlined. The fundamentals of the theory of Tikhonov regularization are then treated and a collection of exercises and a bibliography are provided

  4. An integral equation arising in two group neutron transport theory

    International Nuclear Information System (INIS)

    Cassell, J S; Williams, M M R

    2003-01-01

    An integral equation describing the fuel distribution necessary to maintain a flat flux in a nuclear reactor in two group transport theory is reduced to the solution of a singular integral equation. The formalism developed enables the physical aspects of the problem to be better understood and its relationship with the corresponding diffusion theory model is highlighted. The integral equation is solved by reducing it to a non-singular Fredholm equation which is then evaluated numerically

  5. The Convergence Study of the Homotopy Analysis Method for Solving Nonlinear Volterra-Fredholm Integrodifferential Equations

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    Behzad Ghanbari

    2014-01-01

    Full Text Available We aim to study the convergence of the homotopy analysis method (HAM in short for solving special nonlinear Volterra-Fredholm integrodifferential equations. The sufficient condition for the convergence of the method is briefly addressed. Some illustrative examples are also presented to demonstrate the validity and applicability of the technique. Comparison of the obtained results HAM with exact solution shows that the method is reliable and capable of providing analytic treatment for solving such equations.

  6. Fredholm and Wronskian representations of solutions to the KPI equation and multi-rogue waves

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    Gaillard, Pierre

    2016-06-01

    We construct solutions to the Kadomtsev-Petviashvili equation (KPI) in terms of Fredholm determinants. We deduce solutions written as a quotient of Wronskians of order 2N. These solutions, called solutions of order N, depend on 2N - 1 parameters. When one of these parameters tends to zero, we obtain N order rational solutions expressed as a quotient of two polynomials of degree 2N(N + 1) in x, y, and t depending on 2N - 2 parameters. So we get with this method an infinite hierarchy of solutions to the KPI equation.

  7. Scattering integral equations and four nucleon problem

    International Nuclear Information System (INIS)

    Narodetskii, I.M.

    1980-01-01

    Existing results from the application of integral equation technique to the four-nucleon bound states and scattering are reviewed. The first numerical calculations of the four-body integral equations have been done ten years ago. Yet, it is still widely believed that these equations are too complicated to solve numerically. The purpose of this review is to provide a clear and elementary introduction in the integral equation method and to demonstrate its usefulness in physical applications. The presentation is based on the quasiparticle approach. This permits a simple interpretation of the equations in terms of quasiparticle scattering. The mathematical basis for the quasiparticle approach is the Hilbert-Schmidt method of the Fredholm integral equation theory. The first part of this review contains a detailed discussion of the Hilbert-Schmidt expansion as applied to the 2-particle amplitudes and to the kernel of the four-body equations. The second part contains the discussion of the four-body quasiparticle equations and of the resed forullts obtain bound states and scattering

  8. Fredholm's minors of arbitrary order: their representations as a determinant of resolvents and in terms of free fermions and an explicit formula for their functional derivative

    International Nuclear Information System (INIS)

    Feinberg, Joshua

    2004-01-01

    We study the Fredholm minors associated with a Fredholm equation of the second type. We present a couple of new linear recursion relations involving the nth and (n - 1)th minors, whose solution is a representation of the nth minor as an n x n determinant of resolvents. The latter is given a simple interpretation in terms of a path integral over non-interacting fermions. We also provide an explicit formula for the functional derivative of a Fredholm minor of order n with respect to the kernel. Our formula is a linear combination of the nth and the (n ± 1)th minors

  9. On the asymptotic solution to a class of linear integral equations

    International Nuclear Information System (INIS)

    Gautesen, A.K.

    1988-01-01

    The authors consider Fredholm integral equations of the first kind whose kernels are a function of the difference between two points times a large parameter. Conditions on the kernel are stated in terms of a function corresponding to a Wiener-Hopf factorization of the Fourier transform of the kernel. They give the complete asymptotic expansions of the solution to the integral equations. As applications of the author's results, the author considers the steady-state, acoustical scattering of a plane wave by both a hard strip and a soft strip. The author's results are uniform with respect to the direction of incidence

  10. Lévy flights in an infinite potential well as a hypersingular Fredholm problem.

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    Kirichenko, Elena V; Garbaczewski, Piotr; Stephanovich, Vladimir; Żaba, Mariusz

    2016-05-01

    We study Lévy flights with arbitrary index 0potential well of infinite depth. Such a problem appears in many physical systems ranging from stochastic interfaces to fracture dynamics and multifractality in disordered quantum systems. The major technical tool is a transformation of the eigenvalue problem for initial fractional Schrödinger equation into that for Fredholm integral equation with hypersingular kernel. The latter equation is then solved by means of expansion over the complete set of orthogonal functions in the domain D, reducing the problem to the spectrum of a matrix of infinite dimensions. The eigenvalues and eigenfunctions are then obtained numerically with some analytical results regarding the structure of the spectrum.

  11. Simulation electromagnetic scattering on bodies through integral equation and neural networks methods

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    Lvovich, I. Ya; Preobrazhenskiy, A. P.; Choporov, O. N.

    2018-05-01

    The paper deals with the issue of electromagnetic scattering on a perfectly conducting diffractive body of a complex shape. Performance calculation of the body scattering is carried out through the integral equation method. Fredholm equation of the second time was used for calculating electric current density. While solving the integral equation through the moments method, the authors have properly described the core singularity. The authors determined piecewise constant functions as basic functions. The chosen equation was solved through the moments method. Within the Kirchhoff integral approach it is possible to define the scattered electromagnetic field, in some way related to obtained electrical currents. The observation angles sector belongs to the area of the front hemisphere of the diffractive body. To improve characteristics of the diffractive body, the authors used a neural network. All the neurons contained a logsigmoid activation function and weighted sums as discriminant functions. The paper presents the matrix of weighting factors of the connectionist model, as well as the results of the optimized dimensions of the diffractive body. The paper also presents some basic steps in calculation technique of the diffractive bodies, based on the combination of integral equation and neural networks methods.

  12. Perturbation Results on Semi-Fredholm Operators and Applications

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    Abdelmoumen Boulbeba

    2009-01-01

    Full Text Available We give some results concerning stability in the Fredholm operators and Browder operators set, via the concept of measure of noncompactness. Moreover, we prove some localization results on the essential spectra of bounded operators on Banach space. As application, we describe the essential spectra of weighted shift operators. Finally, we describe the spectra of polynomially compact operators, and we use the obtained results to study the solvability for operator equations in Banach spaces.

  13. Integral equation models for the inverse problem of biological ion channel distributions

    International Nuclear Information System (INIS)

    French, D A; Groetsch, C W

    2007-01-01

    Olfactory cilia are thin hair-like filaments that extend from olfactory receptor neurons into the nasal mucus. Transduction of an odor into an electrical signal is accomplished by a depolarizing influx of ions through cyclic-nucleotide-gated channels in the membrane that forms the lateral surface of the cilium. In an experimental procedure developed by S. Kleene, a cilium is detached at its base and drawn into a recording pipette. The cilium base is then immersed in a bath of a channel activating agent (cAMP) which is allowed to diffuse into the cilium interior, opening channels as it goes and initiating a transmembrane current. The total current is recorded as a function of time and serves as data for a nonlinear integral equation of the first kind modeling the spatial distribution of ion channels along the length of the cilium. We discuss some linear Fredholm integral equations that result from simplifications of this model. A numerical procedure is proposed for a class of integral equations suggested by this simplified model and numerical results using simulated and laboratory data are presented

  14. A Multiple Iterated Integral Inequality and Applications

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    Zongyi Hou

    2014-01-01

    Full Text Available We establish new multiple iterated Volterra-Fredholm type integral inequalities, where the composite function w(u(s of the unknown function u with nonlinear function w in integral functions in [Ma, QH, Pečarić, J: Estimates on solutions of some new nonlinear retarded Volterra-Fredholm type integral inequalities. Nonlinear Anal. 69 (2008 393–407] is changed into the composite functions w1(u(s,w2(u(s,…, wn (u(s of the unknown function u with different nonlinear functions w1,w2,…,wn, respectively. By adopting novel analysis techniques, the upper bounds of the embedded unknown functions are estimated explicitly. The derived results can be applied in the study of solutions of ordinary differential equations and integral equations.

  15. Quarter-Sweep Iteration Concept on Conjugate Gradient Normal Residual Method via Second Order Quadrature - Finite Difference Schemes for Solving Fredholm Integro-Differential Equations

    International Nuclear Information System (INIS)

    Aruchunan, E.

    2015-01-01

    In this paper, we have examined the effectiveness of the quarter-sweep iteration concept on conjugate gradient normal residual (CGNR) iterative method by using composite Simpson's (CS) and finite difference (FD) discretization schemes in solving Fredholm integro-differential equations. For comparison purposes, Gauss- Seidel (GS) and the standard or full- and half-sweep CGNR methods namely FSCGNR and HSCGNR are also presented. To validate the efficacy of the proposed method, several analyses were carried out such as computational complexity and percentage reduction on the proposed and existing methods. (author)

  16. Numerical Solution of Nonlinear Fredholm Integro-Differential Equations Using Spectral Homotopy Analysis Method

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    Z. Pashazadeh Atabakan

    2013-01-01

    Full Text Available Spectral homotopy analysis method (SHAM as a modification of homotopy analysis method (HAM is applied to obtain solution of high-order nonlinear Fredholm integro-differential problems. The existence and uniqueness of the solution and convergence of the proposed method are proved. Some examples are given to approve the efficiency and the accuracy of the proposed method. The SHAM results show that the proposed approach is quite reasonable when compared to homotopy analysis method, Lagrange interpolation solutions, and exact solutions.

  17. Fredholm Modules over Graph C-Algebras

    DEFF Research Database (Denmark)

    Crisp, Tyrone

    2015-01-01

    We present two applications of explicit formulas, due to Cuntz and Krieger, for computations in K-homology of graph C∗-algebras. We prove that every K-homology class for such an algebra is represented by a Fredholm module having finite-rank commutators, and we exhibit generating Fredholm modules...

  18. Solution of the Helmholtz-Poincare Wave Equation using the coupled boundary integral equations and optimal surface eigenfunctions

    International Nuclear Information System (INIS)

    Werby, M.F.; Broadhead, M.K.; Strayer, M.R.; Bottcher, C.

    1992-01-01

    The Helmholtz-Poincarf Wave Equation (H-PWE) arises in many areas of classical wave scattering theory. In particular it can be found for the cases of acoustical scattering from submerged bounded objects and electromagnetic scattering from objects. The extended boundary integral equations (EBIE) method is derived from considering both the exterior and interior solutions of the H-PWECs. This coupled set of expressions has the advantage of not only offering a prescription for obtaining a solution for the exterior scattering problem, but it also obviates the problem of irregular values corresponding to fictitious interior eigenvalues. Once the coupled equations are derived, they can be obtained in matrix form by expanding all relevant terms in partial wave expansions, including a bi-orthogonal expansion of the Green's function. However some freedom in the choice of the surface expansion is available since the unknown surface quantities may be expanded in a variety of ways so long as closure is obtained. Out of many possible choices, we develop an optimal method to obtain such expansions which is based on the optimum eigenfunctions related to the surface of the object. In effect, we convert part of the problem (that associated with the Fredholms integral equation of the first kind) an eigenvalue problem of a related Hermitian operator. The methodology will be explained in detail and examples will be presented

  19. Solvability in the sense of sequences to some non-Fredholm operators

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    Vitaly Volpert

    2013-07-01

    Full Text Available We study the solvability of certain linear nonhomogeneous elliptic problems and show that under reasonable technical conditions the convergence in $L^2(mathbb{R}^d$ of their right sides implies the existence and the convergence in $H^2(mathbb{R}^d$ of the solutions. The equations involve second order differential operators without Fredholm property and we use the methods of spectral and scattering theory for Schrodinger type operators analogously to our preceding work [17].

  20. Remote monitoring of environmental particulate pollution - A problem in inversion of first-kind integral equations

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    Fymat, A. L.

    1975-01-01

    The determination of the microstructure, chemical nature, and dynamical evolution of scattering particulates in the atmosphere is considered. A description is given of indirect sampling techniques which can circumvent most of the difficulties associated with direct sampling techniques, taking into account methods based on scattering, extinction, and diffraction of an incident light beam. Approaches for reconstructing the particulate size distribution from the direct and the scattered radiation are discussed. A new method is proposed for determining the chemical composition of the particulates and attention is given to the relevance of methods of solution involving first kind Fredholm integral equations.

  1. Faddeev equations with an extra resonance channel in muon catalysis

    International Nuclear Information System (INIS)

    Motovilov, A.K.; Kuperin, Yu.A.; Susko, A.A.; Vinitskij, S.I.

    1988-01-01

    A three-body model is applied to derive inhomogeneous integral and differential Faddeev equations with energy-dependent potentials. The integral equations are shown to be Fredholm equations. The stricking probability ω s is determined in terms of the amplitudes of spherical waves in the asymptotics of the exit-channel wave function. The integral representation for ω s is given in terms of the continuum wave functions. To the first Born Approximation for the wave function, this representation yields an explicit expression for ω s through the expansion coefficients of the wave function of dtμ of the initial channel

  2. Evaluation of temperature dependent neutron resonance integrals

    International Nuclear Information System (INIS)

    Menon, S.V.G.; Sahni, D.C.

    1975-01-01

    The Fourier transform method is extended for evaluating temperature dependent resonance integrals and Doppler coefficients. With the temperature dependent cross-sections, the slowing-down equation is transformed into a Fredholm integral equation of second kind. A method of solution is presented using the familiar Gauss-Hermite quadrature formulae. As a byproduct of the above technique, a fast and accurate method for computing the resonance integral J-function is given. (orig.) [de

  3. Study on the Accuracy Improvement of the Second-Kind Fredholm Integral Equations by Using the Buffa-Christiansen Functions with MLFMA

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    Yue-Qian Wu

    2016-01-01

    Full Text Available Former works show that the accuracy of the second-kind integral equations can be improved dramatically by using the rotated Buffa-Christiansen (BC functions as the testing functions, and sometimes their accuracy can be even better than the first-kind integral equations. When the rotated BC functions are used as the testing functions, the discretization error of the identity operators involved in the second-kind integral equations can be suppressed significantly. However, the sizes of spherical objects which were analyzed are relatively small. Numerical capability of the method of moments (MoM for solving integral equations with the rotated BC functions is severely limited. Hence, the performance of BC functions for accuracy improvement of electrically large objects is not studied. In this paper, the multilevel fast multipole algorithm (MLFMA is employed to accelerate iterative solution of the magnetic-field integral equation (MFIE. Then a series of numerical experiments are performed to study accuracy improvement of MFIE in perfect electric conductor (PEC cases with the rotated BC as testing functions. Numerical results show that the effect of accuracy improvement by using the rotated BC as the testing functions is greatly different with curvilinear or plane triangular elements but falls off when the size of the object is large.

  4. on differential operators on w 1,2 space and fredholm operators

    African Journals Online (AJOL)

    A selfadjoint differential operator defined over a closed and bounded interval on Sobolev space which is a dense linear subspace of a Hilbert space over the same interval is considered and shown to be a Fredholm operator with index zero. KEY WORDS: Sobolev space, Hilbert space, dense subspace, Fredholm operator

  5. A hybrid iterative scheme for optimal control problems governed by ...

    African Journals Online (AJOL)

    MRT

    KEY WORDS: Optimal control problem; Fredholm integral equation; ... control problems governed by Fredholm integral and integro-differential equations is given in (Brunner and Yan, ..... The exact optimal trajectory and control functions are. 2.

  6. Effective quadrature formula in solving linear integro-differential equations of order two

    Science.gov (United States)

    Eshkuvatov, Z. K.; Kammuji, M.; Long, N. M. A. Nik; Yunus, Arif A. M.

    2017-08-01

    In this note, we solve general form of Fredholm-Volterra integro-differential equations (IDEs) of order 2 with boundary condition approximately and show that proposed method is effective and reliable. Initially, IDEs is reduced into integral equation of the third kind by using standard integration techniques and identity between multiple and single integrals then truncated Legendre series are used to estimate the unknown function. For the kernel integrals, we have applied Gauss-Legendre quadrature formula and collocation points are chosen as the roots of the Legendre polynomials. Finally, reduce the integral equations of the third kind into the system of algebraic equations and Gaussian elimination method is applied to get approximate solutions. Numerical examples and comparisons with other methods reveal that the proposed method is very effective and dominated others in many cases. General theory of existence of the solution is also discussed.

  7. A degree theory for a class of perturbed Fredholm maps II

    Directory of Open Access Journals (Sweden)

    Calamai Alessandro

    2006-01-01

    Full Text Available In a recent paper we gave a notion of degree for a class of perturbations of nonlinear Fredholm maps of index zero between real infinite dimensional Banach spaces. Our purpose here is to extend that notion in order to include the degree introduced by Nussbaum for local -condensing perturbations of the identity, as well as the degree for locally compact perturbations of Fredholm maps of index zero recently defined by the first and third authors.

  8. Functional Analysis: Entering Hilbert Space

    DEFF Research Database (Denmark)

    Hansen, Vagn Lundsgaard

    in a new chapter on Fredholm theory (Chapter 6). Fredholm theory originates in pioneering work of the Swedish mathematician Erik Ivar Fred-holm on integral equations, which inspired the study of a new class of bounded linear operators, known as Fredholm operators. Chapter 6 presents the basic elements...

  9. Matrix form of Legendre polynomials for solving linear integro-differential equations of high order

    Science.gov (United States)

    Kammuji, M.; Eshkuvatov, Z. K.; Yunus, Arif A. M.

    2017-04-01

    This paper presents an effective approximate solution of high order of Fredholm-Volterra integro-differential equations (FVIDEs) with boundary condition. Legendre truncated series is used as a basis functions to estimate the unknown function. Matrix operation of Legendre polynomials is used to transform FVIDEs with boundary conditions into matrix equation of Fredholm-Volterra type. Gauss Legendre quadrature formula and collocation method are applied to transfer the matrix equation into system of linear algebraic equations. The latter equation is solved by Gauss elimination method. The accuracy and validity of this method are discussed by solving two numerical examples and comparisons with wavelet and methods.

  10. On Dirichlet-to-Neumann Maps and Some Applications to Modified Fredholm Determinants

    OpenAIRE

    Gesztesy, Fritz; Mitrea, Marius; Zinchenko, Maxim

    2010-01-01

    We consider Dirichlet-to-Neumann maps associated with (not necessarily self-adjoint) Schrodinger operators in $L^2(\\Omega; d^n x)$, $n=2,3$, where $\\Omega$ is an open set with a compact, nonempty boundary satisfying certain regularity conditions. As an application we describe a reduction of a certain ratio of modified Fredholm perturbation determinants associated with operators in $L^2(\\Omega; d^n x)$ to modified Fredholm perturbation determinants associated with operators in $L^2(\\partial\\Om...

  11. Fredholm theory in ordered Banach algebras | Benjamin ...

    African Journals Online (AJOL)

    This paper illustrates some initial steps taken in the effort of unifying the theory of positivity in ordered Banach algebas (OBAs) with the general Fred-holm theory in Banach algebras. We introduce here upper Weyl and upper Browder elements in an OBA relative to an arbitrary Banach algebra homomorphism and investigate ...

  12. Integral equations

    CERN Document Server

    Moiseiwitsch, B L

    2005-01-01

    Two distinct but related approaches hold the solutions to many mathematical problems--the forms of expression known as differential and integral equations. The method employed by the integral equation approach specifically includes the boundary conditions, which confers a valuable advantage. In addition, the integral equation approach leads naturally to the solution of the problem--under suitable conditions--in the form of an infinite series.Geared toward upper-level undergraduate students, this text focuses chiefly upon linear integral equations. It begins with a straightforward account, acco

  13. Inverse random source scattering for the Helmholtz equation in inhomogeneous media

    Science.gov (United States)

    Li, Ming; Chen, Chuchu; Li, Peijun

    2018-01-01

    This paper is concerned with an inverse random source scattering problem in an inhomogeneous background medium. The wave propagation is modeled by the stochastic Helmholtz equation with the source driven by additive white noise. The goal is to reconstruct the statistical properties of the random source such as the mean and variance from the boundary measurement of the radiated random wave field at multiple frequencies. Both the direct and inverse problems are considered. We show that the direct problem has a unique mild solution by a constructive proof. For the inverse problem, we derive Fredholm integral equations, which connect the boundary measurement of the radiated wave field with the unknown source function. A regularized block Kaczmarz method is developed to solve the ill-posed integral equations. Numerical experiments are included to demonstrate the effectiveness of the proposed method.

  14. Multidimensional singular integrals and integral equations

    CERN Document Server

    Mikhlin, Solomon Grigorievich; Stark, M; Ulam, S

    1965-01-01

    Multidimensional Singular Integrals and Integral Equations presents the results of the theory of multidimensional singular integrals and of equations containing such integrals. Emphasis is on singular integrals taken over Euclidean space or in the closed manifold of Liapounov and equations containing such integrals. This volume is comprised of eight chapters and begins with an overview of some theorems on linear equations in Banach spaces, followed by a discussion on the simplest properties of multidimensional singular integrals. Subsequent chapters deal with compounding of singular integrals

  15. The method of normal forms for singularly perturbed systems of Fredholm integro-differential equations with rapidly varying kernels

    Energy Technology Data Exchange (ETDEWEB)

    Bobodzhanov, A A; Safonov, V F [National Research University " Moscow Power Engineering Institute" , Moscow (Russian Federation)

    2013-07-31

    The paper deals with extending the Lomov regularization method to classes of singularly perturbed Fredholm-type integro-differential systems, which have not so far been studied. In these the limiting operator is discretely noninvertible. Such systems are commonly known as problems with unstable spectrum. Separating out the essential singularities in the solutions to these problems presents great difficulties. The principal one is to give an adequate description of the singularities induced by 'instability points' of the spectrum. A methodology for separating singularities by using normal forms is developed. It is applied to the above type of systems and is substantiated in these systems. Bibliography: 10 titles.

  16. Accuracy of approximations of solutions to Fredholm equations by kernel methods

    Czech Academy of Sciences Publication Activity Database

    Gnecco, G.; Kůrková, Věra; Sanguineti, M.

    2012-01-01

    Roč. 218, č. 14 (2012), s. 7481-7497 ISSN 0096-3003 R&D Projects: GA ČR GAP202/11/1368; GA MŠk OC10047 Grant - others:CNR-AV ČR(CZ-IT) Project 2010–2012 “Complexity of Neural -Network and Kernel Computational Models Institutional research plan: CEZ:AV0Z10300504 Keywords : approximate solutions to integral equations * radial and kernel-based networks * Gaussian kernels * model complexity * analysis of algorithms Subject RIV: IN - Informatics, Computer Science Impact factor: 1.349, year: 2012

  17. Bethe-Salpeter equation for fermion-antifermion system in the ladder approximation

    International Nuclear Information System (INIS)

    Fukui, Ichio; Seto, Noriaki; Yoshida, Toshihiro.

    1977-01-01

    The Bethe-Salpeter (B-S) equation is important for studying hadron physics. Especially intensive investigation on the fermion-antifermion B-S equation is indispensable for the phenomenological studies of hardrons. However, many components of the B-S amplitude and the Wick-rotated integral kernel of non-Fredholm type have prevented from knowing details the solutions even in the ladder approximation. Some particular solutions are known in case of the vanishing four-momenta of bound states. The B-S equation for the bound state of fermion-anti-fermion system interacting through vector (axial-vector) particle exchange was studied in the ladder approximation with Feynman gauge. The reduced equations were obtained for suitably decomposed amplitude, and it is shown that, in the S-wave case, the coupled equations separate into two parts. In the nonrelativistic limit, large components of the amplitude satisfy the Wick-Cutkosky equation, and small components are expressed in terms of the large ones. Equations are derived for the equal-time amplitudes. (Kobatake, H.)

  18. Integration of Chandrasekhar's integral equation

    International Nuclear Information System (INIS)

    Tanaka, Tasuku

    2003-01-01

    We solve Chandrasekhar's integration equation for radiative transfer in the plane-parallel atmosphere by iterative integration. The primary thrust in radiative transfer has been to solve the forward problem, i.e., to evaluate the radiance, given the optical thickness and the scattering phase function. In the area of satellite remote sensing, our problem is the inverse problem: to retrieve the surface reflectance and the optical thickness of the atmosphere from the radiance measured by satellites. In order to retrieve the optical thickness and the surface reflectance from the radiance at the top-of-the atmosphere (TOA), we should express the radiance at TOA 'explicitly' in the optical thickness and the surface reflectance. Chandrasekhar formalized radiative transfer in the plane-parallel atmosphere in a simultaneous integral equation, and he obtained the second approximation. Since then no higher approximation has been reported. In this paper, we obtain the third approximation of the scattering function. We integrate functions derived from the second approximation in the integral interval from 1 to ∞ of the inverse of the cos of zenith angles. We can obtain the indefinite integral rather easily in the form of a series expansion. However, the integrals at the upper limit, ∞, are not yet known to us. We can assess the converged values of those series expansions at ∞ through calculus. For integration, we choose coupling pairs to avoid unnecessary terms in the outcome of integral and discover that the simultaneous integral equation can be deduced to the mere integral equation. Through algebraic calculation, we obtain the third approximation as a polynomial of the third degree in the atmospheric optical thickness

  19. Explicit frequency equations of free vibration of a nonlocal Timoshenko beam with surface effects

    Science.gov (United States)

    Zhao, Hai-Sheng; Zhang, Yao; Lie, Seng-Tjhen

    2018-02-01

    Considerations of nonlocal elasticity and surface effects in micro- and nanoscale beams are both important for the accurate prediction of natural frequency. In this study, the governing equation of a nonlocal Timoshenko beam with surface effects is established by taking into account three types of boundary conditions: hinged-hinged, clamped-clamped and clamped-hinged ends. For a hinged-hinged beam, an exact and explicit natural frequency equation is obtained. However, for clamped-clamped and clamped-hinged beams, the solutions of corresponding frequency equations must be determined numerically due to their transcendental nature. Hence, the Fredholm integral equation approach coupled with a curve fitting method is employed to derive the approximate fundamental frequency equations, which can predict the frequency values with high accuracy. In short, explicit frequency equations of the Timoshenko beam for three types of boundary conditions are proposed to exhibit directly the dependence of the natural frequency on the nonlocal elasticity, surface elasticity, residual surface stress, shear deformation and rotatory inertia, avoiding the complicated numerical computation.

  20. Some integral formulations occurring in accelerator physics

    International Nuclear Information System (INIS)

    Miano, G.; Verolino, L.; Vaccaro, V.G.

    1995-10-01

    In this paper a powerful and robust analytical-numerical approach to study the electromagnetic interaction between a bunch of particles and the discontinuities of the vacuum chamber of a particle accelerator is discussed. In particular the diffraction of the electromagnetic field created by a bunch of a bunch of charges travelling through an iris and a drift tube is considered. Choosing in both cases a spectral transform of the current density distribution on the scatterer as unknowns, an effective numerical model is obtained. These unknowns have to satisfy a system of dual integral equations. A general procedure to transform this system into only one Fredholm integral equation of the second kind (in the case of the iris) or to a system of linear algebraic equations by means of a Neumann series (in the case of the drift tube) is described. These models allow to compute the longitudinal coupling impedance with a good accuracy either in the low frequency limit or in the high frequency limit

  1. Some integral formulations occurring in accelerator physics

    Energy Technology Data Exchange (ETDEWEB)

    Miano, G; Verolino, L [Naples Univ. (Italy). Dip. di Ingegneria Elettronica; [INFN, Naples (Italy); Vaccaro, V G [Naples Univ. (Italy). Dip. di Scienze Fisiche; [INFN, Naples (Italy)

    1995-10-01

    In this paper a powerful and robust analytical-numerical approach to study the electromagnetic interaction between a bunch of particles and the discontinuities of the vacuum chamber of a particle accelerator is discussed. In particular the diffraction of the electromagnetic field created by a bunch of a bunch of charges travelling through an iris and a drift tube is considered. Choosing in both cases a spectral transform of the current density distribution on the scatterer as unknowns, an effective numerical model is obtained. These unknowns have to satisfy a system of dual integral equations. A general procedure to transform this system into only one Fredholm integral equation of the second kind (in the case of the iris) or to a system of linear algebraic equations by means of a Neumann series (in the case of the drift tube) is described. These models allow to compute the longitudinal coupling impedance with a good accuracy either in the low frequency limit or in the high frequency limit.

  2. Integral equations and their applications

    CERN Document Server

    Rahman, M

    2007-01-01

    For many years, the subject of functional equations has held a prominent place in the attention of mathematicians. In more recent years this attention has been directed to a particular kind of functional equation, an integral equation, wherein the unknown function occurs under the integral sign. The study of this kind of equation is sometimes referred to as the inversion of a definite integral. While scientists and engineers can already choose from a number of books on integral equations, this new book encompasses recent developments including some preliminary backgrounds of formulations of integral equations governing the physical situation of the problems. It also contains elegant analytical and numerical methods, and an important topic of the variational principles. Primarily intended for senior undergraduate students and first year postgraduate students of engineering and science courses, students of mathematical and physical sciences will also find many sections of direct relevance. The book contains eig...

  3. On integrability of the Killing equation

    Science.gov (United States)

    Houri, Tsuyoshi; Tomoda, Kentaro; Yasui, Yukinori

    2018-04-01

    Killing tensor fields have been thought of as describing the hidden symmetry of space(-time) since they are in one-to-one correspondence with polynomial first integrals of geodesic equations. Since many problems in classical mechanics can be formulated as geodesic problems in curved space and spacetime, solving the defining equation for Killing tensor fields (the Killing equation) is a powerful way to integrate equations of motion. Thus it has been desirable to formulate the integrability conditions of the Killing equation, which serve to determine the number of linearly independent solutions and also to restrict the possible forms of solutions tightly. In this paper, we show the prolongation for the Killing equation in a manner that uses Young symmetrizers. Using the prolonged equations, we provide the integrability conditions explicitly.

  4. Linear integral equations and soliton systems

    International Nuclear Information System (INIS)

    Quispel, G.R.W.

    1983-01-01

    A study is presented of classical integrable dynamical systems in one temporal and one spatial dimension. The direct linearizations are given of several nonlinear partial differential equations, for example the Korteweg-de Vries equation, the modified Korteweg-de Vries equation, the sine-Gordon equation, the nonlinear Schroedinger equation, and the equation of motion for the isotropic Heisenberg spin chain; the author also discusses several relations between these equations. The Baecklund transformations of these partial differential equations are treated on the basis of a singular transformation of the measure (or equivalently of the plane-wave factor) occurring in the corresponding linear integral equations, and the Baecklund transformations are used to derive the direct linearization of a chain of so-called modified partial differential equations. Finally it is shown that the singular linear integral equations lead in a natural way to the direct linearizations of various nonlinear difference-difference equations. (Auth.)

  5. Geophysical interpretation using integral equations

    CERN Document Server

    Eskola, L

    1992-01-01

    Along with the general development of numerical methods in pure and applied to apply integral equations to geophysical modelling has sciences, the ability improved considerably within the last thirty years or so. This is due to the successful derivation of integral equations that are applicable to the modelling of complex structures, and efficient numerical algorithms for their solution. A significant stimulus for this development has been the advent of fast digital computers. The purpose of this book is to give an idea of the principles by which boundary-value problems describing geophysical models can be converted into integral equations. The end results are the integral formulas and integral equations that form the theoretical framework for practical applications. The details of mathematical analysis have been kept to a minimum. Numerical algorithms are discussed only in connection with some illustrative examples involving well-documented numerical modelling results. The reader is assu­ med to have a back...

  6. Existence of solutions for nonlinear mixed type integrodifferential equation of second order

    Directory of Open Access Journals (Sweden)

    Haribhau Laxman Tidke

    2010-04-01

    Full Text Available In this paper, we investigate the existence of solutions for nonlinear mixed Volterra-Fredholm integrodifferential equation of second order with nonlocal conditions in Banach spaces. Our analysis is based on Leray-Schauder alternative, rely on a priori bounds of solutions and the inequality established by B. G. Pachpatte.

  7. Integral equation methods for electromagnetics

    CERN Document Server

    Volakis, John

    2012-01-01

    This text/reference is a detailed look at the development and use of integral equation methods for electromagnetic analysis, specifically for antennas and radar scattering. Developers and practitioners will appreciate the broad-based approach to understanding and utilizing integral equation methods and the unique coverage of historical developments that led to the current state-of-the-art. In contrast to existing books, Integral Equation Methods for Electromagnetics lays the groundwork in the initial chapters so students and basic users can solve simple problems and work their way up to the mo

  8. An analytical method for the inverse Cauchy problem of Lame equation in a rectangle

    Science.gov (United States)

    Grigor’ev, Yu

    2018-04-01

    In this paper, we present an analytical computational method for the inverse Cauchy problem of Lame equation in the elasticity theory. A rectangular domain is frequently used in engineering structures and we only consider the analytical solution in a two-dimensional rectangle, wherein a missing boundary condition is recovered from the full measurement of stresses and displacements on an accessible boundary. The essence of the method consists in solving three independent Cauchy problems for the Laplace and Poisson equations. For each of them, the Fourier series is used to formulate a first-kind Fredholm integral equation for the unknown function of data. Then, we use a Lavrentiev regularization method, and the termwise separable property of kernel function allows us to obtain a closed-form regularized solution. As a result, for the displacement components, we obtain solutions in the form of a sum of series with three regularization parameters. The uniform convergence and error estimation of the regularized solutions are proved.

  9. Classical solutions of two dimensional Stokes problems on non smooth domains. 1: The Radon integral operators

    International Nuclear Information System (INIS)

    Lubuma, M.S.

    1991-05-01

    The applicability of the Neumann indirect method of potentials to the Dirichlet and Neumann problems for the two-dimensional Stokes operator on a non smooth boundary Γ is subject to two kinds of sufficient and/or necessary conditions on Γ. The first one, occurring in electrostatic, is equivalent to the boundedness on C(Γ) of the velocity double layer potential W as well as to the existence of jump relations of potentials. The second condition, which forces Γ to be a simple rectifiable curve and which, compared to the Laplacian, is a stronger restriction on the corners of Γ, states that the Fredholm radius of W is greater than 2. Under these conditions, the Radon boundary integral equations defined by the above mentioned jump relations are solvable by the Fredholm theory; the double (for Dirichlet) and the single (for Neumann) layer potentials corresponding to their solutions are classical solutions of the Stokes problems. (author). 48 refs

  10. PREFACE: Symmetries and integrability of difference equations Symmetries and integrability of difference equations

    Science.gov (United States)

    Levi, Decio; Olver, Peter; Thomova, Zora; Winternitz, Pavel

    2009-11-01

    The concept of integrability was introduced in classical mechanics in the 19th century for finite dimensional continuous Hamiltonian systems. It was extended to certain classes of nonlinear differential equations in the second half of the 20th century with the discovery of the inverse scattering transform and the birth of soliton theory. Also at the end of the 19th century Lie group theory was invented as a powerful tool for obtaining exact analytical solutions of large classes of differential equations. Together, Lie group theory and integrability theory in its most general sense provide the main tools for solving nonlinear differential equations. Like differential equations, difference equations play an important role in physics and other sciences. They occur very naturally in the description of phenomena that are genuinely discrete. Indeed, they may actually be more fundamental than differential equations if space-time is actually discrete at very short distances. On the other hand, even when treating continuous phenomena described by differential equations it is very often necessary to resort to numerical methods. This involves a discretization of the differential equation, i.e. a replacement of the differential equation by a difference one. Given the well developed and understood techniques of symmetry and integrability for differential equations a natural question to ask is whether it is possible to develop similar techniques for difference equations. The aim is, on one hand, to obtain powerful methods for solving `integrable' difference equations and to establish practical integrability criteria, telling us when the methods are applicable. On the other hand, Lie group methods can be adapted to solve difference equations analytically. Finally, integrability and symmetry methods can be combined with numerical methods to obtain improved numerical solutions of differential equations. The origin of the SIDE meetings goes back to the early 1990s and the first

  11. Completely integrable operator evolution equations. II

    International Nuclear Information System (INIS)

    Chudnovsky, D.V.

    1979-01-01

    The author continues the investigation of operator classical completely integrable systems. The main attention is devoted to the stationary operator non-linear Schroedinger equation. It is shown that this equation can be used for separation of variables for a large class of completely integrable equations. (Auth.)

  12. Application of the Galerkin's method to the solution of the one-dimensional integral transport equation: generalized collision probabilities taken in account the flux gradient and the linearly anisotropic scattering

    International Nuclear Information System (INIS)

    Sanchez, Richard.

    1975-04-01

    For the one-dimensional geometries, the transport equation with linearly anisotropic scattering can be reduced to a single integral equation; this is a singular-kernel FREDHOLM equation of the second kind. When applying a conventional projective method that of GALERKIN, to the solution of this equation the well-known collision probability algorithm is obtained. Piecewise polynomial expansions are used to represent the flux. In the ANILINE code, the flux is supposed to be linear in plane geometry and parabolic in both cylindrical and spherical geometries. An integral relationship was found between the one-dimensional isotropic and anisotropic kernels; this allows to reduce the new matrix elements (issuing from the anisotropic kernel) to classic collision probabilities of the isotropic scattering equation. For cylindrical and spherical geometries used an approximate representation of the current was used to avoid an additional numerical integration. Reflective boundary conditions were considered; in plane geometry the reflection is supposed specular, for the other geometries the isotropic reflection hypothesis has been adopted. Further, the ANILINE code enables to deal with an incoming isotropic current. Numerous checks were performed in monokinetic theory. Critical radii and albedos were calculated for homogeneous slabs, cylinders and spheres. For heterogeneous media, the thermal utilization factor obtained by this method was compared with the theoretical result based upon a formula by BENOIST. Finally, ANILINE was incorporated into the multigroup APOLLO code, which enabled to analyse the MINERVA experimental reactor in transport theory with 99 groups. The ANILINE method is particularly suited to the treatment of strongly anisotropic media with considerable flux gradients. It is also well adapted to the calculation of reflectors, and in general, to the exact analysis of anisotropic effects in large-sized media [fr

  13. Integral Equation Methods for Electromagnetic and Elastic Waves

    CERN Document Server

    Chew, Weng; Hu, Bin

    2008-01-01

    Integral Equation Methods for Electromagnetic and Elastic Waves is an outgrowth of several years of work. There have been no recent books on integral equation methods. There are books written on integral equations, but either they have been around for a while, or they were written by mathematicians. Much of the knowledge in integral equation methods still resides in journal papers. With this book, important relevant knowledge for integral equations are consolidated in one place and researchers need only read the pertinent chapters in this book to gain important knowledge needed for integral eq

  14. Integrable dissipative nonlinear second order differential equations via factorizations and Abel equations

    Energy Technology Data Exchange (ETDEWEB)

    Mancas, Stefan C. [Department of Mathematics, Embry–Riddle Aeronautical University, Daytona Beach, FL 32114-3900 (United States); Rosu, Haret C., E-mail: hcr@ipicyt.edu.mx [IPICYT, Instituto Potosino de Investigacion Cientifica y Tecnologica, Apdo Postal 3-74 Tangamanga, 78231 San Luis Potosí, SLP (Mexico)

    2013-09-02

    We emphasize two connections, one well known and another less known, between the dissipative nonlinear second order differential equations and the Abel equations which in their first-kind form have only cubic and quadratic terms. Then, employing an old integrability criterion due to Chiellini, we introduce the corresponding integrable dissipative equations. For illustration, we present the cases of some integrable dissipative Fisher, nonlinear pendulum, and Burgers–Huxley type equations which are obtained in this way and can be of interest in applications. We also show how to obtain Abel solutions directly from the factorization of second order nonlinear equations.

  15. On the energy spectrum of the Bethe-Salpeter equation; Ob ehnergeticheskom spektre uravneniya Bete-Solpitera

    Energy Technology Data Exchange (ETDEWEB)

    Dorkin, S M [Dal` nevostochnyj Gosudarstvennyj Univ., Vladivostok (Russian Federation); Kaptar` , L P; Semikh, S S [Joint Inst. for Nuclear Research, Dubna (Russian Federation). Lab. of Theoretical Physics

    1997-12-31

    The problem of calculating the energy spectrum of a two-fermion bound state within the Bethe-Salpeter formalism is discussed. An expansion of the kernel of the spinor-spinor Bethe-Salpeter equation in the ladder approximation is found in terms of a bi-orthogonal basis of the generalized Gilbert-Schmidt series for symmetric equations of the Fredholm type. According to this expansion, a new method of solving the Bethe-Salpeter equation and finding the mass spectrum is proposed. Methodological result of numerical solutions of equations with scalar interaction is presented. (author). 20 refs., 3 figs.

  16. An integral transform of the Salpeter equation

    International Nuclear Information System (INIS)

    Krolikowski, W.

    1980-03-01

    We find a new form of relativistic wave equation for two spin-1/2 particles, which arises by an integral transformation (in the position space) of the wave function in the Salpeter equation. The non-locality involved in this transformation is extended practically over the Compton wavelength of the lighter of two particles. In the case of equal masses the new equation assumes the form of the Breit equation with an effective integral interaction. In the one-body limit it reduces to the Dirac equation also with an effective integral interaction. (author)

  17. On the recovering of a coupled nonlinear Schroedinger potential

    Energy Technology Data Exchange (ETDEWEB)

    Corona, Gulmaro Corona [Area de Analisis Matematico y sus Aplicaciones, Universidad Autonoma Metropolitana, Atzcapotzalco, DF (Mexico)]. E-mail: ccg@hp9000a1.uam.mx

    2000-04-28

    We establish a priori conditions for a Gel'fand-Levitan (GL) integral using some results of the Fredholm theory. As consequence, we obtain a recovering formula for the potential of the coupled nonlinear Schroedinger equations. The remarkable fact is that the recovering formula is given in terms of the solutions of a classical GL-integral equation. (author)

  18. Integration rules for scattering equations

    International Nuclear Information System (INIS)

    Baadsgaard, Christian; Bjerrum-Bohr, N.E.J.; Bourjaily, Jacob L.; Damgaard, Poul H.

    2015-01-01

    As described by Cachazo, He and Yuan, scattering amplitudes in many quantum field theories can be represented as integrals that are fully localized on solutions to the so-called scattering equations. Because the number of solutions to the scattering equations grows quite rapidly, the contour of integration involves contributions from many isolated components. In this paper, we provide a simple, combinatorial rule that immediately provides the result of integration against the scattering equation constraints for any Möbius-invariant integrand involving only simple poles. These rules have a simple diagrammatic interpretation that makes the evaluation of any such integrand immediate. Finally, we explain how these rules are related to the computation of amplitudes in the field theory limit of string theory.

  19. Project ''REMAD'', Neutron spectrometry. Code ''SOHO - phase II''. First part ''FORMULATION''

    International Nuclear Information System (INIS)

    Zaborowski, H.L.

    1982-08-01

    ''SOHO'' is a new neutron spectrometric code with large potential applications besides the BONNER spheres system for which he has been initially developped. This code makes use of a new and original iteractive procedure for the search of approximate solutions from the Fredholm integral Equation of the First kind whose Resolution Function is experimentally and/or mathematically defined (e.g. the Log-Normal Hypothesis for the BONNER Spheres). Upon discretization of the Fredholm integral Equation we get systems of non-exact homogeneous linear equations: Q X = e whose approximate solutions are given when: Q X → 0. We show that the iterative procedure converges absolutely leading to the existence of the approximate solutions, independently of the kind of initialization used. For the applications to health physics of the BONNER sphere ''SOHO'' has been programmed on a HP - 41 CV calculator [fr

  20. Application of Fourier transforms for microwave radiometric inversions

    Science.gov (United States)

    Holmes, J. J.; Balanis, C. A.; Truman, W. M.

    1975-01-01

    Existing microwave radiometer technology now provides a suitable method for remote determination of the ocean surface's absolute brightness temperature. To extract the brightness temperature of the water from the antenna temperature, an unstable Fredholm integral equation of the first kind is solved. Fourier transform techniques are used to invert the integral after it is placed into a cross correlation form. Application and verification of the methods to a two-dimensional modeling of a laboratory wave tank system are included. The instability of the ill-posed Fredholm equation is examined and a restoration procedure is included which smooths the resulting oscillations. With the recent availability and advances of fast Fourier transform (FFT) techniques, the method presented becomes very attractive in the evaluation of large quantities of data.

  1. Completely integrable operator evolutionary equations

    International Nuclear Information System (INIS)

    Chudnovsky, D.V.

    1979-01-01

    The authors present natural generalizations of classical completely integrable equations where the functions are replaced by arbitrary operators. Among these equations are the non-linear Schroedinger, the Korteweg-de Vries, and the modified KdV equations. The Lax representation and the Baecklund transformations are presented. (Auth.)

  2. Asymptotic integration of differential and difference equations

    CERN Document Server

    Bodine, Sigrun

    2015-01-01

    This book presents the theory of asymptotic integration for both linear differential and difference equations. This type of asymptotic analysis is based on some fundamental principles by Norman Levinson. While he applied them to a special class of differential equations, subsequent work has shown that the same principles lead to asymptotic results for much wider classes of differential and also difference equations. After discussing asymptotic integration in a unified approach, this book studies how the application of these methods provides several new insights and frequent improvements to results found in earlier literature. It then continues with a brief introduction to the relatively new field of asymptotic integration for dynamic equations on time scales. Asymptotic Integration of Differential and Difference Equations is a self-contained and clearly structured presentation of some of the most important results in asymptotic integration and the techniques used in this field. It will appeal to researchers i...

  3. Transformation properties of the integrable evolution equations

    International Nuclear Information System (INIS)

    Konopelchenko, B.G.

    1981-01-01

    Group-theoretical properties of partial differential equations integrable by the inverse scattering transform method are discussed. It is shown that nonlinear transformations typical to integrable equations (symmetry groups, Baecklund-transformations) and these equations themselves are contained in a certain universal nonlinear transformation group. (orig.)

  4. Application of wavelets to singular integral scattering equations

    International Nuclear Information System (INIS)

    Kessler, B.M.; Payne, G.L.; Polyzou, W.N.

    2004-01-01

    The use of orthonormal wavelet basis functions for solving singular integral scattering equations is investigated. It is shown that these basis functions lead to sparse matrix equations which can be solved by iterative techniques. The scaling properties of wavelets are used to derive an efficient method for evaluating the singular integrals. The accuracy and efficiency of the wavelet transforms are demonstrated by solving the two-body T-matrix equation without partial wave projection. The resulting matrix equation which is characteristic of multiparticle integral scattering equations is found to provide an efficient method for obtaining accurate approximate solutions to the integral equation. These results indicate that wavelet transforms may provide a useful tool for studying few-body systems

  5. Application and sensitivity investigation of Fourier transforms for microwave radiometric inversions

    Science.gov (United States)

    Holmes, J. J.; Balanis, C. A.

    1974-01-01

    Existing microwave radiometer technology now provides a suitable method for remote determination of the ocean surface's absolute brightness temperature. To extract the brightness temperature of the water from the antenna temperature equation, an unstable Fredholm integral equation of the first kind was solved. Fast Fourier Transform techniques were used to invert the integral after it is placed into a cross-correlation form. Application and verification of the methods to a two-dimensional modeling of a laboratory wave tank system were included. The instability of the Fredholm equation was then demonstrated and a restoration procedure was included which smooths the resulting oscillations. With the recent availability and advances of Fast Fourier Transform techniques, the method presented becomes very attractive in the evaluation of large quantities of data. Actual radiometric measurements of sea water are inverted using the restoration method, incorporating the advantages of the Fast Fourier Transform algorithm for computations.

  6. Darboux invariants of integrable equations with variable spectral parameters

    International Nuclear Information System (INIS)

    Shin, H J

    2008-01-01

    The Darboux transformation for integrable equations with variable spectral parameters is introduced. Darboux invariant quantities are calculated, which are used in constructing the Lax pair of integrable equations. This approach serves as a systematic method for constructing inhomogeneous integrable equations and their soliton solutions. The structure functions of variable spectral parameters determine the integrability and nonlinear coupling terms. Three cases of integrable equations are treated as examples of this approach

  7. On the Use of Importance Sampling in Particle Transport Problems

    Energy Technology Data Exchange (ETDEWEB)

    Eriksson, B

    1965-06-15

    The idea of importance sampling is applied to the problem of solving integral equations of Fredholm's type. Especially Bolzmann's neutron transport equation is taken into consideration. For the solution of the latter equation, an importance sampling technique is derived from some simple transformations at the original transport equation into a similar equation. Examples of transformations are given, which have been used with great success in practice.

  8. On the Use of Importance Sampling in Particle Transport Problems

    International Nuclear Information System (INIS)

    Eriksson, B.

    1965-06-01

    The idea of importance sampling is applied to the problem of solving integral equations of Fredholm's type. Especially Bolzmann's neutron transport equation is taken into consideration. For the solution of the latter equation, an importance sampling technique is derived from some simple transformations at the original transport equation into a similar equation. Examples of transformations are given, which have been used with great success in practice

  9. New multidimensional partially integrable generalization of S-integrable N-wave equation

    International Nuclear Information System (INIS)

    Zenchuk, A. I.

    2007-01-01

    This paper develops a modification of the dressing method based on the inhomogeneous linear integral equation with integral operator having nonempty kernel. The method allows one to construct the systems of multidimensional partial differential equations having differential polynomial structure in any dimension n. The associated solution space is not full, although it is parametrized by certain number of arbitrary functions of (n-1) variables. We consider four-dimensional generalization of the classical (2+1)-dimensional S-integrable N-wave equation as an example

  10. Integrable discretizations of the short pulse equation

    International Nuclear Information System (INIS)

    Feng Baofeng; Maruno, Ken-ichi; Ohta, Yasuhiro

    2010-01-01

    In this paper, we propose integrable semi-discrete and full-discrete analogues of the short pulse (SP) equation. The key construction is the bilinear form and determinant structure of solutions of the SP equation. We also give the determinant formulas of N-soliton solutions of the semi-discrete and full-discrete analogues of the SP equations, from which the multi-loop and multi-breather solutions can be generated. In the continuous limit, the full-discrete SP equation converges to the semi-discrete SP equation, and then to the continuous SP equation. Based on the semi-discrete SP equation, an integrable numerical scheme, i.e. a self-adaptive moving mesh scheme, is proposed and used for the numerical computation of the short pulse equation.

  11. Integrable discretization s of derivative nonlinear Schroedinger equations

    International Nuclear Information System (INIS)

    Tsuchida, Takayuki

    2002-01-01

    We propose integrable discretizations of derivative nonlinear Schroedinger (DNLS) equations such as the Kaup-Newell equation, the Chen-Lee-Liu equation and the Gerdjikov-Ivanov equation by constructing Lax pairs. The discrete DNLS systems admit the reduction of complex conjugation between two dependent variables and possess bi-Hamiltonian structure. Through transformations of variables and reductions, we obtain novel integrable discretizations of the nonlinear Schroedinger (NLS), modified KdV (mKdV), mixed NLS, matrix NLS, matrix KdV, matrix mKdV, coupled NLS, coupled Hirota, coupled Sasa-Satsuma and Burgers equations. We also discuss integrable discretizations of the sine-Gordon equation, the massive Thirring model and their generalizations. (author)

  12. A Comprehensive Review of Boundary Integral Formulations of Acoustic Scattering Problems

    Directory of Open Access Journals (Sweden)

    S.I. Zaman

    2000-12-01

    Full Text Available This is a review presenting an overview of the developments in boundary integral formulations of the acoustic scattering problems. Generally, the problem is formulated in one of two ways viz. Green’s representation formula, and the Layer-theoretic formulation utilizing either a simple-layer or a double-layer potential. The review presents and expounds the major contributions in this area over the last four decades. The need for a robust and improved formulation of the exterior scattering problem (Neumann or Dirichlet arose due to the fact that the classical formulation failed to yield a unique solution at (acoustic wave-numbers which correspond to eigenvalues (eigenfrequencies of the corresponding interior scattering problem. Moreover, this correlation becomes more pronounced as the wave-numbers become larger i.e. as the (acoustic frequency increases. The robust integral formulations which are discussed here yield Fredholms integral equations of the second kind which are more amenable to computation than the first kind. However, the integral equation involves a hypersingular kernel which creates ill-conditioning in the final matrix representation. This is circumvented by a regularisation technique. An extensive useful list of references is also presented here for researchers in this area.

  13. Integrable peakon equations with cubic nonlinearity

    International Nuclear Information System (INIS)

    Hone, Andrew N W; Wang, J P

    2008-01-01

    We present a new integrable partial differential equation found by Vladimir Novikov. Like the Camassa-Holm and Degasperis-Procesi equations, this new equation admits peaked soliton (peakon) solutions, but it has nonlinear terms that are cubic, rather than quadratic. We give a matrix Lax pair for V Novikov's equation, and show how it is related by a reciprocal transformation to a negative flow in the Sawada-Kotera hierarchy. Infinitely many conserved quantities are found, as well as a bi-Hamiltonian structure. The latter is used to obtain the Hamiltonian form of the finite-dimensional system for the interaction of N peakons, and the two-body dynamics (N = 2) is explicitly integrated. Finally, all of this is compared with some analogous results for another cubic peakon equation derived by Zhijun Qiao. (fast track communication)

  14. Picard-Fuchs equations of dimensionally regulated Feynman integrals

    Energy Technology Data Exchange (ETDEWEB)

    Zayadeh, Raphael

    2013-12-15

    This thesis is devoted to studying differential equations of Feynman integrals. A Feynman integral depends on a dimension D. For integer values of D it can be written as a projective integral, which is called the Feynman parameter prescription. A major complication arises from the fact that for some values of D the integral can diverge. This problem is solved within dimensional regularization by continuing the integral as a meromorphic function on the complex plane and replacing the ill-defined quantity by a Laurent series in a dimensional regularization parameter. All terms in such a Laurent expansion are periods in the sense of Kontsevich and Zagier. We describe a new method to compute differential equations of Feynman integrals. So far, the standard has been to use integration-by-parts (IBP) identities to obtain coupled systems of linear differential equations for the master integrals. Our method is based on the theory of Picard-Fuchs equations. In the case we are interested in, that of projective and quasiprojective families, a Picard-Fuchs equation can be computed by means of the Griffiths-Dwork reduction. We describe a method that is designed for fixed integer dimension. After a suitable integer shift of dimension we obtain a period of a family of hypersurfaces, hence a Picard-Fuchs equation. This equation is inhomogeneous because the domain of integration has a boundary and we only obtain a relative cycle. As a second step we shift back the dimension using Tarasov's generalized dimensional recurrence relations. Furthermore, we describe a method to directly compute the differential equation for general D without shifting the dimension. This is based on the Griffiths-Dwork reduction. The success of this method depends on the ability to solve large systems of linear equations. We give examples of two and three-loop graphs. Tarasov classifies two-loop two-point functions and we give differential equations for these. For us the most interesting example is

  15. Picard-Fuchs equations of dimensionally regulated Feynman integrals

    International Nuclear Information System (INIS)

    Zayadeh, Raphael

    2013-12-01

    This thesis is devoted to studying differential equations of Feynman integrals. A Feynman integral depends on a dimension D. For integer values of D it can be written as a projective integral, which is called the Feynman parameter prescription. A major complication arises from the fact that for some values of D the integral can diverge. This problem is solved within dimensional regularization by continuing the integral as a meromorphic function on the complex plane and replacing the ill-defined quantity by a Laurent series in a dimensional regularization parameter. All terms in such a Laurent expansion are periods in the sense of Kontsevich and Zagier. We describe a new method to compute differential equations of Feynman integrals. So far, the standard has been to use integration-by-parts (IBP) identities to obtain coupled systems of linear differential equations for the master integrals. Our method is based on the theory of Picard-Fuchs equations. In the case we are interested in, that of projective and quasiprojective families, a Picard-Fuchs equation can be computed by means of the Griffiths-Dwork reduction. We describe a method that is designed for fixed integer dimension. After a suitable integer shift of dimension we obtain a period of a family of hypersurfaces, hence a Picard-Fuchs equation. This equation is inhomogeneous because the domain of integration has a boundary and we only obtain a relative cycle. As a second step we shift back the dimension using Tarasov's generalized dimensional recurrence relations. Furthermore, we describe a method to directly compute the differential equation for general D without shifting the dimension. This is based on the Griffiths-Dwork reduction. The success of this method depends on the ability to solve large systems of linear equations. We give examples of two and three-loop graphs. Tarasov classifies two-loop two-point functions and we give differential equations for these. For us the most interesting example is the two

  16. Some New Integrable Equations from the Self-Dual Yang-Mills Equations

    International Nuclear Information System (INIS)

    Ivanova, T.A.; Popov, A.D.

    1994-01-01

    Using the symmetry reductions of the self-dual Yang-Mills (SDYM) equations in (2+2) dimensions, we introduce new integrable equations which are 'deformations' of the chiral model in (2+1) dimensions, generalized nonlinear Schroedinger, Korteweg-de Vries, Toda lattice, Garnier, Euler-Arnold, generalized Calogero-Moser and Euler-Calogero-Moser equations. The Lax pairs for all of these equations are derived by the symmetry reductions of the Lax pair for the SDYM equations. 34 refs

  17. Baecklund transformations for integrable lattice equations

    International Nuclear Information System (INIS)

    Atkinson, James

    2008-01-01

    We give new Baecklund transformations (BTs) for some known integrable (in the sense of being multidimensionally consistent) quadrilateral lattice equations. As opposed to the natural auto-BT inherent in every such equation, these BTs are of two other kinds. Specifically, it is found that some equations admit additional auto-BTs (with Baecklund parameter), whilst some pairs of apparently distinct equations admit a BT which connects them

  18. PREFACE: Symmetries and Integrability of Difference Equations

    Science.gov (United States)

    Doliwa, Adam; Korhonen, Risto; Lafortune, Stéphane

    2007-10-01

    The notion of integrability was first introduced in the 19th century in the context of classical mechanics with the definition of Liouville integrability for Hamiltonian flows. Since then, several notions of integrability have been introduced for partial and ordinary differential equations. Closely related to integrability theory is the symmetry analysis of nonlinear evolution equations. Symmetry analysis takes advantage of the Lie group structure of a given equation to study its properties. Together, integrability theory and symmetry analysis provide the main method by which nonlinear evolution equations can be solved explicitly. Difference equations (DE), like differential equations, are important in numerous fields of science and have a wide variety of applications in such areas as mathematical physics, computer visualization, numerical analysis, mathematical biology, economics, combinatorics, and quantum field theory. It is thus crucial to develop tools to study and solve DEs. While the theory of symmetry and integrability for differential equations is now largely well-established, this is not yet the case for discrete equations. Although over recent years there has been significant progress in the development of a complete analytic theory of difference equations, further tools are still needed to fully understand, for instance, the symmetries, asymptotics and the singularity structure of difference equations. The series of SIDE meetings on Symmetries and Integrability of Difference Equations started in 1994. Its goal is to provide a platform for an international and interdisciplinary communication for researchers working in areas associated with integrable discrete systems, such as classical and quantum physics, computer science and numerical analysis, mathematical biology and economics, discrete geometry and combinatorics, theory of special functions, etc. The previous SIDE meetings took place in Estérel near Montréal, Canada (1994), at the University of

  19. Counting master integrals. Integration by parts vs. functional equations

    International Nuclear Information System (INIS)

    Kniehl, Bernd A.; Tarasov, Oleg V.

    2016-01-01

    We illustrate the usefulness of functional equations in establishing relationships between master integrals under the integration-by-parts reduction procedure by considering a certain two-loop propagator-type diagram as an example.

  20. Integrable coupling system of fractional soliton equation hierarchy

    Energy Technology Data Exchange (ETDEWEB)

    Yu Fajun, E-mail: yfajun@163.co [College of Maths and Systematic Science, Shenyang Normal University, Shenyang 110034 (China)

    2009-10-05

    In this Letter, we consider the derivatives and integrals of fractional order and present a class of the integrable coupling system of the fractional order soliton equations. The fractional order coupled Boussinesq and KdV equations are the special cases of this class. Furthermore, the fractional AKNS soliton equation hierarchy is obtained.

  1. Identification of the material properties in nonuniform nanostructures

    International Nuclear Information System (INIS)

    Bao, Gang; Xu, Xiang

    2015-01-01

    This paper is concerned with addressing two significant challenges arising from quantifying mechanical properties of nanomaterials, namely nonuniformity of the nanomaterial and the high noise level of measurements. For nonuniformity, an explicit solution is derived for the general Euler–Bernoulli equation in terms of the Green function for the Poisson equation. Then, by examining a stochastic source, the systematic error may be removed from measurements, which leads to more accurate estimation of mechanical properties. Based on Itô integral properties, three deterministic Fredholm integral equations can be deduced to extract the stiffness and the structure of the random source from measured data. To overcome ill-posedness and high nonlinearity in solving the Fredholm equations, a Tikhonov regularization method is developed with an a priori strategy of choosing the regularization parameter. Moreover, under a regularity assumption for the stiffness coefficient and structures of the random source, the convergence rate can be obtained in the sense of probability. Numerical examples are presented to illustrate the validity and effectiveness of the novel model and regularization method. (paper)

  2. Inequalities for differential and integral equations

    CERN Document Server

    Ames, William F

    1997-01-01

    Inequalities for Differential and Integral Equations has long been needed; it contains material which is hard to find in other books. Written by a major contributor to the field, this comprehensive resource contains many inequalities which have only recently appeared in the literature and which can be used as powerful tools in the development of applications in the theory of new classes of differential and integral equations. For researchers working in this area, it will be a valuable source of reference and inspiration. It could also be used as the text for an advanced graduate course.Key Features* Covers a variety of linear and nonlinear inequalities which find widespread applications in the theory of various classes of differential and integral equations* Contains many inequalities which have only recently appeared in literature and cannot yet be found in other books* Provides a valuable reference to engineers and graduate students

  3. Asymptotics for the Fredholm determinant of the sine kernel on a union of intervals

    Science.gov (United States)

    Widom, Harold

    1995-07-01

    In the bulk scaling limit of the Gaussian Unitary Ensemble of hermitian matrices the probability that an interval of length s contains no eigenvalues is the Fredholm determinant of the sine kernel{sin (x - y)}/{π (x - y)} over this interval. A formal asymptotic expansion for the determinant as s tends to infinity was obtained by Dyson. In this paper we replace a single interval of length s by sJ, where J is a union of m intervals and present a proof of the asymptotics up to second order. The logarithmic derivative with respect to s of the determinant equals a constant (expressible in terms of hyperelliptic integrals) times s, plus a bounded oscillatory function of s (zero if m=1, periodic if m=2, and in general expressible in terms of the solution of a Jacobi inversion problem), plus o(1). Also determined are the asymptotics of the trace of the resolvent operator, which is the ratio in the same model of the probability that the set contains exactly one eigenvalue to the probability that it contains none. The proofs use ideas from orthogonal polynomial theory.

  4. Bounded solutions for fuzzy differential and integral equations

    Energy Technology Data Exchange (ETDEWEB)

    Nieto, Juan J. [Departamento de Analisis Matematico Facultad de Matematicas Universidad de Santiago de Compostela, 15782 (Spain)] e-mail: amnieto@usc.es; Rodriguez-Lopez, Rosana [Departamento de Analisis Matematico Facultad de Matematicas Universidad de Santiago de Compostela, 15782 (Spain)] e-mail: amrosana@usc.es

    2006-03-01

    We find sufficient conditions for the boundness of every solution of first-order fuzzy differential equations as well as certain fuzzy integral equations. Our results are based on several theorems concerning crisp differential and integral inequalities.

  5. Algorithms For Integrating Nonlinear Differential Equations

    Science.gov (United States)

    Freed, A. D.; Walker, K. P.

    1994-01-01

    Improved algorithms developed for use in numerical integration of systems of nonhomogenous, nonlinear, first-order, ordinary differential equations. In comparison with integration algorithms, these algorithms offer greater stability and accuracy. Several asymptotically correct, thereby enabling retention of stability and accuracy when large increments of independent variable used. Accuracies attainable demonstrated by applying them to systems of nonlinear, first-order, differential equations that arise in study of viscoplastic behavior, spread of acquired immune-deficiency syndrome (AIDS) virus and predator/prey populations.

  6. Integral equations for four identical particles in angular momentum representation

    International Nuclear Information System (INIS)

    Kharchenko, V.F.; Shadchin, S.A.

    1975-01-01

    In integral equations of motion for a system of four identical spinless particles with central pair interactions, transition is realized from the representation of relative Jacobi momenta to the representation of their moduli and relative angular moments. As a result, the variables associated with the rotation of the system as a whole are separated in the equations. The integral equations of motion for four particles are reduced to the form of an infinite system of three-demensional integral equations. The four-particle kinematic factors contained in integral kernels are expressed in terms of three-particle type kinematic factors. In the case of separable two-particle interaction, the equations of motion for four particles have the form of an infinite system of two-dimensional integral equations

  7. Asymptotics for the Fredholm Determinant of the Sine Kernel on a Union of Intervals

    OpenAIRE

    Widom, Harold

    1994-01-01

    In the bulk scaling limit of the Gaussian Unitary Ensemble of Hermitian matrices the probability that an interval of length $s$ contains no eigenvalues is the Fredholm determinant of the sine kernel $\\sin(x-y)\\over\\pi(x-y)$ over this interval. A formal asymptotic expansion for the determinant as $s$ tends to infinity was obtained by Dyson. In this paper we replace a single interval of length $s$ by $sJ$ where $J$ is a union of $m$ intervals and present a proof of the asymptotics up to second ...

  8. Feynman integrals and difference equations

    International Nuclear Information System (INIS)

    Moch, S.; Schneider, C.

    2007-09-01

    We report on the calculation of multi-loop Feynman integrals for single-scale problems by means of difference equations in Mellin space. The solution to these difference equations in terms of harmonic sums can be constructed algorithmically over difference fields, the so-called ΠΣ * -fields. We test the implementation of the Mathematica package Sigma on examples from recent higher order perturbative calculations in Quantum Chromodynamics. (orig.)

  9. Transport methods: general. 8. Formulation of Transport Equation in a Split Form

    International Nuclear Information System (INIS)

    Stancic, V.

    2001-01-01

    The singular eigenfunction expansion method has enabled the application of functional analysis methods in transport theory. However, when applying it, the users were discouraged, since in most problems, including slab problems, an extra problem has occurred. It appears necessary to solve the Fredholm integral equation in order to determine the expansion coefficients. There are several reasons for this difficulty. One reason might be the use of the full-range expansion techniques even in the regions where the function is singular. Such an example is the free boundary condition that requires the distribution to be equal to zero. Moreover, at μ = 0, the transport equation becomes an integral one. Both reasons motivated us to redefine the transport equation in a more natural way. Similar to scattering theory, here we define the flux distribution as a direct sum of forward- and backward-directed neutrons, e.g., μ ≥ 0 and μ < 0, respectively. As a result, the plane geometry transport equation is being split into coupled-pair equations. Further, using an appropriate transformation, this pair of equations reduces to a self-adjoint one having the same form as the known full-range single flux. It is interesting that all the methods of full-range theory are applicable here provided the flux as well as the transformed transport operator are two-dimensional matrices. Applying this to the slab problem, we find explicit expressions for reflected and transmitted particles caused by an arbitrary plane source. That is the news in this paper. Because of space constraints, only fundamentals of this approach will be presented here. We assume that the reader is familiar with this field; therefore, the applications are noted only at the end. (author)

  10. Variational Integrals of a Class of Nonhomogeneous -Harmonic Equations

    Directory of Open Access Journals (Sweden)

    Guanfeng Li

    2014-01-01

    Full Text Available We introduce a class of variational integrals whose Euler equations are nonhomogeneous -harmonic equations. We investigate the relationship between the minimization problem and the Euler equation and give a simple proof of the existence of some nonhomogeneous -harmonic equations by applying direct methods of the calculus of variations. Besides, we establish some interesting results on variational integrals.

  11. Instabilities in numerical solutions to Fredholm and Volterra integral equations of the first kind. Resolution by Tchebycheff polynomials. Application to photonuclear cross-sections; Instabilite des solutions numeriques d'equations integrales de Fredholm et Volterra de premiere espece. Resolution par les polynomes de Tchebycheff. Application aux sections efficaces photonucleaires

    Energy Technology Data Exchange (ETDEWEB)

    Moriceau, Y [Commissariat a l' Energie Atomique, Centre d' Etudes de Limeil, 94 - Villeneuve-Saint-Georges (France)

    1968-03-01

    It is well known, if not well explained, that photo cross-sections curves depend on numerical resolution; as well as many other physical solutions from integral equations of the first kind, they are oscillating. In the first part of this report, a typical example points out how oscillations are growing. In the second part, a new method is explained yielding a smooth resolution. From experimental data on equidistant intervals, we build functions expanded in Tchebycheff polynomials; the solution is of this kind. Then, the third part points out that semi-analytical resolutions of this problem are illusive. (author) [French] C'est un fait reconnu mais mal explique, que les courbes de sections efficaces photonucleaires dependent de la resolution numerique adoptee. Beaucoup d'autres solutions physiques extraites d'une equation integrale de 1ere espece sont dans ce cas; elles sont arbitraires et oscillatoires. Dans la 1ere partie de ce rapport, on montre, dans un cas particulier typique, comment se forment les oscillations. Dans la 2eme partie, on presente une methode originale qui permet d'obtenir une resolution exempte d'oscillations. A partir de donnees experimentales a intervalles equidistants, on construit des fonctions developpees en polynomes de Tchebycheff; la solution est de ce type. Enfin, on montre dans la 3eme partie que les resolutions semi-analytiques de ce probleme sont illusoires. (auteur)

  12. A hierarchy of Liouville integrable discrete Hamiltonian equations

    Energy Technology Data Exchange (ETDEWEB)

    Xu Xixiang [College of Science, Shandong University of Science and Technology, Qingdao 266510 (China)], E-mail: xixiang_xu@yahoo.com.cn

    2008-05-12

    Based on a discrete four-by-four matrix spectral problem, a hierarchy of Lax integrable lattice equations with two potentials is derived. Two Hamiltonian forms are constructed for each lattice equation in the resulting hierarchy by means of the discrete variational identity. A strong symmetry operator of the resulting hierarchy is given. Finally, it is shown that the resulting lattice equations are all Liouville integrable discrete Hamiltonian systems.

  13. Method of mechanical quadratures for solving singular integral equations of various types

    Science.gov (United States)

    Sahakyan, A. V.; Amirjanyan, H. A.

    2018-04-01

    The method of mechanical quadratures is proposed as a common approach intended for solving the integral equations defined on finite intervals and containing Cauchy-type singular integrals. This method can be used to solve singular integral equations of the first and second kind, equations with generalized kernel, weakly singular equations, and integro-differential equations. The quadrature rules for several different integrals represented through the same coefficients are presented. This allows one to reduce the integral equations containing integrals of different types to a system of linear algebraic equations.

  14. Feynman integrals and difference equations

    Energy Technology Data Exchange (ETDEWEB)

    Moch, S. [Deutsches Elektronen-Synchrotron (DESY), Zeuthen (Germany); Schneider, C. [Johannes Kepler Univ., Linz (Austria). Research Inst. for Symbolic Computation

    2007-09-15

    We report on the calculation of multi-loop Feynman integrals for single-scale problems by means of difference equations in Mellin space. The solution to these difference equations in terms of harmonic sums can be constructed algorithmically over difference fields, the so-called {pi}{sigma}{sup *}-fields. We test the implementation of the Mathematica package Sigma on examples from recent higher order perturbative calculations in Quantum Chromodynamics. (orig.)

  15. Integral equation hierarchy for continuum percolation

    International Nuclear Information System (INIS)

    Given, J.A.

    1988-01-01

    In this thesis a projection operator technique is presented that yields hierarchies of integral equations satisfied exactly by the n-point connectedness functions in a continuum version of the site-bond percolation problem. The n-point connectedness functions carry the same structural information for a percolation problem as then-point correlation functions do for a thermal problem. This method extends the Potts model mapping of Fortuin and Kastelyn to the continuum by exploiting an s-state generalization of the Widom-Rowlinson model, a continuum model for phase separation. The projection operator technique is used to produce an integral equation hierarchy for percolation similar to the Born-Green heirarchy. The Kirkwood superposition approximation (SA) is extended to percolation in order to close this hierarchy and yield a nonlinear integral equation for the two-point connectedness function. The fact that this function, in the SA, is the analytic continuation to negative density of the two-point correlation function in a corresponding thermal problem is discussed. The BGY equation for percolation is solved numerically, both by an expansion in powers of the density, and by an iterative technique due to Kirkwood. It is argued both analytically and numerically, that the BYG equation for percolation, unlike its thermal counterpart, shows non-classical critical behavior, with η = 1 and γ = 0.05 ± .1. Finally a sequence of refinements to the superposition approximations based in the theory of fluids by Rice and Lekner is discussed

  16. Analysis of the neutron slowing down equation

    International Nuclear Information System (INIS)

    Sengupta, A.; Karnick, H.

    1978-01-01

    The infinite series solution of the elementary neutron slowing down equation is studied using the theory of entire functions of exponential type and nonharmonic Fourier series. It is shown from Muntz--Szasz and Paley--Wiener theorems, that the set of exponentials ]exp(ilambda/sub n/u) ]/sup infinity//sub n/=-infinity, where ]lambda/sub n/]/sup infinity//sub n/=-infinity are the roots of the transcendental equation in slowing down theory, is complete and forms a basis in a lethargy interval epsilon. This distinctive role of the maximum lethargy change per collision is due to the Fredholm character of the slowing down operator which need not be quasinilpotent. The discontinuities in the derivatives of the collision density are examined by treating the slowing down equation in its differential-difference form. The solution (Hilbert) space is the union of a countable number of subspaces L 2 (-epsilon/2, epsilon/2) over each of which the exponential functions are complete

  17. Master equations and the theory of stochastic path integrals

    Science.gov (United States)

    Weber, Markus F.; Frey, Erwin

    2017-04-01

    This review provides a pedagogic and self-contained introduction to master equations and to their representation by path integrals. Since the 1930s, master equations have served as a fundamental tool to understand the role of fluctuations in complex biological, chemical, and physical systems. Despite their simple appearance, analyses of master equations most often rely on low-noise approximations such as the Kramers-Moyal or the system size expansion, or require ad-hoc closure schemes for the derivation of low-order moment equations. We focus on numerical and analytical methods going beyond the low-noise limit and provide a unified framework for the study of master equations. After deriving the forward and backward master equations from the Chapman-Kolmogorov equation, we show how the two master equations can be cast into either of four linear partial differential equations (PDEs). Three of these PDEs are discussed in detail. The first PDE governs the time evolution of a generalized probability generating function whose basis depends on the stochastic process under consideration. Spectral methods, WKB approximations, and a variational approach have been proposed for the analysis of the PDE. The second PDE is novel and is obeyed by a distribution that is marginalized over an initial state. It proves useful for the computation of mean extinction times. The third PDE describes the time evolution of a ‘generating functional’, which generalizes the so-called Poisson representation. Subsequently, the solutions of the PDEs are expressed in terms of two path integrals: a ‘forward’ and a ‘backward’ path integral. Combined with inverse transformations, one obtains two distinct path integral representations of the conditional probability distribution solving the master equations. We exemplify both path integrals in analysing elementary chemical reactions. Moreover, we show how a well-known path integral representation of averaged observables can be recovered from

  18. Master equations and the theory of stochastic path integrals.

    Science.gov (United States)

    Weber, Markus F; Frey, Erwin

    2017-04-01

    This review provides a pedagogic and self-contained introduction to master equations and to their representation by path integrals. Since the 1930s, master equations have served as a fundamental tool to understand the role of fluctuations in complex biological, chemical, and physical systems. Despite their simple appearance, analyses of master equations most often rely on low-noise approximations such as the Kramers-Moyal or the system size expansion, or require ad-hoc closure schemes for the derivation of low-order moment equations. We focus on numerical and analytical methods going beyond the low-noise limit and provide a unified framework for the study of master equations. After deriving the forward and backward master equations from the Chapman-Kolmogorov equation, we show how the two master equations can be cast into either of four linear partial differential equations (PDEs). Three of these PDEs are discussed in detail. The first PDE governs the time evolution of a generalized probability generating function whose basis depends on the stochastic process under consideration. Spectral methods, WKB approximations, and a variational approach have been proposed for the analysis of the PDE. The second PDE is novel and is obeyed by a distribution that is marginalized over an initial state. It proves useful for the computation of mean extinction times. The third PDE describes the time evolution of a 'generating functional', which generalizes the so-called Poisson representation. Subsequently, the solutions of the PDEs are expressed in terms of two path integrals: a 'forward' and a 'backward' path integral. Combined with inverse transformations, one obtains two distinct path integral representations of the conditional probability distribution solving the master equations. We exemplify both path integrals in analysing elementary chemical reactions. Moreover, we show how a well-known path integral representation of averaged observables can be recovered from them. Upon

  19. Abel integral equations analysis and applications

    CERN Document Server

    Gorenflo, Rudolf

    1991-01-01

    In many fields of application of mathematics, progress is crucially dependent on the good flow of information between (i) theoretical mathematicians looking for applications, (ii) mathematicians working in applications in need of theory, and (iii) scientists and engineers applying mathematical models and methods. The intention of this book is to stimulate this flow of information. In the first three chapters (accessible to third year students of mathematics and physics and to mathematically interested engineers) applications of Abel integral equations are surveyed broadly including determination of potentials, stereology, seismic travel times, spectroscopy, optical fibres. In subsequent chapters (requiring some background in functional analysis) mapping properties of Abel integral operators and their relation to other integral transforms in various function spaces are investi- gated, questions of existence and uniqueness of solutions of linear and nonlinear Abel integral equations are treated, and for equatio...

  20. A note on the Pfaffian integration theorem

    International Nuclear Information System (INIS)

    Borodin, Alexei; Kanzieper, Eugene

    2007-01-01

    Two alternative, fairly compact proofs are presented of the Pfaffian integration theorem that surfaced in the recent studies of spectral properties of Ginibre's Orthogonal Ensemble. The first proof is based on a concept of the Fredholm Pfaffian; the second proof is purely linear algebraic. (fast track communication)

  1. Fuchs indices and the first integrals of nonlinear differential equations

    International Nuclear Information System (INIS)

    Kudryashov, Nikolai A.

    2005-01-01

    New method of finding the first integrals of nonlinear differential equations in polynomial form is presented. Basic idea of our approach is to use the scaling of solution of nonlinear differential equation and to find the dimensions of arbitrary constants in the Laurent expansion of the general solution. These dimensions allows us to obtain the scalings of members for the first integrals of nonlinear differential equations. Taking the polynomials with unknown coefficients into account we present the algorithm of finding the first integrals of nonlinear differential equations in the polynomial form. Our method is applied to look for the first integrals of eight nonlinear ordinary differential equations of the fourth order. The general solution of one of the fourth order ordinary differential equations is given

  2. On a Volterra Stieltjes integral equation

    Directory of Open Access Journals (Sweden)

    P. T. Vaz

    1990-01-01

    Full Text Available The paper deals with a study of linear Volterra integral equations involving Lebesgue-Stieltjes integrals in two independent variables. The authors prove an existence theorem using the Banach fixed-point principle. An explicit example is also considered.

  3. Transmission problem for the Laplace equation and the integral equation method

    Czech Academy of Sciences Publication Activity Database

    Medková, Dagmar

    2012-01-01

    Roč. 387, č. 2 (2012), s. 837-843 ISSN 0022-247X Institutional research plan: CEZ:AV0Z10190503 Keywords : transmission problem * Laplace equation * boundary integral equation Subject RIV: BA - General Mathematics Impact factor: 1.050, year: 2012 http://www.sciencedirect.com/science/article/pii/S0022247X11008985

  4. Numerov iteration method for second order integral-differential equation

    International Nuclear Information System (INIS)

    Zeng Fanan; Zhang Jiaju; Zhao Xuan

    1987-01-01

    In this paper, Numerov iterative method for second order integral-differential equation and system of equations are constructed. Numerical examples show that this method is better than direct method (Gauss elimination method) in CPU time and memoy requireing. Therefore, this method is an efficient method for solving integral-differential equation in nuclear physics

  5. An integrable semi-discretization of the Boussinesq equation

    International Nuclear Information System (INIS)

    Zhang, Yingnan; Tian, Lixin

    2016-01-01

    Highlights: • A new integrable semi-discretization of the Boussinesq equation is present. • A Bäcklund transformation and a Lax pair for the differential-difference system is derived by using Hirota's bilinear method. • The soliton solutions of 'good' Boussinesq equation and numerical algorithms are investigated. - Abstract: In this paper, we present an integrable semi-discretization of the Boussinesq equation. Different from other discrete analogues, we discretize the ‘time’ variable and get an integrable differential-difference system. Under a standard limitation, the differential-difference system converges to the continuous Boussinesq equation such that the discrete system can be used to design numerical algorithms. Using Hirota's bilinear method, we find a Bäcklund transformation and a Lax pair of the differential-difference system. For the case of ‘good’ Boussinesq equation, we investigate the soliton solutions of its discrete analogue and design numerical algorithms. We find an effective way to reduce the phase shift caused by the discretization. The numerical results coincide with our analysis.

  6. ON ASYMTOTIC APPROXIMATIONS OF FIRST INTEGRALS FOR DIFFERENTIAL AND DIFFERENCE EQUATIONS

    Directory of Open Access Journals (Sweden)

    W.T. van Horssen

    2007-04-01

    Full Text Available In this paper the concept of integrating factors for differential equations and the concept of invariance factors for difference equations to obtain first integrals or invariants will be presented. It will be shown that all integrating factors have to satisfya system of partial differential equations, and that all invariance factors have to satisfy a functional equation. In the period 1997-2001 a perturbation method based on integrating vectors was developed to approximate first integrals for systems of ordinary differential equations. This perturbation method will be reviewed shortly. Also in the paper the first results in the development of a perturbation method for difference equations based on invariance factors will be presented.

  7. Minimally coupled N-particle scattering integral equations

    International Nuclear Information System (INIS)

    Kowalski, K.L.

    1977-01-01

    A concise formalism is developed which permits the efficient representation and generalization of several known techniques for deriving connected-kernel N-particle scattering integral equations. The methods of Kouri, Levin, and Tobocman and Bencze and Redish which lead to minimally coupled integral equations are of special interest. The introduction of channel coupling arrays is characterized in a general manner and the common base of this technique and that of the so-called channel coupling scheme is clarified. It is found that in the Bencze-Redish formalism a particular coupling array has a crucial function but one different from that of the arrays employed by Kouri, Levin, and Tobocman. The apparent dependence of the proof of the minimality of the Bencze-Redish integral equations upon the form of the inhomogeneous term in these equations is eliminated. This is achieved by an investigation of the full (nonminimal) Bencze-Redish kernel. It is shown that the second power of this operator is connected, a result which is needed for the full applicability of the Bencze-Redish formalism. This is used to establish the relationship between the existence of solutions to the homogeneous form of the minimal equations and eigenvalues of the full Bencze-Redish kernel

  8. Optimal estimation of the intensity function of a spatial point process

    DEFF Research Database (Denmark)

    Guan, Yongtao; Jalilian, Abdollah; Waagepetersen, Rasmus

    easily computable estimating functions. We derive the optimal estimating function in a class of first-order estimating functions. The optimal estimating function depends on the solution of a certain Fredholm integral equation and reduces to the likelihood score in case of a Poisson process. We discuss...

  9. Analytical- and numerical investigations concerning the radiance distribution at the boundaries of an anisotropically scattering slab, cylinder, or sphere.

    NARCIS (Netherlands)

    Hoenders, BJ; Benaron, DA; Chance, B; Muller, GJ; Katzir, A

    1996-01-01

    We will present two methods leading to the solution for the problem of anisotropical light scattering by arbitrarily shaped bodies. Each method converts the equation of radiative transfer and the boundary conditions valid at the boundary of the scattering medium into a set of Fredholm integral

  10. ON DIFFERENTIAL EQUATIONS, INTEGRABLE SYSTEMS, AND GEOMETRY

    OpenAIRE

    Enrique Gonzalo Reyes Garcia

    2004-01-01

    ON DIFFERENTIAL EQUATIONS, INTEGRABLE SYSTEMS, AND GEOMETRY Equations in partial derivatives appeared in the 18th century as essential tools for the analytic study of physical models and, later, they proved to be fundamental for the progress of mathematics. For example, fundamental results of modern differential geometry are based on deep theorems on differential equations. Reciprocally, it is possible to study differential equations through geometrical means just like it was done by o...

  11. A Generalized Analytic Operator-Valued Function Space Integral and a Related Integral Equation

    International Nuclear Information System (INIS)

    Chang, K.S.; Kim, B.S.; Park, C.H.; Ryu, K.S.

    2003-01-01

    We introduce a generalized Wiener measure associated with a Gaussian Markov process and define a generalized analytic operator-valued function space integral as a bounded linear operator from L p into L p-ci r cumflexprime (1< p ≤ 2) by the analytic continuation of the generalized Wiener integral. We prove the existence of the integral for certain functionals which involve some Borel measures. Also we show that the generalized analytic operator-valued function space integral satisfies an integral equation related to the generalized Schroedinger equation. The resulting theorems extend the theory of operator-valued function space integrals substantially and previous theorems about these integrals are generalized by our results

  12. An integrable semi-discretization of the Boussinesq equation

    Energy Technology Data Exchange (ETDEWEB)

    Zhang, Yingnan, E-mail: ynzhang@njnu.edu.cn [Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, Jiangsu (China); Tian, Lixin, E-mail: tianlixin@njnu.edu.cn [Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, Jiangsu (China); Nonlinear Scientific Research Center, Jiangsu University, Zhenjiang, Jiangsu (China)

    2016-10-23

    Highlights: • A new integrable semi-discretization of the Boussinesq equation is present. • A Bäcklund transformation and a Lax pair for the differential-difference system is derived by using Hirota's bilinear method. • The soliton solutions of 'good' Boussinesq equation and numerical algorithms are investigated. - Abstract: In this paper, we present an integrable semi-discretization of the Boussinesq equation. Different from other discrete analogues, we discretize the ‘time’ variable and get an integrable differential-difference system. Under a standard limitation, the differential-difference system converges to the continuous Boussinesq equation such that the discrete system can be used to design numerical algorithms. Using Hirota's bilinear method, we find a Bäcklund transformation and a Lax pair of the differential-difference system. For the case of ‘good’ Boussinesq equation, we investigate the soliton solutions of its discrete analogue and design numerical algorithms. We find an effective way to reduce the phase shift caused by the discretization. The numerical results coincide with our analysis.

  13. Normal and adjoint integral and integrodifferential neutron transport equations. Pt. 2

    International Nuclear Information System (INIS)

    Velarde, G.

    1976-01-01

    Using the simplifying hypotheses of the integrodifferential Boltzmann equations of neutron transport, given in JEN 334 report, several integral equations, and theirs adjoint ones, are obtained. Relations between the different normal and adjoint eigenfunctions are established and, in particular, proceeding from the integrodifferential Boltzmann equation it's found out the relation between the solutions of the adjoint equation of its integral one, and the solutions of the integral equation of its adjoint one (author)

  14. On a new series of integrable nonlinear evolution equations

    International Nuclear Information System (INIS)

    Ichikawa, Y.H.; Wadati, Miki; Konno, Kimiaki; Shimizu, Tohru.

    1980-10-01

    Recent results of our research are surveyed in this report. The derivative nonlinear Schroedinger equation for the circular polarized Alfven wave admits the spiky soliton solutions for the plane wave boundary condition. The nonlinear equation for complex amplitude associated with the carrier wave is shown to be a generalized nonlinear Schroedinger equation, having the ordinary cubic nonlinear term and the derivative of cubic nonlinear term. A generalized scheme of the inverse scattering transformation has confirmed that superposition of the A-K-N-S scheme and the K-N scheme for the component equations valids for the generalized nonlinear Schroedinger equation. Then, two types of new integrable nonlinear evolution equation have been derived from our scheme of the inverse scattering transformation. One is the type of nonlinear Schroedinger equation, while the other is the type of Korteweg-de Vries equation. Brief discussions are presented for physical phenomena, which could be accounted by the second type of the new integrable nonlinear evolution equation. Lastly, the stationary solitary wave solutions have been constructed for the integrable nonlinear evolution equation of the second type. These solutions have peculiar structure that they are singular and discrete. It is a new challenge to construct singular potentials by the inverse scattering transformation. (author)

  15. Rocking Rotation of a Rigid Disk Embedded in a Transversely Isotropic Half-Space

    Directory of Open Access Journals (Sweden)

    Seyed Ahmadi

    2014-06-01

    Full Text Available The asymmetric problem of rocking rotation of a circular rigid disk embedded in a finite depth of a transversely isotropic half-space is analytically addressed. The rigid disk is assumed to be in frictionless contact with the elastic half-space. By virtue of appropriate Green's functions, the mixed boundary value problem is written as a dual integral equation. Employing further mathematical techniques, the integral equation is reduced to a well-known Fredholm integral equation of the second kind. The results related to the contact stress distribution across the disk region and the equivalent rocking stiffness of the system are expressed in terms of the solution of the obtained Fredholm  integral  equation. When the rigid disk is located on the surface or at the remote boundary, the exact closed-form solutions are presented. For verification purposes, the limiting case of an isotropic half-space is considered and the results are verified with those available in the literature. The jump behavior in the results at the edge of the rigid disk for the case of an infinitesimal embedment is highlighted analytically for the first time. Selected numerical results are depicted for the contact stress distribution across the disk region, rocking stiffness of the system, normal stress, and displacement components along the radial axis. Moreover, effects of anisotropy on the rocking stiffness factor are discussed in detail.

  16. Development of the unfolding procedures in fast neutron scintillation spectrometry; Razvoj unfolding procedura u scintilacionoj spektrometriji brzih neutrona

    Energy Technology Data Exchange (ETDEWEB)

    Marinkovic, P [Elektrotehnicki fakultet, Belgrade (Yugoslavia)

    1988-07-01

    Two unfolding procedures have been developed for obtaining fast neutron spectrum from proton-recoil spectrum assigned for spectrometry with organic scintillators. First is the method of differentiation of proton-recoil spectrum, and the second is the method based on solution of integral equation of Fredholm of first kind. (author)

  17. On solvability of some quadratic functional-integral equation in Banach algebra

    International Nuclear Information System (INIS)

    Darwish, M.A.

    2007-08-01

    Using the technique of a suitable measure of non-compactness in Banach algebra, we prove an existence theorem for some functional-integral equations which contain, as particular cases, a lot of integral and functional-integral equations that arise in many branches of nonlinear analysis and its applications. Also, the famous Chandrasekhar's integral equation is considered as a special case. (author)

  18. Integrable boundary conditions and modified Lax equations

    International Nuclear Information System (INIS)

    Avan, Jean; Doikou, Anastasia

    2008-01-01

    We consider integrable boundary conditions for both discrete and continuum classical integrable models. Local integrals of motion generated by the corresponding 'transfer' matrices give rise to time evolution equations for the initial Lax operator. We systematically identify the modified Lax pairs for both discrete and continuum boundary integrable models, depending on the classical r-matrix and the boundary matrix

  19. Integral equation for inhomogeneous condensed bosons generalizing the Gross-Pitaevskii differential equation

    International Nuclear Information System (INIS)

    Angilella, G.G.N.; Pucci, R.; March, N.H.

    2004-01-01

    We give here the derivation of a Gross-Pitaevskii-type equation for inhomogeneous condensed bosons. Instead of the original Gross-Pitaevskii differential equation, we obtain an integral equation that implies less restrictive assumptions than are made in the very recent study of Pieri and Strinati [Phys. Rev. Lett. 91, 030401 (2003)]. In particular, the Thomas-Fermi approximation and the restriction to small spatial variations of the order parameter invoked in their study are avoided

  20. Numerical solution of boundary-integral equations for molecular electrostatics.

    Science.gov (United States)

    Bardhan, Jaydeep P

    2009-03-07

    Numerous molecular processes, such as ion permeation through channel proteins, are governed by relatively small changes in energetics. As a result, theoretical investigations of these processes require accurate numerical methods. In the present paper, we evaluate the accuracy of two approaches to simulating boundary-integral equations for continuum models of the electrostatics of solvation. The analysis emphasizes boundary-element method simulations of the integral-equation formulation known as the apparent-surface-charge (ASC) method or polarizable-continuum model (PCM). In many numerical implementations of the ASC/PCM model, one forces the integral equation to be satisfied exactly at a set of discrete points on the boundary. We demonstrate in this paper that this approach to discretization, known as point collocation, is significantly less accurate than an alternative approach known as qualocation. Furthermore, the qualocation method offers this improvement in accuracy without increasing simulation time. Numerical examples demonstrate that electrostatic part of the solvation free energy, when calculated using the collocation and qualocation methods, can differ significantly; for a polypeptide, the answers can differ by as much as 10 kcal/mol (approximately 4% of the total electrostatic contribution to solvation). The applicability of the qualocation discretization to other integral-equation formulations is also discussed, and two equivalences between integral-equation methods are derived.

  1. Irreducibility and co-primeness as an integrability criterion for discrete equations

    International Nuclear Information System (INIS)

    Kanki, Masataka; Mada, Jun; Mase, Takafumi; Tokihiro, Tetsuji

    2014-01-01

    We study the Laurent property, the irreducibility and co-primeness of discrete integrable and non-integrable equations. First we study a discrete integrable equation related to the Somos-4 sequence, and also a non-integrable equation as a comparison. We prove that the conditions of irreducibility and co-primeness hold only in the integrable case. Next, we generalize our previous results on the singularities of the discrete Korteweg–de Vries (dKdV) equation. In our previous paper (Kanki et al 2014 J. Phys. A: Math. Theor. 47 065201) we described the singularity confinement test (one of the integrability criteria) using the Laurent property, and the irreducibility, and co-primeness of the terms in the bilinear dKdV equation, in which we only considered simplified boundary conditions. This restriction was needed to obtain simple (monomial) relations between the bilinear form and the nonlinear form of the dKdV equation. In this paper, we prove the co-primeness of the terms in the nonlinear dKdV equation for general initial conditions and boundary conditions, by using the localization of Laurent rings and the interchange of the axes. We assert that co-primeness of the terms can be used as a new integrability criterion, which is a mathematical re-interpretation of the confinement of singularities in the case of discrete equations. (paper)

  2. Set-Valued Stochastic Equation with Set-Valued Square Integrable Martingale

    Directory of Open Access Journals (Sweden)

    Li Jun-Gang

    2017-01-01

    Full Text Available In this paper, we shall introduce the stochastic integral of a stochastic process with respect to set-valued square integrable martingale. Then we shall give the Aumann integral measurable theorem, and give the set-valued stochastic Lebesgue integral and set-valued square integrable martingale integral equation. The existence and uniqueness of solution to set-valued stochastic integral equation are proved. The discussion will be useful in optimal control and mathematical finance in psychological factors.

  3. Setting and solving several factorization problems for integral operators

    International Nuclear Information System (INIS)

    Engibaryan, N B

    2000-01-01

    The problem of factorization I-K=(I-U - )(I-U + ), is considered. Here I is the identity operator, K is a fixed integral operator of Fredholm type: (Kf)(x)=∫ a b k(x,t)f(t)dt, -∞≤a ± are unknown upper and lower Volterra operators. Classes of generalized Volterra operators U ± are introduced such that I-U ± are not necessarily invertible operators in the spaces of functions on (a,b) under consideration. A combination of the method of non-linear factorization equations and a priori estimates brings forth new results on the existence and properties of the solution to this problem for k≥0, both in the subcritical case μ + and U - vanish on some parts S - and S + of the domain S=(a,b) 2 such that S + union S - =S

  4. Babenko’s Approach to Abel’s Integral Equations

    Directory of Open Access Journals (Sweden)

    Chenkuan Li

    2018-03-01

    Full Text Available The goal of this paper is to investigate the following Abel’s integral equation of the second kind: y ( t + λ Γ ( α ∫ 0 t ( t − τ α − 1 y ( τ d τ = f ( t , ( t > 0 and its variants by fractional calculus. Applying Babenko’s approach and fractional integrals, we provide a general method for solving Abel’s integral equation and others with a demonstration of different types of examples by showing convergence of series. In particular, we extend this equation to a distributional space for any arbitrary α ∈ R by fractional operations of generalized functions for the first time and obtain several new and interesting results that cannot be realized in the classical sense or by the Laplace transform.

  5. Symbolic-Numeric Integration of the Dynamical Cosserat Equations

    KAUST Repository

    Lyakhov, Dmitry A.

    2017-08-29

    We devise a symbolic-numeric approach to the integration of the dynamical part of the Cosserat equations, a system of nonlinear partial differential equations describing the mechanical behavior of slender structures, like fibers and rods. This is based on our previous results on the construction of a closed form general solution to the kinematic part of the Cosserat system. Our approach combines methods of numerical exponential integration and symbolic integration of the intermediate system of nonlinear ordinary differential equations describing the dynamics of one of the arbitrary vector-functions in the general solution of the kinematic part in terms of the module of the twist vector-function. We present an experimental comparison with the well-established generalized \\\\alpha -method illustrating the computational efficiency of our approach for problems in structural mechanics.

  6. Symbolic-Numeric Integration of the Dynamical Cosserat Equations

    KAUST Repository

    Lyakhov, Dmitry A.; Gerdt, Vladimir P.; Weber, Andreas G.; Michels, Dominik L.

    2017-01-01

    We devise a symbolic-numeric approach to the integration of the dynamical part of the Cosserat equations, a system of nonlinear partial differential equations describing the mechanical behavior of slender structures, like fibers and rods. This is based on our previous results on the construction of a closed form general solution to the kinematic part of the Cosserat system. Our approach combines methods of numerical exponential integration and symbolic integration of the intermediate system of nonlinear ordinary differential equations describing the dynamics of one of the arbitrary vector-functions in the general solution of the kinematic part in terms of the module of the twist vector-function. We present an experimental comparison with the well-established generalized \\alpha -method illustrating the computational efficiency of our approach for problems in structural mechanics.

  7. Determination of crystal oscillatory spectra by internal friction data spectroscopic analysis

    International Nuclear Information System (INIS)

    Zaykin, Yu.A.

    1998-01-01

    Technique for relaxation spectra determination on the basis of internal friction averaging over relaxation frequencies is developed. It is shown that mathematically the problem is reduced to solution of the first type Fredholm integral equation. Impurity oscillatory spectra in alpha-iron, molybdenum and Fe-Cr-Ni alloy are obtained. (author)

  8. Integrability of the Gross-Pitaevskii equation with Feshbach resonance management

    International Nuclear Information System (INIS)

    Zhao Dun; Luo Honggang; Chai Huayue

    2008-01-01

    In this Letter we study the integrability of a class of Gross-Pitaevskii equations managed by Feshbach resonance in an expulsive parabolic external potential. By using WTC test, we find a condition under which the Gross-Pitaevskii equation is completely integrable. Under the present model, this integrability condition is completely consistent with that proposed by Serkin, Hasegawa, and Belyaeva [V.N. Serkin, A. Hasegawa, T.L. Belyaeva, Phys. Rev. Lett. 98 (2007) 074102]. Furthermore, this integrability can also be explicitly shown by a transformation, which can convert the Gross-Pitaevskii equation into the well-known standard nonlinear Schroedinger equation. By this transformation, each exact solution of the standard nonlinear Schroedinger equation can be converted into that of the Gross-Pitaevskii equation, which builds a systematical connection between the canonical solitons and the so-called nonautonomous ones. The finding of this transformation has a significant contribution to understanding the essential properties of the nonautonomous solitons and the dynamics of the Bose-Einstein condensates by using the Feshbach resonance technique

  9. Comments on the integrability of the loop-space chiral equations

    International Nuclear Information System (INIS)

    Gu, C.; Wang, L.L.C.

    1980-01-01

    A demonstration is given how the ordinary space chiral equations provide the existence conditions for the infinite number of conserved currents and how these currents are related to the so-called inverse-scattering equations, whose integrability is provided by the original chiral equations. Loop-space chiral equations are introduced. The integrability conditions of the non-local currents in two possible different situations are discussed. In the first case, the generating functions are functionals of the loop alone. The integrability conditions are not satisfied and higher order conserved non-local currents do not exist. In the second case, the generating functions are functionals of the loop as well as a parameter the integrability conditions at a restricted point of the parameter are satisfied, however there is an infinite fold of arbitrariness. It indicates that additional guiding principles are needed in addition to the original loop-space chiral equation in order to uniquely determine the infinite conserved non-local currents as functionals of the loop and the parameter

  10. First integrals of the axisymmetric shape equation of lipid membranes

    Science.gov (United States)

    Zhang, Yi-Heng; McDargh, Zachary; Tu, Zhan-Chun

    2018-03-01

    The shape equation of lipid membranes is a fourth-order partial differential equation. Under the axisymmetric condition, this equation was transformed into a second-order ordinary differential equation (ODE) by Zheng and Liu (Phys. Rev. E 48 2856 (1993)). Here we try to further reduce this second-order ODE to a first-order ODE. First, we invert the usual process of variational calculus, that is, we construct a Lagrangian for which the ODE is the corresponding Euler–Lagrange equation. Then, we seek symmetries of this Lagrangian according to the Noether theorem. Under a certain restriction on Lie groups of the shape equation, we find that the first integral only exists when the shape equation is identical to the Willmore equation, in which case the symmetry leading to the first integral is scale invariance. We also obtain the mechanical interpretation of the first integral by using the membrane stress tensor. Project supported by the National Natural Science Foundation of China (Grant No. 11274046) and the National Science Foundation of the United States (Grant No. 1515007).

  11. Kwong-Wong-type integral equation on time scales

    Directory of Open Access Journals (Sweden)

    Baoguo Jia

    2011-09-01

    Full Text Available Consider the second-order nonlinear dynamic equation $$ [r(tx^Delta(ho(t]^Delta+p(tf(x(t=0, $$ where $p(t$ is the backward jump operator. We obtain a Kwong-Wong-type integral equation, that is: If $x(t$ is a nonoscillatory solution of the above equation on $[T_0,infty$, then the integral equation $$ frac{r^sigma(tx^Delta(t}{f(x^sigma(t} =P^sigma(t+int^infty_{sigma(t}frac{r^sigma(s [int^1_0f'(x_h(sdh][x^Delta(s]^2}{f(x(s f(x^sigma(s}Delta s $$ is satisfied for $tgeq T_0$, where $P^sigma(t=int^infty_{sigma(t}p(sDelta s$, and $x_h(s=x(s+hmu(sx^Delta(s$. As an application, we show that the superlinear dynamic equation $$ [r(tx^{Delta}(ho(t]^Delta+p(tf(x(t=0, $$ is oscillatory, under certain conditions.

  12. Complete integrability of the difference evolution equations

    International Nuclear Information System (INIS)

    Gerdjikov, V.S.; Ivanov, M.I.; Kulish, P.P.

    1980-01-01

    The class of exactly solvable nonlinear difference evolution equations (DEE) related to the discrete analog of the one-dimensional Dirac problem L is studied. For this starting from L we construct a special linear non-local operator Λ and obtain the expansions of w and σ 3 deltaw over its eigenfunctions, w being the potential in L. This allows us to obtain compact expressions for the integrals of motion and to prove that these DEE are completely integrable Hamiltonian systems. Moreover, it is shown that there exists a hierarchy of Hamiltonian structures, generated by Λ, and the action-angle variables are explicity calculated. As particular cases the difference analog of the non-linear Schroedinger equation and the modified Korteweg-de-Vries equation are considered. The quantization of these Hamiltonian system through the use of the quantum inverse scattering method is briefly discussed [ru

  13. Initial states in integrable quantum field theory quenches from an integral equation hierarchy

    Directory of Open Access Journals (Sweden)

    D.X. Horváth

    2016-01-01

    Full Text Available We consider the problem of determining the initial state of integrable quantum field theory quenches in terms of the post-quench eigenstates. The corresponding overlaps are a fundamental input to most exact methods to treat integrable quantum quenches. We construct and examine an infinite integral equation hierarchy based on the form factor bootstrap, proposed earlier as a set of conditions determining the overlaps. Using quenches of the mass and interaction in Sinh-Gordon theory as a concrete example, we present theoretical arguments that the state has the squeezed coherent form expected for integrable quenches, and supporting an Ansatz for the solution of the hierarchy. Moreover we also develop an iterative method to solve numerically the lowest equation of the hierarchy. The iterative solution along with extensive numerical checks performed using the next equation of the hierarchy provides a strong numerical evidence that the proposed Ansatz gives a very good approximation for the solution.

  14. Initial states in integrable quantum field theory quenches from an integral equation hierarchy

    Energy Technology Data Exchange (ETDEWEB)

    Horváth, D.X., E-mail: esoxluciuslinne@gmail.com [MTA-BME “Momentum” Statistical Field Theory Research Group, Budafoki út 8, 1111 Budapest (Hungary); Department of Theoretical Physics, Budapest University of Technology and Economics, Budafoki út 8, 1111 Budapest (Hungary); Sotiriadis, S., E-mail: sotiriad@sissa.it [SISSA and INFN, Via Bonomea 265, 34136 Trieste (Italy); Takács, G., E-mail: takacsg@eik.bme.hu [MTA-BME “Momentum” Statistical Field Theory Research Group, Budafoki út 8, 1111 Budapest (Hungary); Department of Theoretical Physics, Budapest University of Technology and Economics, Budafoki út 8, 1111 Budapest (Hungary)

    2016-01-15

    We consider the problem of determining the initial state of integrable quantum field theory quenches in terms of the post-quench eigenstates. The corresponding overlaps are a fundamental input to most exact methods to treat integrable quantum quenches. We construct and examine an infinite integral equation hierarchy based on the form factor bootstrap, proposed earlier as a set of conditions determining the overlaps. Using quenches of the mass and interaction in Sinh-Gordon theory as a concrete example, we present theoretical arguments that the state has the squeezed coherent form expected for integrable quenches, and supporting an Ansatz for the solution of the hierarchy. Moreover we also develop an iterative method to solve numerically the lowest equation of the hierarchy. The iterative solution along with extensive numerical checks performed using the next equation of the hierarchy provides a strong numerical evidence that the proposed Ansatz gives a very good approximation for the solution.

  15. Coupling Integrable Couplings of an Equation Hierarchy

    International Nuclear Information System (INIS)

    Wang Hui; Xia Tie-Cheng

    2013-01-01

    Based on a kind of Lie algebra G proposed by Zhang, one isospectral problem is designed. Under the framework of zero curvature equation, a new kind of integrable coupling of an equation hierarchy is generated using the methods proposed by Ma and Gao. With the help of variational identity, we get the Hamiltonian structure of the hierarchy. (general)

  16. Multi-component bi-Hamiltonian Dirac integrable equations

    Energy Technology Data Exchange (ETDEWEB)

    Ma Wenxiu [Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620-5700 (United States)], E-mail: mawx@math.usf.edu

    2009-01-15

    A specific matrix iso-spectral problem of arbitrary order is introduced and an associated hierarchy of multi-component Dirac integrable equations is constructed within the framework of zero curvature equations. The bi-Hamiltonian structure of the obtained Dirac hierarchy is presented be means of the variational trace identity. Two examples in the cases of lower order are computed.

  17. Quadratic algebras in the noncommutative integration method of wave equation

    International Nuclear Information System (INIS)

    Varaksin, O.L.

    1995-01-01

    The paper deals with the investigation of applications of the method of noncommutative integration of linear differential equations by partial derivatives. Nontrivial example was taken for integration of three-dimensions wave equation with the use of non-Abelian quadratic algebras

  18. Analytic solution of integral equations for molecular fluids

    International Nuclear Information System (INIS)

    Cummings, P.T.

    1984-01-01

    We review some recent progress in the analytic solution of integral equations for molecular fluids. The site-site Ornstein-Zernike (SSOZ) equation with approximate closures appropriate to homonuclear diatomic fluids both with and without attractive dispersion-like interactions has recently been solved in closed form analytically. In this paper, the close relationship between the SSOZ equation for homonuclear dumbells and the usual Ornstein-Zernike (OZ) equation for atomic fluids is carefully elucidated. This relationship is a key motivation for the analytic solutions of the SSOZ equation that have been obtained to date. (author)

  19. Differential equations and integrable models: the SU(3) case

    International Nuclear Information System (INIS)

    Dorey, Patrick; Tateo, Roberto

    2000-01-01

    We exhibit a relationship between the massless a 2 (2) integrable quantum field theory and a certain third-order ordinary differential equation, thereby extending a recent result connecting the massless sine-Gordon model to the Schroedinger equation. This forms part of a more general correspondence involving A 2 -related Bethe ansatz systems and third-order differential equations. A non-linear integral equation for the generalised spectral problem is derived, and some numerical checks are performed. Duality properties are discussed, and a simple variant of the non-linear equation is suggested as a candidate to describe the finite volume ground state energies of minimal conformal field theories perturbed by the operators phi 12 , phi 21 and phi 15 . This is checked against previous results obtained using the thermodynamic Bethe ansatz

  20. Partially integrable nonlinear equations with one higher symmetry

    International Nuclear Information System (INIS)

    Mikhailov, A V; Novikov, V S; Wang, J P

    2005-01-01

    In this letter, we present a family of second order in time nonlinear partial differential equations, which have only one higher symmetry. These equations are not integrable, but have a solution depending on one arbitrary function. (letter to the editor)

  1. On the integrability of the generalized Fisher-type nonlinear diffusion equations

    International Nuclear Information System (INIS)

    Wang Dengshan; Zhang Zhifei

    2009-01-01

    In this paper, the geometric integrability and Lax integrability of the generalized Fisher-type nonlinear diffusion equations with modified diffusion in (1+1) and (2+1) dimensions are studied by the pseudo-spherical surface geometry method and prolongation technique. It is shown that the (1+1)-dimensional Fisher-type nonlinear diffusion equation is geometrically integrable in the sense of describing a pseudo-spherical surface of constant curvature -1 only for m = 2, and the generalized Fisher-type nonlinear diffusion equations in (1+1) and (2+1) dimensions are Lax integrable only for m = 2. This paper extends the results in Bindu et al 2001 (J. Phys. A: Math. Gen. 34 L689) and further provides the integrability information of (1+1)- and (2+1)-dimensional Fisher-type nonlinear diffusion equations for m = 2

  2. Numerical integration of asymptotic solutions of ordinary differential equations

    Science.gov (United States)

    Thurston, Gaylen A.

    1989-01-01

    Classical asymptotic analysis of ordinary differential equations derives approximate solutions that are numerically stable. However, the analysis also leads to tedious expansions in powers of the relevant parameter for a particular problem. The expansions are replaced with integrals that can be evaluated by numerical integration. The resulting numerical solutions retain the linear independence that is the main advantage of asymptotic solutions. Examples, including the Falkner-Skan equation from laminar boundary layer theory, illustrate the method of asymptotic analysis with numerical integration.

  3. Energy preserving integration of bi-Hamiltonian partial differential equations

    NARCIS (Netherlands)

    Karasozen, B.; Simsek, G.

    2013-01-01

    The energy preserving average vector field (AVF) integrator is applied to evolutionary partial differential equations (PDEs) in bi-Hamiltonian form with nonconstant Poisson structures. Numerical results for the Korteweg de Vries (KdV) equation and for the Ito type coupled KdV equation confirm the

  4. Simplifying Differential Equations for Multiscale Feynman Integrals beyond Multiple Polylogarithms.

    Science.gov (United States)

    Adams, Luise; Chaubey, Ekta; Weinzierl, Stefan

    2017-04-07

    In this Letter we exploit factorization properties of Picard-Fuchs operators to decouple differential equations for multiscale Feynman integrals. The algorithm reduces the differential equations to blocks of the size of the order of the irreducible factors of the Picard-Fuchs operator. As a side product, our method can be used to easily convert the differential equations for Feynman integrals which evaluate to multiple polylogarithms to an ϵ form.

  5. Discrete Painlevé equations: an integrability paradigm

    International Nuclear Information System (INIS)

    Grammaticos, B; Ramani, A

    2014-01-01

    In this paper we present a review of results on discrete Painlevé equations. We begin with an introduction which serves as a refresher on the continuous Painlevé equations. Next, in the first, main part of the paper, we introduce the discrete Painlevé equations, the various methods for their derivation, and their properties as well as their classification scheme. Along the way we present a brief summary of the two major discrete integrability detectors and of Quispel–Roberts–Thompson mapping, which plays a primordial role in the derivation of discrete Painlevé equations. The second part of the paper is more technical and focuses on the presentation of new results on what are called asymmetric discrete Painlevé equations. (comment)

  6. Lectures on differential equations for Feynman integrals

    International Nuclear Information System (INIS)

    Henn, Johannes M

    2015-01-01

    Over the last year significant progress was made in the understanding of the computation of Feynman integrals using differential equations (DE). These lectures give a review of these developments, while not assuming any prior knowledge of the subject. After an introduction to DE for Feynman integrals, we point out how they can be simplified using algorithms available in the mathematical literature. We discuss how this is related to a recent conjecture for a canonical form of the equations. We also discuss a complementary approach that is based on properties of the space–time loop integrands, and explain how the ideas of leading singularities and d-log representations can be used to find an optimal basis for the DE. Finally, as an application of these ideas we show how single-scale integrals can be bootstrapped using the Drinfeld associator of a DE. (topical review)

  7. Bargmann Symmetry Constraint for a Family of Liouville Integrable Differential-Difference Equations

    International Nuclear Information System (INIS)

    Xu Xixiang

    2012-01-01

    A family of integrable differential-difference equations is derived from a new matrix spectral problem. The Hamiltonian forms of obtained differential-difference equations are constructed. The Liouville integrability for the obtained integrable family is proved. Then, Bargmann symmetry constraint of the obtained integrable family is presented by binary nonliearization method of Lax pairs and adjoint Lax pairs. Under this Bargmann symmetry constraints, an integrable symplectic map and a sequences of completely integrable finite-dimensional Hamiltonian systems in Liouville sense are worked out, and every integrable differential-difference equations in the obtained family is factored by the integrable symplectic map and a completely integrable finite-dimensional Hamiltonian system. (general)

  8. Integrable semi-discretizations of the reduced Ostrovsky equation

    International Nuclear Information System (INIS)

    Feng, Bao-Feng; Maruno, Ken-ichi; Ohta, Yasuhiro

    2015-01-01

    Based on our previous work on the reduced Ostrovsky equation (J. Phys. A: Math. Theor. 45 355203), we construct its integrable semi-discretizations. Since the reduced Ostrovsky equation admits two alternative representations, one being its original form, the other the differentiated form (the short wave limit of the Degasperis–Procesi equation) two semi-discrete analogues of the reduced Ostrovsky equation are constructed possessing the same N-loop soliton solution. The relationship between these two versions of semi-discretizations is also clarified. (paper)

  9. Block-pulse functions approach to numerical solution of Abel’s integral equation

    Directory of Open Access Journals (Sweden)

    Monireh Nosrati Sahlan

    2015-12-01

    Full Text Available This study aims to present a computational method for solving Abel’s integral equation of the second kind. The introduced method is based on the use of Block-pulse functions (BPFs via collocation method. Abel’s integral equations as singular Volterra integral equations are hard and heavy in computation, but because of the properties of BPFs, as is reported in examples, this method is more efficient and more accurate than some other methods for solving this class of integral equations. On the other hand, the benefit of this method is low cost of computing operations. The applied method transforms the singular integral equation into triangular linear algebraic system that can be solved easily. An error analysis is worked out and applications are demonstrated through illustrative examples.

  10. Integration of differential equations by the pseudo-linear (PL) approximation

    International Nuclear Information System (INIS)

    Bonalumi, Riccardo A.

    1998-01-01

    A new method of integrating differential equations was originated with the technique of approximately calculating the integrals called the pseudo-linear (PL) procedure: this method is A-stable. This article contains the following examples: 1st order ordinary differential equations (ODEs), 2nd order linear ODEs, stiff system of ODEs (neutron kinetics), one-dimensional parabolic (diffusion) partial differential equations. In this latter case, this PL method coincides with the Crank-Nicholson method

  11. On the complete integrability of the discrete Nahm equations

    International Nuclear Information System (INIS)

    Murray, M.K.

    2000-01-01

    The discrete Nahm equations, a system of matrix valued difference equations, arose in the work of Braam and Austin on half-integral mass hyperbolic monopoles. We show that the discrete Nahm equations are completely integrable in a natural sense: to any solution we can associate a spectral curve and a holomorphic line-bundle over the spectral curve, such that the discrete-time DN evolution corresponds to walking in the Jacobian of the spectral curve in a straight line through the line-bundle with steps of a fixed size. Some of the implications for hyperbolic monopoles are also discussed. (orig.)

  12. Splines and their reciprocal-bases in volume-integral equations

    International Nuclear Information System (INIS)

    Sabbagh, H.A.

    1993-01-01

    The authors briefly outline the use of higher-order splines and their reciprocal-bases in discretizing the volume-integral equations of electromagnetics. The discretization is carried out by means of the method of moments, in which the expansion functions are the higher-order splines, and the testing functions are the corresponding reciprocal-basis functions. These functions satisfy an orthogonality condition with respect to the spline expansion functions. Thus, the method is not Galerkin, but the structure of the resulting equations is quite regular, nevertheless. The theory is applied to the volume-integral equations for the unknown current density, or unknown electric field, within a scattering body, and to the equations for eddy-current nondestructive evaluation. Numerical techniques for computing the matrix elements are also given

  13. Evaluating four-loop conformal Feynman integrals by D-dimensional differential equations

    Science.gov (United States)

    Eden, Burkhard; Smirnov, Vladimir A.

    2016-10-01

    We evaluate a four-loop conformal integral, i.e. an integral over four four-dimensional coordinates, by turning to its dimensionally regularized version and applying differential equations for the set of the corresponding 213 master integrals. To solve these linear differential equations we follow the strategy suggested by Henn and switch to a uniformly transcendental basis of master integrals. We find a solution to these equations up to weight eight in terms of multiple polylogarithms. Further, we present an analytical result for the given four-loop conformal integral considered in four-dimensional space-time in terms of single-valued harmonic polylogarithms. As a by-product, we obtain analytical results for all the other 212 master integrals within dimensional regularization, i.e. considered in D dimensions.

  14. Evaluating four-loop conformal Feynman integrals by D-dimensional differential equations

    Energy Technology Data Exchange (ETDEWEB)

    Eden, Burkhard [Institut für Mathematik und Physik, Humboldt-Universität zu Berlin,Zum großen Windkanal 6, 12489 Berlin (Germany); Smirnov, Vladimir A. [Skobeltsyn Institute of Nuclear Physics, Moscow State University,119992 Moscow (Russian Federation)

    2016-10-21

    We evaluate a four-loop conformal integral, i.e. an integral over four four-dimensional coordinates, by turning to its dimensionally regularized version and applying differential equations for the set of the corresponding 213 master integrals. To solve these linear differential equations we follow the strategy suggested by Henn and switch to a uniformly transcendental basis of master integrals. We find a solution to these equations up to weight eight in terms of multiple polylogarithms. Further, we present an analytical result for the given four-loop conformal integral considered in four-dimensional space-time in terms of single-valued harmonic polylogarithms. As a by-product, we obtain analytical results for all the other 212 master integrals within dimensional regularization, i.e. considered in D dimensions.

  15. Unconditionally stable integration of Maxwell's equations

    NARCIS (Netherlands)

    J.G. Verwer (Jan); M.A. Botchev

    2008-01-01

    htmlabstractNumerical integration of Maxwell''s equations is often based on explicit methods accepting a stability step size restriction. In literature evidence is given that there is also a need for unconditionally stable methods, as exemplified by the successful alternating direction

  16. Is Yang-Mills equation a totally integrable system. Lecture III

    International Nuclear Information System (INIS)

    Chau Wang, L.L.

    1981-01-01

    Topics covered include: loop-space formulation of gauge theory - loop-space chiral equation; two dimensional chiral equation - conservation laws, linear system and integrability; and parallel development for the loop-space chiral equation - subtlety

  17. Adaptive integral equation methods in transport theory

    International Nuclear Information System (INIS)

    Kelley, C.T.

    1992-01-01

    In this paper, an adaptive multilevel algorithm for integral equations is described that has been developed with the Chandrasekhar H equation and its generalizations in mind. The algorithm maintains good performance when the Frechet derivative of the nonlinear map is singular at the solution, as happens in radiative transfer with conservative scattering and in critical neutron transport. Numerical examples that demonstrate the algorithm's effectiveness are presented

  18. Integration of equations of parabolic type by the method of nets

    CERN Document Server

    Saul'Yev, V K; Stark, M; Ulam, S

    1964-01-01

    International Series of Monographs in Pure and Applied Mathematics, Volume 54: Integration of Equations of Parabolic Type by the Method of Nets deals with solving parabolic partial differential equations using the method of nets. The first part of this volume focuses on the construction of net equations, with emphasis on the stability and accuracy of the approximating net equations. The method of nets or method of finite differences (used to define the corresponding numerical method in ordinary differential equations) is one of many different approximate methods of integration of partial diff

  19. Geometry, Heat Equation and Path Integrals on the Poincare Upper Half-Plane

    OpenAIRE

    Reijiro, KUBO; Research Institute for Theoretical Physics Hiroshima University

    1988-01-01

    Geometry, heat equation and Feynman's path integrals are studied on the Poincare upper half-plane. The fundamental solution to the heat equation ∂f/∂t=Δ_Hf is expressed in terms of a path integral defined on the upper half-plane. It is shown that Kac's statement that Feynman's path integral satisfies the Schrodinger equation is also valid for our case.

  20. Unconditionally stable integration of Maxwell's equations

    NARCIS (Netherlands)

    Verwer, J.G.; Bochev, Mikhail A.

    Numerical integration of Maxwell's equations is often based on explicit methods accepting a stability step size restriction. In literature evidence is given that there is also a need for unconditionally stable methods, as exemplified by the successful alternating direction implicit finite difference

  1. Unconditionally stable integration of Maxwell's equations

    NARCIS (Netherlands)

    J.G. Verwer (Jan); M.A. Botchev

    2009-01-01

    textabstractNumerical integration of Maxwell’s equations is often based on explicit methods accepting a stability step size restriction. In literature evidence is given that there is also a need for unconditionally stable methods, as exemplified by the successful alternating direction implicit –

  2. Numerical solution of the potential problem by integral equations without Green's functions

    International Nuclear Information System (INIS)

    De Mey, G.

    1977-01-01

    An integral equation technique will be presented to solve Laplace's equation in a two-dimensional area S. The Green's function has been replaced by a particular solution of Laplace equation in order to establish the integral equation. It is shown that accurate results can be obtained provided the pivotal elimination method is used to solve the linear algebraic set

  3. Integral propagator solvers for Vlasov-Fokker-Planck equations

    International Nuclear Information System (INIS)

    Donoso, J M; Rio, E del

    2007-01-01

    We briefly discuss the use of short-time integral propagators on solving the so-called Vlasov-Fokker-Planck equation for the dynamics of a distribution function. For this equation, the diffusion tensor is singular and the usual Gaussian representation of the short-time propagator is no longer valid. However, we prove that the path-integral approach on solving the equation is, in fact, reliable by means of our generalized propagator, which is obtained through the construction of an auxiliary solvable Fokker-Planck equation. The new representation of the grid-free advancing scheme describes the inherent cross- and self-diffusion processes, in both velocity and configuration spaces, in a natural manner, although these processes are not explicitly depicted in the differential equation. We also show that some splitting methods, as well as some finite-difference schemes, could fail in describing the aforementioned diffusion processes, governed in the whole phase space only by the velocity diffusion tensor. The short-time transition probability offers a stable and robust numerical algorithm that preserves the distribution positiveness and its norm, ensuring the smoothness of the evolving solution at any time step. (fast track communication)

  4. The Integral Equation Method and the Neumann Problem for the Poisson Equation on NTA Domains

    Czech Academy of Sciences Publication Activity Database

    Medková, Dagmar

    2009-01-01

    Roč. 63, č. 21 (2009), s. 227-247 ISSN 0378-620X Institutional research plan: CEZ:AV0Z10190503 Keywords : Poisson equation * Neumann problem * integral equation method Subject RIV: BA - General Mathematics Impact factor: 0.477, year: 2009

  5. Geometry, heat equation and path integrals on the Poincare upper half-plane

    International Nuclear Information System (INIS)

    Kubo, Reijiro.

    1987-08-01

    Geometry, heat equation and Feynman's path integrals are studied on the Poincare upper half-plane. The fundamental solution to the heat equation δf/δt = Δ H f is expressed in terms of a path integral defined on the upper half-plane. It is shown that Kac's proof that Feynman's path integral satisfies the Schroedinger equation is also valid for our case. (author)

  6. On discrete 2D integrable equations of higher order

    International Nuclear Information System (INIS)

    Adler, V E; Postnikov, V V

    2014-01-01

    We study two-dimensional discrete integrable equations of order 1 with respect to one independent variable and m with respect to another one. A generalization of the multidimensional consistency property is proposed for this type of equations. The examples are related to the Bäcklund–Darboux transformations for the lattice equations of Bogoyavlensky type. (paper)

  7. Integration of the three-dimensional Vlasov equation for a magnetized plasma

    International Nuclear Information System (INIS)

    Cheng, C.Z.

    1976-04-01

    A second order splitting scheme is developed to integrate the three dimensional Vlasov equation for a plasma in a magnetic field. The integration of the Vlasov equation is divided into a series of intermediate steps and Fourier interpolation and the ASD method with a third order Taylor expansion are used to integrate the fractional equations. Numerical experiments related to cyclotron waves in 2 and 2 1 / 2 D are demonstrated with high accuracy and efficiency. The computer storage requirements are modest; for example, a typical 2D nonlinear electron plasma simulation requires only 4000 ''particles.''

  8. Monograph - The Numerical Integration of Ordinary Differential Equations.

    Science.gov (United States)

    Hull, T. E.

    The materials presented in this monograph are intended to be included in a course on ordinary differential equations at the upper division level in a college mathematics program. These materials provide an introduction to the numerical integration of ordinary differential equations, and they can be used to supplement a regular text on this…

  9. Abecedarian School on Symmetries and Integrability of Difference Equations (ASIDE) & SIDE 12 International Conference Symmetries and Integrability of Difference Equations

    CERN Document Server

    Rebelo, Raphaël; Winternitz, Pavel

    2017-01-01

    This book shows how Lie group and integrability techniques, originally developed for differential equations, have been adapted to the case of difference equations. Difference equations are playing an increasingly important role in the natural sciences. Indeed, many phenomena are inherently discrete and thus naturally described by difference equations. More fundamentally, in subatomic physics, space-time may actually be discrete. Differential equations would then just be approximations of more basic discrete ones. Moreover, when using differential equations to analyze continuous processes, it is often necessary to resort to numerical methods. This always involves a discretization of the differential equations involved, thus replacing them by difference ones. Each of the nine peer-reviewed chapters in this volume serves as a self-contained treatment of a topic, containing introductory material as well as the latest research results and exercises. Each chapter is presented by one or more early career researchers...

  10. On monotonic solutions of an integral equation of Abel type

    International Nuclear Information System (INIS)

    Darwish, Mohamed Abdalla

    2007-08-01

    We present an existence theorem of monotonic solutions for a quadratic integral equation of Abel type in C[0, 1]. The famous Chandrasekhar's integral equation is considered as a special case. The concept of measure of noncompactness and a fi xed point theorem due to Darbo are the main tools in carrying out our proof. (author)

  11. Alternative integral equations and perturbation expansions for self-coupled scalar fields

    International Nuclear Information System (INIS)

    Ford, L.H.

    1985-01-01

    It is shown that the theory of a self-coupled scalar field may be expressed in terms of a class of integral equations which include the Yang-Feldman equation as a particular case. Other integral equations in this class could be used to generate alternative perturbation expansions which contain a nonanalytic dependence upon the coupling constant and are less ultraviolet divergent than the conventional perturbation expansion. (orig.)

  12. Geometrical-integrability constraints and equations of motion in four plus extended super spaces

    International Nuclear Information System (INIS)

    Chau, L.L.

    1987-01-01

    It is pointed out that many equations of motion in physics, including gravitational and Yang-Mills equations, have a common origin: i.e. they are the results of certain geometrical integrability conditions. These integrability conditions lead to linear systems and conservation laws that are important in integrating these equations of motion

  13. Differential equations for loop integrals in Baikov representation

    Science.gov (United States)

    Bosma, Jorrit; Larsen, Kasper J.; Zhang, Yang

    2018-05-01

    We present a proof that differential equations for Feynman loop integrals can always be derived in Baikov representation without involving dimension-shift identities. We moreover show that in a large class of two- and three-loop diagrams it is possible to avoid squared propagators in the intermediate steps of setting up the differential equations.

  14. Integral equations for free-molecule ow in MEMS: recent advancements

    Directory of Open Access Journals (Sweden)

    Fedeli Patrick

    2017-03-01

    Full Text Available We address a Boundary Integral Equation (BIE approach for the analysis of gas dissipation in near-vacuum for Micro Electro Mechanical Systems (MEMS. Inspired by an analogy with the radiosity equation in computer graphics, we discuss an efficient way to compute the visible domain of integration. Moreover, we tackle the issue of near singular integrals by developing a set of analytical formulas for planar polyhedral domains. Finally a validation with experimental results taken from the literature is presented.

  15. Nonlinear integral equations for the sausage model

    Science.gov (United States)

    Ahn, Changrim; Balog, Janos; Ravanini, Francesco

    2017-08-01

    The sausage model, first proposed by Fateev, Onofri, and Zamolodchikov, is a deformation of the O(3) sigma model preserving integrability. The target space is deformed from the sphere to ‘sausage’ shape by a deformation parameter ν. This model is defined by a factorizable S-matrix which is obtained by deforming that of the O(3) sigma model by a parameter λ. Clues for the deformed sigma model are provided by various UV and IR information through the thermodynamic Bethe ansatz (TBA) analysis based on the S-matrix. Application of TBA to the sausage model is, however, limited to the case of 1/λ integer where the coupled integral equations can be truncated to a finite number. In this paper, we propose a finite set of nonlinear integral equations (NLIEs), which are applicable to generic value of λ. Our derivation is based on T-Q relations extracted from the truncated TBA equations. For a consistency check, we compute next-leading order corrections of the vacuum energy and extract the S-matrix information in the IR limit. We also solved the NLIE both analytically and numerically in the UV limit to get the effective central charge and compared with that of the zero-mode dynamics to obtain exact relation between ν and λ. Dedicated to the memory of Petr Petrovich Kulish.

  16. Feynman path integral related to stochastic schroedinger equation

    International Nuclear Information System (INIS)

    Belavkin, V.P.; Smolyanov, O.G.

    1998-01-01

    The derivation of the Schroedinger equation describing the continuous measurement process is presented. The representation of the solution of the stochastic Schroedinger equation for continuous measurements is obtained by means of the Feynman path integral. The connection with the heuristic approach to the description of continuous measurements is considered. The connection with the Senon paradox is established [ru

  17. On the structure of the commutative Z2 graded algebra valued integrable equations

    International Nuclear Information System (INIS)

    Konopelchenko, B.G.

    1980-01-01

    Partial differential equations integrable by the linear matrix spectral problem of arbitrary order are considered for the case that the 'potentials' take their values in the commutative infinte-dimensional Z 2 graded algebra (superalgebra). The general form of the integrable equations and their Baecklund transformations are found. The infinite sets of the integrals of the motion are constructed. The hamiltonian character of the integrable equations is proved. (orig.)

  18. Set-valued and fuzzy stochastic integral equations driven by semimartingales under Osgood condition

    Directory of Open Access Journals (Sweden)

    Malinowski Marek T.

    2015-01-01

    Full Text Available We analyze the set-valued stochastic integral equations driven by continuous semimartingales and prove the existence and uniqueness of solutions to such equations in the framework of the hyperspace of nonempty, bounded, convex and closed subsets of the Hilbert space L2 (consisting of square integrable random vectors. The coefficients of the equations are assumed to satisfy the Osgood type condition that is a generalization of the Lipschitz condition. Continuous dependence of solutions with respect to data of the equation is also presented. We consider equations driven by semimartingale Z and equations driven by processes A;M from decomposition of Z, where A is a process of finite variation and M is a local martingale. These equations are not equivalent. Finally, we show that the analysis of the set-valued stochastic integral equations can be extended to a case of fuzzy stochastic integral equations driven by semimartingales under Osgood type condition. To obtain our results we use the set-valued and fuzzy Maruyama type approximations and Bihari’s inequality.

  19. Alternate Solution to Generalized Bernoulli Equations via an Integrating Factor: An Exact Differential Equation Approach

    Science.gov (United States)

    Tisdell, C. C.

    2017-01-01

    Solution methods to exact differential equations via integrating factors have a rich history dating back to Euler (1740) and the ideas enjoy applications to thermodynamics and electromagnetism. Recently, Azevedo and Valentino presented an analysis of the generalized Bernoulli equation, constructing a general solution by linearizing the problem…

  20. ICM: an Integrated Compartment Method for numerically solving partial differential equations

    Energy Technology Data Exchange (ETDEWEB)

    Yeh, G.T.

    1981-05-01

    An integrated compartment method (ICM) is proposed to construct a set of algebraic equations from a system of partial differential equations. The ICM combines the utility of integral formulation of finite element approach, the simplicity of interpolation of finite difference approximation, and the flexibility of compartment analyses. The integral formulation eases the treatment of boundary conditions, in particular, the Neumann-type boundary conditions. The simplicity of interpolation provides great economy in computation. The flexibility of discretization with irregular compartments of various shapes and sizes offers advantages in resolving complex boundaries enclosing compound regions of interest. The basic procedures of ICM are first to discretize the region of interest into compartments, then to apply three integral theorems of vectors to transform the volume integral to the surface integral, and finally to use interpolation to relate the interfacial values in terms of compartment values to close the system. The Navier-Stokes equations are used as an example of how to derive the corresponding ICM alogrithm for a given set of partial differential equations. Because of the structure of the algorithm, the basic computer program remains the same for cases in one-, two-, or three-dimensional problems.

  1. Multi-symplectic variational integrators for nonlinear Schrödinger equations with variable coefficients

    International Nuclear Information System (INIS)

    Liao Cui-Cui; Cui Jin-Chao; Liang Jiu-Zhen; Ding Xiao-Hua

    2016-01-01

    In this paper, we propose a variational integrator for nonlinear Schrödinger equations with variable coefficients. It is shown that our variational integrator is naturally multi-symplectic. The discrete multi-symplectic structure of the integrator is presented by a multi-symplectic form formula that can be derived from the discrete Lagrangian boundary function. As two examples of nonlinear Schrödinger equations with variable coefficients, cubic nonlinear Schrödinger equations and Gross–Pitaevskii equations are extensively studied by the proposed integrator. Our numerical simulations demonstrate that the integrator is capable of preserving the mass, momentum, and energy conservation during time evolutions. Convergence tests are presented to verify that our integrator has second-order accuracy both in time and space. (paper)

  2. Localization of the eigenvalues of linear integral equations with applications to linear ordinary differential equations.

    Science.gov (United States)

    Sloss, J. M.; Kranzler, S. K.

    1972-01-01

    The equivalence of a considered integral equation form with an infinite system of linear equations is proved, and the localization of the eigenvalues of the infinite system is expressed. Error estimates are derived, and the problems of finding upper bounds and lower bounds for the eigenvalues are solved simultaneously.

  3. Integral equation approach to time-dependent kinematic dynamos in finite domains

    International Nuclear Information System (INIS)

    Xu Mingtian; Stefani, Frank; Gerbeth, Gunter

    2004-01-01

    The homogeneous dynamo effect is at the root of cosmic magnetic field generation. With only a very few exceptions, the numerical treatment of homogeneous dynamos is carried out in the framework of the differential equation approach. The present paper tries to facilitate the use of integral equations in dynamo research. Apart from the pedagogical value to illustrate dynamo action within the well-known picture of the Biot-Savart law, the integral equation approach has a number of practical advantages. The first advantage is its proven numerical robustness and stability. The second and perhaps most important advantage is its applicability to dynamos in arbitrary geometries. The third advantage is its intimate connection to inverse problems relevant not only for dynamos but also for technical applications of magnetohydrodynamics. The paper provides the first general formulation and application of the integral equation approach to time-dependent kinematic dynamos, with stationary dynamo sources, in finite domains. The time dependence is restricted to the magnetic field, whereas the velocity or corresponding mean-field sources of dynamo action are supposed to be stationary. For the spherically symmetric α 2 dynamo model it is shown how the general formulation is reduced to a coupled system of two radial integral equations for the defining scalars of the poloidal and toroidal field components. The integral equation formulation for spherical dynamos with general stationary velocity fields is also derived. Two numerical examples - the α 2 dynamo model with radially varying α and the Bullard-Gellman model - illustrate the equivalence of the approach with the usual differential equation method. The main advantage of the method is exemplified by the treatment of an α 2 dynamo in rectangular domains

  4. Green function of the double-fractional Fokker-Planck equation: Path integral and stochastic differential equations

    Science.gov (United States)

    Kleinert, H.; Zatloukal, V.

    2013-11-01

    The statistics of rare events, the so-called black-swan events, is governed by non-Gaussian distributions with heavy power-like tails. We calculate the Green functions of the associated Fokker-Planck equations and solve the related stochastic differential equations. We also discuss the subject in the framework of path integration.

  5. Integrability and structural stability of solutions to the Ginzburg-Landau equation

    Science.gov (United States)

    Keefe, Laurence R.

    1986-01-01

    The integrability of the Ginzburg-Landau equation is studied to investigate if the existence of chaotic solutions found numerically could have been predicted a priori. The equation is shown not to possess the Painleveproperty, except for a special case of the coefficients that corresponds to the integrable, nonlinear Schroedinger (NLS) equation. Regarding the Ginzburg-Landau equation as a dissipative perturbation of the NLS, numerical experiments show all but one of a family of two-tori solutions, possessed by the NLS under particular conditions, to disappear under real perturbations to the NLS coefficients of O(10 to the -6th).

  6. One-way spatial integration of hyperbolic equations

    Science.gov (United States)

    Towne, Aaron; Colonius, Tim

    2015-11-01

    In this paper, we develop and demonstrate a method for constructing well-posed one-way approximations of linear hyperbolic systems. We use a semi-discrete approach that allows the method to be applied to a wider class of problems than existing methods based on analytical factorization of idealized dispersion relations. After establishing the existence of an exact one-way equation for systems whose coefficients do not vary along the axis of integration, efficient approximations of the one-way operator are constructed by generalizing techniques previously used to create nonreflecting boundary conditions. When physically justified, the method can be applied to systems with slowly varying coefficients in the direction of integration. To demonstrate the accuracy and computational efficiency of the approach, the method is applied to model problems in acoustics and fluid dynamics via the linearized Euler equations; in particular we consider the scattering of sound waves from a vortex and the evolution of hydrodynamic wavepackets in a spatially evolving jet. The latter problem shows the potential of the method to offer a systematic, convergent alternative to ad hoc regularizations such as the parabolized stability equations.

  7. Integrable discretizations of the (2+1)-dimensional sinh-Gordon equation

    International Nuclear Information System (INIS)

    Hu, Xing-Biao; Yu, Guo-Fu

    2007-01-01

    In this paper, we propose two semi-discrete equations and one fully discrete equation and study them by Hirota's bilinear method. These equations have continuum limits into a system which admits the (2+1)-dimensional generalization of the sinh-Gordon equation. As a result, two integrable semi-discrete versions and one fully discrete version for the sinh-Gordon equation are found. Baecklund transformations, nonlinear superposition formulae, determinant solution and Lax pairs for these discrete versions are presented

  8. A New Algorithm for System of Integral Equations

    Directory of Open Access Journals (Sweden)

    Abdujabar Rasulov

    2014-01-01

    Full Text Available We develop a new algorithm to solve the system of integral equations. In this new method no need to use matrix weights. Beacause of it, we reduce computational complexity considerable. Using the new algorithm it is also possible to solve an initial boundary value problem for system of parabolic equations. To verify the efficiency, the results of computational experiments are given.

  9. On integration of the first order differential equations in a finite terms

    International Nuclear Information System (INIS)

    Malykh, M D

    2017-01-01

    There are several approaches to the description of the concept called briefly as integration of the first order differential equations in a finite terms or symbolical integration. In the report three of them are considered: 1.) finding of a rational integral (Beaune or Poincaré problem), 2.) integration by quadratures and 3.) integration when the general solution of given differential equation is an algebraical function of a constant (Painlevé problem). Their realizations in Sage are presented. (paper)

  10. Polynomial solutions of nonlinear integral equations

    International Nuclear Information System (INIS)

    Dominici, Diego

    2009-01-01

    We analyze the polynomial solutions of a nonlinear integral equation, generalizing the work of Bender and Ben-Naim (2007 J. Phys. A: Math. Theor. 40 F9, 2008 J. Nonlinear Math. Phys. 15 (Suppl. 3) 73). We show that, in some cases, an orthogonal solution exists and we give its general form in terms of kernel polynomials

  11. Polynomial solutions of nonlinear integral equations

    Energy Technology Data Exchange (ETDEWEB)

    Dominici, Diego [Department of Mathematics, State University of New York at New Paltz, 1 Hawk Dr. Suite 9, New Paltz, NY 12561-2443 (United States)], E-mail: dominicd@newpaltz.edu

    2009-05-22

    We analyze the polynomial solutions of a nonlinear integral equation, generalizing the work of Bender and Ben-Naim (2007 J. Phys. A: Math. Theor. 40 F9, 2008 J. Nonlinear Math. Phys. 15 (Suppl. 3) 73). We show that, in some cases, an orthogonal solution exists and we give its general form in terms of kernel polynomials.

  12. Thermodynamically self-consistent integral equations and the structure of liquid metals

    International Nuclear Information System (INIS)

    Pastore, G.; Kahl, G.

    1987-01-01

    We discuss the application of the new thermodynamically self-consistent integral equations for the determination of the structural properties of liquid metals. We present a detailed comparison of the structure (S(q) and g(r)) for models of liquid alkali metals as obtained from two thermodynamically self-consistent integral equations and some published exact computer simulation results; the range of states extends from the triple point to the expanded metal. The theories which only impose thermodynamic self-consistency without any fitting of external data show an excellent agreement with the simulation results, thus demonstrating that this new type of integral equation is definitely superior to the conventional ones (hypernetted chain, Percus-Yevick, mean spherical approximation, etc). (author)

  13. Distribution theory for Schrödinger’s integral equation

    NARCIS (Netherlands)

    Lange, R.J.

    2015-01-01

    Much of the literature on point interactions in quantum mechanics has focused on the differential form of Schrödinger's equation. This paper, in contrast, investigates the integral form of Schrödinger's equation. While both forms are known to be equivalent for smooth potentials, this is not true for

  14. Unconditionally stable integration of Maxwell’s equations

    NARCIS (Netherlands)

    Verwer, J.G.; Botchev, M.A.

    2009-01-01

    Numerical integration of Maxwell’s equations is often based on explicit methods accepting a stability step size restriction. In literature evidence is given that there is also a need for unconditionally stable methods, as exemplified by the successful alternating direction implicit - finite

  15. Bound states and scattering in four-body systems

    International Nuclear Information System (INIS)

    Narodetsky, I.M.

    1979-01-01

    It is the purpose of this review to provide the clear and elementary introduction in the integral equation method and to demonstrate explicitely its usefulness for the physical applications. The existing results concerning the application of the integral equation technique for the four-nucleon bound states and scattering are reviewed.The treatment is based on the quasiparticle approach that permits the simple interpretation of the equations in terms of quasiparticle scattering. The mathematical basis for the quasiparticle approach is the Hilbert-Schmidt theorem of the Fredholm integral equation theory. This paper contains the detailed discussion of the Hilbert-Schmidt expansion as applied to the 2-particle amplitudes and to the 3 + 1 and 2 + 2 amplitudes which are the kernels of the four-body equations. The review contains essentially the discussion of the four-body quasiparticle equations and results obtained for bound states and scattering

  16. Multi-group neutron transport theory

    International Nuclear Information System (INIS)

    Zelazny, R.; Kuszell, A.

    1962-01-01

    Multi-group neutron transport theory. In the paper the general theory of the application of the K. M. Case method to N-group neutron transport theory in plane geometry is given. The eigenfunctions (distributions) for the system of Boltzmann equations have been derived and the completeness theorem has been proved. By means of general solution two examples important for reactor and shielding calculations are given: the solution of a critical and albedo problem for a slab. In both cases the system of singular integral equations for expansion coefficients into a full set of eigenfunction distributions has been reduced to the system of Fredholm-type integral equations. Some results can be applied also to some spherical problems. (author) [fr

  17. A new integral method for solving the point reactor neutron kinetics equations

    International Nuclear Information System (INIS)

    Li Haofeng; Chen Wenzhen; Luo Lei; Zhu Qian

    2009-01-01

    A numerical integral method that efficiently provides the solution of the point kinetics equations by using the better basis function (BBF) for the approximation of the neutron density in one time step integrations is described and investigated. The approach is based on an exact analytic integration of the neutron density equation, where the stiffness of the equations is overcome by the fully implicit formulation. The procedure is tested by using a variety of reactivity functions, including step reactivity insertion, ramp input and oscillatory reactivity changes. The solution of the better basis function method is compared to other analytical and numerical solutions of the point reactor kinetics equations. The results show that selecting a better basis function can improve the efficiency and accuracy of this integral method. The better basis function method can be used in real time forecasting for power reactors in order to prevent reactivity accidents.

  18. Lagrangian structures, integrability and chaos for 3D dynamical equations

    International Nuclear Information System (INIS)

    Bustamante, Miguel D; Hojman, Sergio A

    2003-01-01

    In this paper, we consider the general setting for constructing action principles for three-dimensional first-order autonomous equations. We present the results for some integrable and non-integrable cases of the Lotka-Volterra equation, and show Lagrangian descriptions which are valid for systems satisfying Shil'nikov criteria on the existence of strange attractors, though chaotic behaviour has not been verified up to now. The Euler-Lagrange equations we get for these systems usually present 'time reparametrization' invariance, though other kinds of invariance may be found according to the kernel of the associated symplectic 2-form. The formulation of a Hamiltonian structure (Poisson brackets and Hamiltonians) for these systems from the Lagrangian viewpoint leads to a method of finding new constants of the motion starting from known ones, which is applied to some systems found in the literature known to possess a constant of the motion, to find the other and thus showing their integrability. In particular, we show that the so-called ABC system is completely integrable if it possesses one constant of the motion

  19. Universal and integrable nonlinear evolution systems of equations in 2+1 dimensions

    International Nuclear Information System (INIS)

    Maccari, A.

    1997-01-01

    Integrable systems of nonlinear partial differential equations (PDEs) are obtained from integrable equations in 2+1 dimensions, by means of a reduction method of broad applicability based on Fourier expansion and spatio endash temporal rescalings, which is asymptotically exact in the limit of weak nonlinearity. The integrability by the spectral transform is explicitly demonstrated, because the corresponding Lax pairs have been derived, applying the same reduction method to the Lax pair of the initial equation. These systems of nonlinear PDEs are likely to be of applicative relevance and have a open-quotes universalclose quotes character, inasmuch as they may be derived from a very large class of nonlinear evolution equations with a linear dispersive part. copyright 1997 American Institute of Physics

  20. Integration Processes of Delay Differential Equation Based on Modified Laguerre Functions

    Directory of Open Access Journals (Sweden)

    Yeguo Sun

    2012-01-01

    Full Text Available We propose long-time convergent numerical integration processes for delay differential equations. We first construct an integration process based on modified Laguerre functions. Then we establish its global convergence in certain weighted Sobolev space. The proposed numerical integration processes can also be used for systems of delay differential equations. We also developed a technique for refinement of modified Laguerre-Radau interpolations. Lastly, numerical results demonstrate the spectral accuracy of the proposed method and coincide well with analysis.

  1. Stability of negative solitary waves for an integrable modified Camassa-Holm equation

    International Nuclear Information System (INIS)

    Yin Jiuli; Tian Lixin; Fan Xinghua

    2010-01-01

    In this paper, we prove that the modified Camassa-Holm equation is Painleve integrable. We also study the orbital stability problem of negative solitary waves for this integrable equation. It is shown that the negative solitary waves are stable for arbitrary wave speed of propagation.

  2. Fibonacci-regularization method for solving Cauchy integral equations of the first kind

    Directory of Open Access Journals (Sweden)

    Mohammad Ali Fariborzi Araghi

    2017-09-01

    Full Text Available In this paper, a novel scheme is proposed to solve the first kind Cauchy integral equation over a finite interval. For this purpose, the regularization method is considered. Then, the collocation method with Fibonacci base function is applied to solve the obtained second kind singular integral equation. Also, the error estimate of the proposed scheme is discussed. Finally, some sample Cauchy integral equations stem from the theory of airfoils in fluid mechanics are presented and solved to illustrate the importance and applicability of the given algorithm. The tables in the examples show the efficiency of the method.

  3. Field Method for Integrating the First Order Differential Equation

    Institute of Scientific and Technical Information of China (English)

    JIA Li-qun; ZHENG Shi-wang; ZHANG Yao-yu

    2007-01-01

    An important modern method in analytical mechanics for finding the integral, which is called the field-method, is used to research the solution of a differential equation of the first order. First, by introducing an intermediate variable, a more complicated differential equation of the first order can be expressed by two simple differential equations of the first order, then the field-method in analytical mechanics is introduced for solving the two differential equations of the first order. The conclusion shows that the field-method in analytical mechanics can be fully used to find the solutions of a differential equation of the first order, thus a new method for finding the solutions of the first order is provided.

  4. On an integrable deformed affinsphären equation. A reciprocal gasdynamic connection

    International Nuclear Information System (INIS)

    Rogers, C.; Huang, Yehui

    2012-01-01

    The integrable affinsphären equation originally arose in a geometric context but has an interesting gasdynamic connection. Here, an integrable deformed version of the affinsphären equation is derived in a novel manner via the action of reciprocal transformations on a related anisentropic gasdynamics system. A linear representation for the deformed affinsphären equation is constructed by means of the reciprocal transformations. The latter are then employed to derive a class of exact solutions in parametric form. -- Highlights: ► A deformed affinsphären equation is derived via a reciprocal transformation. ► A linear representation for the deformed affinsphären equation is constructed. ► A class of exact solutions of the deformed affinsphären equation is presented.

  5. Analysis and regularization of the thin-wire integral equation with reduced kernel

    NARCIS (Netherlands)

    Beurden, van M.C.; Tijhuis, A.G.

    2007-01-01

    For the straight wire, modeled as a hollow tube, we establish a conditional equivalence relation between the integral equations with exact and reduced kernel. This relation allows us to examine the existence and uniqueness conditions for the integral equation with reduced kernel, based on a local

  6. Derivation of the Schrodinger Equation from the Hamilton-Jacobi Equation in Feynman's Path Integral Formulation of Quantum Mechanics

    Science.gov (United States)

    Field, J. H.

    2011-01-01

    It is shown how the time-dependent Schrodinger equation may be simply derived from the dynamical postulate of Feynman's path integral formulation of quantum mechanics and the Hamilton-Jacobi equation of classical mechanics. Schrodinger's own published derivations of quantum wave equations, the first of which was also based on the Hamilton-Jacobi…

  7. Numerical Simulation of Antennas with Improved Integral Equation Method

    International Nuclear Information System (INIS)

    Ma Ji; Fang Guang-You; Lu Wei

    2015-01-01

    Simulating antennas around a conducting object is a challenge task in computational electromagnetism, which is concerned with the behaviour of electromagnetic fields. To analyze this model efficiently, an improved integral equation-fast Fourier transform (IE-FFT) algorithm is presented in this paper. The proposed scheme employs two Cartesian grids with different size and location to enclose the antenna and the other object, respectively. On the one hand, IE-FFT technique is used to store matrix in a sparse form and accelerate the matrix-vector multiplication for each sub-domain independently. On the other hand, the mutual interaction between sub-domains is taken as the additional exciting voltage in each matrix equation. By updating integral equations several times, the whole electromagnetic system can achieve a stable status. Finally, the validity of the presented method is verified through the analysis of typical antennas in the presence of a conducting object. (paper)

  8. Dhage Iteration Method for Generalized Quadratic Functional Integral Equations

    Directory of Open Access Journals (Sweden)

    Bapurao C. Dhage

    2015-01-01

    Full Text Available In this paper we prove the existence as well as approximations of the solutions for a certain nonlinear generalized quadratic functional integral equation. An algorithm for the solutions is developed and it is shown that the sequence of successive approximations starting at a lower or upper solution converges monotonically to the solutions of related quadratic functional integral equation under some suitable mixed hybrid conditions. We rely our main result on Dhage iteration method embodied in a recent hybrid fixed point theorem of Dhage (2014 in partially ordered normed linear spaces. An example is also provided to illustrate the abstract theory developed in the paper.

  9. TBA-like integral equations from quantized mirror curves

    Energy Technology Data Exchange (ETDEWEB)

    Okuyama, Kazumi [Department of Physics, Shinshu University,Matsumoto 390-8621 (Japan); Zakany, Szabolcs [Département de Physique Théorique, Université de Genève,Genève, CH-1211 (Switzerland)

    2016-03-15

    Quantizing the mirror curve of certain toric Calabi-Yau (CY) three-folds leads to a family of trace class operators. The resolvent function of these operators is known to encode topological data of the CY. In this paper, we show that in certain cases, this resolvent function satisfies a system of non-linear integral equations whose structure is very similar to the Thermodynamic Bethe Ansatz (TBA) systems. This can be used to compute spectral traces, both exactly and as a semiclassical expansion. As a main example, we consider the system related to the quantized mirror curve of local ℙ{sup 2}. According to a recent proposal, the traces of this operator are determined by the refined BPS indices of the underlying CY. We use our non-linear integral equations to test that proposal.

  10. TBA-like integral equations from quantized mirror curves

    Science.gov (United States)

    Okuyama, Kazumi; Zakany, Szabolcs

    2016-03-01

    Quantizing the mirror curve of certain toric Calabi-Yau (CY) three-folds leads to a family of trace class operators. The resolvent function of these operators is known to encode topological data of the CY. In this paper, we show that in certain cases, this resolvent function satisfies a system of non-linear integral equations whose structure is very similar to the Thermodynamic Bethe Ansatz (TBA) systems. This can be used to compute spectral traces, both exactly and as a semiclassical expansion. As a main example, we consider the system related to the quantized mirror curve of local P2. According to a recent proposal, the traces of this operator are determined by the refined BPS indices of the underlying CY. We use our non-linear integral equations to test that proposal.

  11. Solving Abel’s Type Integral Equation with Mikusinski’s Operator of Fractional Order

    Directory of Open Access Journals (Sweden)

    Ming Li

    2013-01-01

    Full Text Available This paper gives a novel explanation of the integral equation of Abel’s type from the point of view of Mikusinski’s operational calculus. The concept of the inverse of Mikusinski’s operator of fractional order is introduced for constructing a representation of the solution to the integral equation of Abel’s type. The proof of the existence of the inverse of the fractional Mikusinski operator is presented, providing an alternative method of treating the integral equation of Abel’s type.

  12. Integrability of a system of two nonlinear Schroedinger equations

    International Nuclear Information System (INIS)

    Zhukhunashvili, V.Z.

    1989-01-01

    In recent years the inverse scattering method has achieved significant successes in the integration of nonlinear models that arise in different branches of physics. However, its region of applicability is still restricted, i.e., not all nonlinear models can be integrated. In view of the great mathematical difficulties that arise in integration, it is clearly worth testing a model for integrability before turning to integration. Such a possibility is provided by the Zakharov-Schulman method. The question of the integrability of a system of two nonlinear Schroedinger equations is resolved. It is shown that the previously known cases exhaust all integrable variants

  13. On integrability conditions of the equations of nonsymmetrical chiral field on SO(4)

    International Nuclear Information System (INIS)

    Tskhakaya, D.D.

    1990-01-01

    Possibility of integrating the equations of nonsymmetrical chiral field on SO(4) by means of the inverse scattering method is investigated. Maximal number of the motion integrals is found for the corresponding system of ordinary differential equations

  14. Solving differential equations for Feynman integrals by expansions near singular points

    Science.gov (United States)

    Lee, Roman N.; Smirnov, Alexander V.; Smirnov, Vladimir A.

    2018-03-01

    We describe a strategy to solve differential equations for Feynman integrals by powers series expansions near singular points and to obtain high precision results for the corresponding master integrals. We consider Feynman integrals with two scales, i.e. non-trivially depending on one variable. The corresponding algorithm is oriented at situations where canonical form of the differential equations is impossible. We provide a computer code constructed with the help of our algorithm for a simple example of four-loop generalized sunset integrals with three equal non-zero masses and two zero masses. Our code gives values of the master integrals at any given point on the real axis with a required accuracy and a given order of expansion in the regularization parameter ɛ.

  15. Constructing New Discrete Integrable Coupling System for Soliton Equation by Kronecker Product

    International Nuclear Information System (INIS)

    Yu Fajun; Zhang Hongqing

    2008-01-01

    It is shown that the Kronecker product can be applied to constructing new discrete integrable coupling system of soliton equation hierarchy in this paper. A direct application to the fractional cubic Volterra lattice spectral problem leads to a novel integrable coupling system of soliton equation hierarchy. It is also indicated that the study of discrete integrable couplings by using the Kronecker product is an efficient and straightforward method. This method can be used generally

  16. CALL FOR PAPERS: Special issue on Symmetries and Integrability of Difference Equations

    Science.gov (United States)

    Doliwa, Adam; Korhonen, Risto; Lafortune, Stephane

    2006-10-01

    This is a call for contributions to a special issue of Journal of Physics A: Mathematical and General entitled `Special issue on Symmetries and Integrability of Difference Equations' as featured at the SIDE VII meeting held during July 2006 in Melbourne (http://web.maths.unsw.edu.au/%7Eschief/side/side.html). Participants at that meeting, as well as other researchers working in the field of difference equations and discrete systems, are invited to submit a research paper to this issue. This meeting was the seventh of a series of biennial meetings devoted to the study of integrable difference equations and related topics. The notion of integrability was first introduced in the 19th century in the context of classical mechanics with the definition of Liouville integrability for Hamiltonian flows. Since then, several notions of integrability have been introduced for partial and ordinary differential equations. Closely related to integrability theory is the symmetry analysis of nonlinear evolution equations. Symmetry analysis takes advantage of the Lie group structure of a given equation to study its properties. Together, integrability theory and symmetry analysis provide the main method by which nonlinear evolution equations can be solved explicitly. Difference equations, just as differential equations, are important in numerous fields of science and have a wide variety of applications in such areas as: mathematical physics, computer visualization, numerical analysis, mathematical biology, economics, combinatorics, quantum field theory, etc. It is thus crucial to develop tools to study and solve difference equations. While the theory of symmetry and integrability for differential equations is now well-established, this is not yet the case for discrete equations. The situation has undergone impressive development in recent years and has affected a broad range of fields, including the theory of special functions, quantum integrable systems, numerical analysis, cellular

  17. Recursive integral equations with positive kernel for lattice calculations

    International Nuclear Information System (INIS)

    Illuminati, F.; Isopi, M.

    1990-11-01

    A Kirkwood-Salzburg integral equation, with positive defined kernel, for the states of lattice models of statistical mechanics and quantum field theory is derived. The equation is defined in the thermodynamic limit, and its iterative solution is convergent. Moreover, positivity leads to an exact a priori bound on the iteration. The equation's relevance as a reliable algorithm for lattice calculations is therefore suggested, and it is illustrated with a simple application. It should provide a viable alternative to Monte Carlo methods for models of statistical mechanics and lattice gauge theories. 10 refs

  18. Poisson's theorem and integrals of KdV equation

    International Nuclear Information System (INIS)

    Tasso, H.

    1978-01-01

    Using Poisson's theorem it is proved that if F = integral sub(-infinity)sup(+infinity) T(u,usub(x),...usub(n,t))dx is an invariant functional of KdV equation, then integral sub(-infinity)sup(+infinity) delta F/delta u dx integral sub(-infinity)sup(+infinity) delta T/delta u dx is also an invariant functional. In the case of a polynomial T, one finds in a simple way the known recursion ΔTr/Δu = Tsub(r-1). This note gives an example of the usefulness of Poisson's theorem. (author)

  19. Lax Pairs for Discrete Integrable Equations via Darboux Transformations

    International Nuclear Information System (INIS)

    Cao Ce-Wen; Zhang Guang-Yao

    2012-01-01

    A method is developed to construct discrete Lax pairs using Darboux transformations. More kinds of Lax pairs are found for some newly appeared discrete integrable equations, including the H1, the special H3 and the Q1 models in the Adler—Bobenko—Suris list and the closely related discrete and semi-discrete pKdV, pMKdV, SG and Liouville equations. (general)

  20. Non-integrability of time-dependent spherically symmetric Yang-Mills equations

    Energy Technology Data Exchange (ETDEWEB)

    Matinyan, S G; Prokhorenko, E B; Savvidy, G K

    1988-03-07

    The integrability of time-dependent spherically symmetric Yang-Mills equations is studied using the Fermi-Pasta-Ulam method. It is shown that the motion of this system is ergodic, while the system itself is non-integrable, i.e. manifests dynamical chaos.

  1. Method to improve the evaluation of a combination track-etch dosimeter/spectrometer

    International Nuclear Information System (INIS)

    Brackenbush, L.W.; Parkhurst, M.A.; Hadlock, D.E.; Faust, L.G.

    1983-09-01

    A paper is summarized which describes a method of determining the neutron energy spectrum through spectrum unfolding techniques to more accurately assess the dose equivalent from track-etch dosimeters. A mathematical technique is described which can be used in conjunction with the neutron detectors to more accurately estimate neutron dose equivalent. The technique is based upon solutions to a system of Fredholm integral equations of the first type

  2. Heat conduction problem of an evaporating liquid wedge

    Directory of Open Access Journals (Sweden)

    Tomas Barta

    2015-02-01

    Full Text Available We consider the stationary heat transfer near the contact line of an evaporating liquid wedge surrounded by the atmosphere of its pure vapor. In a simplified setting, the problem reduces to the Laplace equation in a half circle, subject to a non-homogeneous and singular boundary condition. By classical tools (conformal mapping, Green's function, we reformulate the problem as an integral equation for the unknown Neumann boundary condition in the setting of appropriate fractional Sobolev and weighted space. The unique solvability is then obtained by means of the Fredholm theorem.

  3. Mesh adaptation technique for Fourier-domain fluorescence lifetime imaging

    International Nuclear Information System (INIS)

    Soloviev, Vadim Y.

    2006-01-01

    A novel adaptive mesh technique in the Fourier domain is introduced for problems in fluorescence lifetime imaging. A dynamical adaptation of the three-dimensional scheme based on the finite volume formulation reduces computational time and balances the ill-posed nature of the inverse problem. Light propagation in the medium is modeled by the telegraph equation, while the lifetime reconstruction algorithm is derived from the Fredholm integral equation of the first kind. Stability and computational efficiency of the method are demonstrated by image reconstruction of two spherical fluorescent objects embedded in a tissue phantom

  4. A New time Integration Scheme for Cahn-hilliard Equations

    KAUST Repository

    Schaefer, R.

    2015-06-01

    In this paper we present a new integration scheme that can be applied to solving difficult non-stationary non-linear problems. It is obtained by a successive linearization of the Crank- Nicolson scheme, that is unconditionally stable, but requires solving non-linear equation at each time step. We applied our linearized scheme for the time integration of the challenging Cahn-Hilliard equation, modeling the phase separation in fluids. At each time step the resulting variational equation is solved using higher-order isogeometric finite element method, with B- spline basis functions. The method was implemented in the PETIGA framework interfaced via the PETSc toolkit. The GMRES iterative solver was utilized for the solution of a resulting linear system at every time step. We also apply a simple adaptivity rule, which increases the time step size when the number of GMRES iterations is lower than 30. We compared our method with a non-linear, two stage predictor-multicorrector scheme, utilizing a sophisticated step length adaptivity. We controlled the stability of our simulations by monitoring the Ginzburg-Landau free energy functional. The proposed integration scheme outperforms the two-stage competitor in terms of the execution time, at the same time having a similar evolution of the free energy functional.

  5. A New time Integration Scheme for Cahn-hilliard Equations

    KAUST Repository

    Schaefer, R.; Smol-ka, M.; Dalcin, L; Paszyn'ski, M.

    2015-01-01

    In this paper we present a new integration scheme that can be applied to solving difficult non-stationary non-linear problems. It is obtained by a successive linearization of the Crank- Nicolson scheme, that is unconditionally stable, but requires solving non-linear equation at each time step. We applied our linearized scheme for the time integration of the challenging Cahn-Hilliard equation, modeling the phase separation in fluids. At each time step the resulting variational equation is solved using higher-order isogeometric finite element method, with B- spline basis functions. The method was implemented in the PETIGA framework interfaced via the PETSc toolkit. The GMRES iterative solver was utilized for the solution of a resulting linear system at every time step. We also apply a simple adaptivity rule, which increases the time step size when the number of GMRES iterations is lower than 30. We compared our method with a non-linear, two stage predictor-multicorrector scheme, utilizing a sophisticated step length adaptivity. We controlled the stability of our simulations by monitoring the Ginzburg-Landau free energy functional. The proposed integration scheme outperforms the two-stage competitor in terms of the execution time, at the same time having a similar evolution of the free energy functional.

  6. Any order approximate analytical solution of the nonlinear Volterra's integral equation for accelerator dynamic systems

    International Nuclear Information System (INIS)

    Liu Chunliang; Xie Xi; Chen Yinbao

    1991-01-01

    The universal nonlinear dynamic system equation is equivalent to its nonlinear Volterra's integral equation, and any order approximate analytical solution of the nonlinear Volterra's integral equation is obtained by exact analytical method, thus giving another derivation procedure as well as another computation algorithm for the solution of the universal nonlinear dynamic system equation

  7. Integrability of the one dimensional Schrödinger equation

    Science.gov (United States)

    Combot, Thierry

    2018-02-01

    We present a definition of integrability for the one-dimensional Schrödinger equation, which encompasses all known integrable systems, i.e., systems for which the spectrum can be explicitly computed. For this, we introduce the class of rigid functions, built as Liouvillian functions, but containing all solutions of rigid differential operators in the sense of Katz, and a notion of natural of boundary conditions. We then make a complete classification of rational integrable potentials. Many new integrable cases are found, some of them physically interesting.

  8. On the use of the Lie group technique for differential equations with a small parameter: Approximate solutions and integrable equations

    International Nuclear Information System (INIS)

    Burde, G.I.

    2002-01-01

    A new approach to the use of the Lie group technique for partial and ordinary differential equations dependent on a small parameter is developed. In addition to determining approximate solutions to the perturbed equation, the approach allows constructing integrable equations that have solutions with (partially) prescribed features. Examples of application of the approach to partial differential equations are given

  9. Existence Results for Some Nonlinear Functional-Integral Equations in Banach Algebra with Applications

    Directory of Open Access Journals (Sweden)

    Lakshmi Narayan Mishra

    2016-04-01

    Full Text Available In the present manuscript, we prove some results concerning the existence of solutions for some nonlinear functional-integral equations which contains various integral and functional equations that considered in nonlinear analysis and its applications. By utilizing the techniques of noncompactness measures, we operate the fixed point theorems such as Darbo's theorem in Banach algebra concerning the estimate on the solutions. The results obtained in this paper extend and improve essentially some known results in the recent literature. We also provide an example of nonlinear functional-integral equation to show the ability of our main result.

  10. The multidensity integral equation approach in the theory of complex liquids

    International Nuclear Information System (INIS)

    Holovko, M.F.

    2001-01-01

    Recent development of the multi-density integral equation approach and its application to the statistical mechanical modelling of a different type of association and clusterization in liquids and solutions are reviewed. The effects of dimerization, polymerization and network formation are discussed. The numerical and analytical solutions of the integral equations in the multi-density formalism for pair correlation functions are used for the description of structural and thermodynamical properties of ionic solutions, polymers and network forming fluids

  11. The ATOMFT integrator - Using Taylor series to solve ordinary differential equations

    Science.gov (United States)

    Berryman, Kenneth W.; Stanford, Richard H.; Breckheimer, Peter J.

    1988-01-01

    This paper discusses the application of ATOMFT, an integration package based on Taylor series solution with a sophisticated user interface. ATOMFT has the capabilities to allow the implementation of user defined functions and the solution of stiff and algebraic equations. Detailed examples, including the solutions to several astrodynamics problems, are presented. Comparisons with its predecessor ATOMCC and other modern integrators indicate that ATOMFT is a fast, accurate, and easy method to use to solve many differential equation problems.

  12. The Neumann Type Systems and Algebro-Geometric Solutions of a System of Coupled Integrable Equations

    International Nuclear Information System (INIS)

    Chen Jinbing; Qiao Zhijun

    2011-01-01

    A system of (1+1)-dimensional coupled integrable equations is decomposed into a pair of new Neumann type systems that separate the spatial and temporal variables for this system over a symplectic submanifold. Then, the Neumann type flows associated with the coupled integrable equations are integrated on the complex tour of a Riemann surface. Finally, the algebro-geometric solutions expressed by Riemann theta functions of the system of coupled integrable equations are obtained by means of the Jacobi inversion.

  13. High-precision numerical integration of equations in dynamics

    Science.gov (United States)

    Alesova, I. M.; Babadzanjanz, L. K.; Pototskaya, I. Yu.; Pupysheva, Yu. Yu.; Saakyan, A. T.

    2018-05-01

    An important requirement for the process of solving differential equations in Dynamics, such as the equations of the motion of celestial bodies and, in particular, the motion of cosmic robotic systems is high accuracy at large time intervals. One of effective tools for obtaining such solutions is the Taylor series method. In this connection, we note that it is very advantageous to reduce the given equations of Dynamics to systems with polynomial (in unknowns) right-hand sides. This allows us to obtain effective algorithms for finding the Taylor coefficients, a priori error estimates at each step of integration, and an optimal choice of the order of the approximation used. In the paper, these questions are discussed and appropriate algorithms are considered.

  14. A predictor-corrector scheme for solving the Volterra integral equation

    KAUST Repository

    Al Jarro, Ahmed

    2011-08-01

    The occurrence of late time instabilities is a common problem of almost all time marching methods developed for solving time domain integral equations. Implicit marching algorithms are now considered stable with various efforts that have been developed for removing low and high frequency instabilities. On the other hand, literature on stabilizing explicit schemes, which might be considered more efficient since they do not require a matrix inversion at each time step, is practically non-existent. In this work, a stable but still explicit predictor-corrector scheme is proposed for solving the Volterra integral equation and its efficacy is verified numerically. © 2011 IEEE.

  15. Path integral solution of linear second order partial differential equations I: the general construction

    International Nuclear Information System (INIS)

    LaChapelle, J.

    2004-01-01

    A path integral is presented that solves a general class of linear second order partial differential equations with Dirichlet/Neumann boundary conditions. Elementary kernels are constructed for both Dirichlet and Neumann boundary conditions. The general solution can be specialized to solve elliptic, parabolic, and hyperbolic partial differential equations with boundary conditions. This extends the well-known path integral solution of the Schroedinger/diffusion equation in unbounded space. The construction is based on a framework for functional integration introduced by Cartier/DeWitt-Morette

  16. Numerical Study of Two-Dimensional Volterra Integral Equations by RDTM and Comparison with DTM

    Directory of Open Access Journals (Sweden)

    Reza Abazari

    2013-01-01

    Full Text Available The two-dimensional Volterra integral equations are solved using more recent semianalytic method, the reduced differential transform method (the so-called RDTM, and compared with the differential transform method (DTM. The concepts of DTM and RDTM are briefly explained, and their application to the two-dimensional Volterra integral equations is studied. The results obtained by DTM and RDTM together are compared with exact solution. As an important result, it is depicted that the RDTM results are more accurate in comparison with those obtained by DTM applied to the same Volterra integral equations. The numerical results reveal that the RDTM is very effective, convenient, and quite accurate compared to the other kind of nonlinear integral equations. It is predicted that the RDTM can be found widely applicable in engineering sciences.

  17. An algorithm of computing inhomogeneous differential equations for definite integrals

    OpenAIRE

    Nakayama, Hiromasa; Nishiyama, Kenta

    2010-01-01

    We give an algorithm to compute inhomogeneous differential equations for definite integrals with parameters. The algorithm is based on the integration algorithm for $D$-modules by Oaku. Main tool in the algorithm is the Gr\\"obner basis method in the ring of differential operators.

  18. On Generating Discrete Integrable Systems via Lie Algebras and Commutator Equations

    International Nuclear Information System (INIS)

    Zhang Yu-Feng; Tam, Honwah

    2016-01-01

    In the paper, we introduce the Lie algebras and the commutator equations to rewrite the Tu-d scheme for generating discrete integrable systems regularly. By the approach the various loop algebras of the Lie algebra A_1 are defined so that the well-known Toda hierarchy and a novel discrete integrable system are obtained, respectively. A reduction of the later hierarchy is just right the famous Ablowitz–Ladik hierarchy. Finally, via two different enlarging Lie algebras of the Lie algebra A_1, we derive two resulting differential-difference integrable couplings of the Toda hierarchy, of course, they are all various discrete expanding integrable models of the Toda hierarchy. When the introduced spectral matrices are higher degrees, the way presented in the paper is more convenient to generate discrete integrable equations than the Tu-d scheme by using the software Maple. (paper)

  19. Higher-Order Integral Equation Methods in Computational Electromagnetics

    DEFF Research Database (Denmark)

    Jørgensen, Erik; Meincke, Peter

    Higher-order integral equation methods have been investigated. The study has focused on improving the accuracy and efficiency of the Method of Moments (MoM) applied to electromagnetic problems. A new set of hierarchical Legendre basis functions of arbitrary order is developed. The new basis...

  20. Approximate solutions for the two-dimensional integral transport equation. The critically mixed methods of resolution

    International Nuclear Information System (INIS)

    Sanchez, Richard.

    1980-11-01

    This work is divided into two part the first part (note CEA-N-2165) deals with the solution of complex two-dimensional transport problems, the second one treats the critically mixed methods of resolution. These methods are applied for one-dimensional geometries with highly anisotropic scattering. In order to simplify the set of integral equation provided by the integral transport equation, the integro-differential equation is used to obtain relations that allow to lower the number of integral equation to solve; a general mathematical and numerical study is presented [fr

  1. Stochastic integration and differential equations

    CERN Document Server

    Protter, Philip E

    2003-01-01

    It has been 15 years since the first edition of Stochastic Integration and Differential Equations, A New Approach appeared, and in those years many other texts on the same subject have been published, often with connections to applications, especially mathematical finance. Yet in spite of the apparent simplicity of approach, none of these books has used the functional analytic method of presenting semimartingales and stochastic integration. Thus a 2nd edition seems worthwhile and timely, though it is no longer appropriate to call it "a new approach". The new edition has several significant changes, most prominently the addition of exercises for solution. These are intended to supplement the text, but lemmas needed in a proof are never relegated to the exercises. Many of the exercises have been tested by graduate students at Purdue and Cornell Universities. Chapter 3 has been completely redone, with a new, more intuitive and simultaneously elementary proof of the fundamental Doob-Meyer decomposition theorem, t...

  2. Retarded potentials and time domain boundary integral equations a road map

    CERN Document Server

    Sayas, Francisco-Javier

    2016-01-01

    This book offers a thorough and self-contained exposition of the mathematics of time-domain boundary integral equations associated to the wave equation, including applications to scattering of acoustic and elastic waves. The book offers two different approaches for the analysis of these integral equations, including a systematic treatment of their numerical discretization using Galerkin (Boundary Element) methods in the space variables and Convolution Quadrature in the time variable. The first approach follows classical work started in the late eighties, based on Laplace transforms estimates. This approach has been refined and made more accessible by tailoring the necessary mathematical tools, avoiding an excess of generality. A second approach contains a novel point of view that the author and some of his collaborators have been developing in recent years, using the semigroup theory of evolution equations to obtain improved results. The extension to electromagnetic waves is explained in one of the appendices...

  3. A new model for the collective beam-beam interaction

    Energy Technology Data Exchange (ETDEWEB)

    Ellison, J.A.; Sobol, A.V. [New Mexico Univ., Albuquerque, NM (United States); Vogt, M. [Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany)

    2006-09-15

    The Collective Beam-Beam interaction is studied in the framework of maps with a ''kick-lattice'' model in 4-D phase space. A novel approach to the classical method of averaging is used to derive an approximate map which is equivalent to a flow within the averaging approximation. The flow equation is a continuous-time Vlasov equation which we call the averaged Vlasov equation, the new model of this paper. The power of this approach is evidenced by the fact that the averaged Vlasov equation has exact equilibria and the associated lineralized equations have uncoupled azimuthal Fourier modes. The equation for the Fourier modes leads to a Fredholm integral equation of the third kind and the setting is ready-made for the development of a weakly nonlinear theory to study the coupling of the {pi} and {sigma} modes. The {pi} and {sigma} modes are calculated from the third kind integral equation and results are compared with the kick-lattice model. (orig.)

  4. A new model for the collective beam-beam interaction

    International Nuclear Information System (INIS)

    Ellison, J.A.; Sobol, A.V.; Vogt, M.

    2006-09-01

    The Collective Beam-Beam interaction is studied in the framework of maps with a ''kick-lattice'' model in 4-D phase space. A novel approach to the classical method of averaging is used to derive an approximate map which is equivalent to a flow within the averaging approximation. The flow equation is a continuous-time Vlasov equation which we call the averaged Vlasov equation, the new model of this paper. The power of this approach is evidenced by the fact that the averaged Vlasov equation has exact equilibria and the associated lineralized equations have uncoupled azimuthal Fourier modes. The equation for the Fourier modes leads to a Fredholm integral equation of the third kind and the setting is ready-made for the development of a weakly nonlinear theory to study the coupling of the π and σ modes. The π and σ modes are calculated from the third kind integral equation and results are compared with the kick-lattice model. (orig.)

  5. APPLICATION OF BOUNDARY INTEGRAL EQUATION METHOD FOR THERMOELASTICITY PROBLEMS

    Directory of Open Access Journals (Sweden)

    Vorona Yu.V.

    2015-12-01

    Full Text Available Boundary Integral Equation Method is used for solving analytically the problems of coupled thermoelastic spherical wave propagation. The resulting mathematical expressions coincide with the solutions obtained in a conventional manner.

  6. Canonical algorithms for numerical integration of charged particle motion equations

    Science.gov (United States)

    Efimov, I. N.; Morozov, E. A.; Morozova, A. R.

    2017-02-01

    A technique for numerically integrating the equation of charged particle motion in a magnetic field is considered. It is based on the canonical transformations of the phase space in Hamiltonian mechanics. The canonical transformations make the integration process stable against counting error accumulation. The integration algorithms contain a minimum possible amount of arithmetics and can be used to design accelerators and devices of electron and ion optics.

  7. Integral equations with difference kernels on finite intervals

    CERN Document Server

    Sakhnovich, Lev A

    2015-01-01

    This book focuses on solving integral equations with difference kernels on finite intervals. The corresponding problem on the semiaxis was previously solved by N. Wiener–E. Hopf and by M.G. Krein. The problem on finite intervals, though significantly more difficult, may be solved using our method of operator identities. This method is also actively employed in inverse spectral problems, operator factorization and nonlinear integral equations. Applications of the obtained results to optimal synthesis, light scattering, diffraction, and hydrodynamics problems are discussed in this book, which also describes how the theory of operators with difference kernels is applied to stable processes and used to solve the famous M. Kac problems on stable processes. In this second edition these results are extensively generalized and include the case of all Levy processes. We present the convolution expression for the well-known Ito formula of the generator operator, a convolution expression that has proven to be fruitful...

  8. Integrable discretizations and self-adaptive moving mesh method for a coupled short pulse equation

    International Nuclear Information System (INIS)

    Feng, Bao-Feng; Chen, Junchao; Chen, Yong; Maruno, Ken-ichi; Ohta, Yasuhiro

    2015-01-01

    In the present paper, integrable semi-discrete and fully discrete analogues of a coupled short pulse (CSP) equation are constructed. The key to the construction are the bilinear forms and determinant structure of the solutions of the CSP equation. We also construct N-soliton solutions for the semi-discrete and fully discrete analogues of the CSP equations in the form of Casorati determinants. In the continuous limit, we show that the fully discrete CSP equation converges to the semi-discrete CSP equation, then further to the continuous CSP equation. Moreover, the integrable semi-discretization of the CSP equation is used as a self-adaptive moving mesh method for numerical simulations. The numerical results agree with the analytical results very well. (paper)

  9. Integrability and Poisson Structures of Three Dimensional Dynamical Systems and Equations of Hydrodynamic Type

    Science.gov (United States)

    Gumral, Hasan

    Poisson structure of completely integrable 3 dimensional dynamical systems can be defined in terms of an integrable 1-form. We take advantage of this fact and use the theory of foliations in discussing the geometrical structure underlying complete and partial integrability. We show that the Halphen system can be formulated in terms of a flat SL(2,R)-valued connection and belongs to a non-trivial Godbillon-Vey class. On the other hand, for the Euler top and a special case of 3-species Lotka-Volterra equations which are contained in the Halphen system as limiting cases, this structure degenerates into the form of globally integrable bi-Hamiltonian structures. The globally integrable bi-Hamiltonian case is a linear and the sl_2 structure is a quadratic unfolding of an integrable 1-form in 3 + 1 dimensions. We complete the discussion of the Hamiltonian structure of 2-component equations of hydrodynamic type by presenting the Hamiltonian operators for Euler's equation and a continuum limit of Toda lattice. We present further infinite sequences of conserved quantities for shallow water equations and show that their generalizations by Kodama admit bi-Hamiltonian structure. We present a simple way of constructing the second Hamiltonian operators for N-component equations admitting some scaling properties. The Kodama reduction of the dispersionless-Boussinesq equations and the Lax reduction of the Benney moment equations are shown to be equivalent by a symmetry transformation. They can be cast into the form of a triplet of conservation laws which enable us to recognize a non-trivial scaling symmetry. The resulting bi-Hamiltonian structure generates three infinite sequences of conserved densities.

  10. Discrete integrable couplings associated with Toda-type lattice and two hierarchies of discrete soliton equations

    International Nuclear Information System (INIS)

    Zhang Yufeng; Fan Engui; Zhang Yongqing

    2006-01-01

    With the help of two semi-direct sum Lie algebras, an efficient way to construct discrete integrable couplings is proposed. As its applications, the discrete integrable couplings of the Toda-type lattice equations are obtained. The approach can be devoted to establishing other discrete integrable couplings of the discrete lattice integrable hierarchies of evolution equations

  11. Elliptic Euler–Poisson–Darboux equation, critical points and integrable systems

    International Nuclear Information System (INIS)

    Konopelchenko, B G; Ortenzi, G

    2013-01-01

    The structure and properties of families of critical points for classes of functions W(z, z-bar ) obeying the elliptic Euler–Poisson–Darboux equation E(1/2, 1/2) are studied. General variational and differential equations governing the dependence of critical points in variational (deformation) parameters are found. Explicit examples of the corresponding integrable quasi-linear differential systems and hierarchies are presented. There are the extended dispersionless Toda/nonlinear Schrödinger hierarchies, the ‘inverse’ hierarchy and equations associated with the real-analytic Eisenstein series E(β, β-bar ;1/2) among them. The specific bi-Hamiltonian structure of these equations is also discussed. (paper)

  12. Solution of four-nucleon integral equations using the effective UPA

    International Nuclear Information System (INIS)

    Perne, R.; Sandhas, W.

    1978-01-01

    In the three-body case it is standard to either solve the (two-dimensional) Faddeev equations directly, or to reduce them first to one-dimensional equations by means of separable approximation (expansion) of the underlying two-body interactions. The basic four-body operator identities are reduced by the latter treatment to effective three-body equations only. These may be handled like their genuine three-body analoga, i.e., by directly solving them, or by expanding the effective interactions ocurring into separable terms. Such a procedure provides us in a second step with one-dimensional integral equations for the four-body problem, too. (orig./WL) [de

  13. A calderón multiplicative preconditioner for the combined field integral equation

    KAUST Repository

    Bagci, Hakan; Andriulli, Francesco P.; Cools, Kristof; Olyslager, Femke; Michielssen, Eric

    2009-01-01

    A Calderón multiplicative preconditioner (CMP) for the combined field integral equation (CFIE) is developed. Just like with previously proposed Caldern-preconditioned CFIEs, a localization procedure is employed to ensure that the equation

  14. An efficient explicit marching on in time solver for magnetic field volume integral equation

    KAUST Repository

    Sayed, Sadeed Bin; Ulku, H. Arda; Bagci, Hakan

    2015-01-01

    An efficient explicit marching on in time (MOT) scheme for solving the magnetic field volume integral equation is proposed. The MOT system is cast in the form of an ordinary differential equation and is integrated in time using a PE(CE)m multistep

  15. An integral equation-based numerical solver for Taylor states in toroidal geometries

    Science.gov (United States)

    O'Neil, Michael; Cerfon, Antoine J.

    2018-04-01

    We present an algorithm for the numerical calculation of Taylor states in toroidal and toroidal-shell geometries using an analytical framework developed for the solution to the time-harmonic Maxwell equations. Taylor states are a special case of what are known as Beltrami fields, or linear force-free fields. The scheme of this work relies on the generalized Debye source representation of Maxwell fields and an integral representation of Beltrami fields which immediately yields a well-conditioned second-kind integral equation. This integral equation has a unique solution whenever the Beltrami parameter λ is not a member of a discrete, countable set of resonances which physically correspond to spontaneous symmetry breaking. Several numerical examples relevant to magnetohydrodynamic equilibria calculations are provided. Lastly, our approach easily generalizes to arbitrary geometries, both bounded and unbounded, and of varying genus.

  16. Integral equation models for image restoration: high accuracy methods and fast algorithms

    International Nuclear Information System (INIS)

    Lu, Yao; Shen, Lixin; Xu, Yuesheng

    2010-01-01

    Discrete models are consistently used as practical models for image restoration. They are piecewise constant approximations of true physical (continuous) models, and hence, inevitably impose bottleneck model errors. We propose to work directly with continuous models for image restoration aiming at suppressing the model errors caused by the discrete models. A systematic study is conducted in this paper for the continuous out-of-focus image models which can be formulated as an integral equation of the first kind. The resulting integral equation is regularized by the Lavrentiev method and the Tikhonov method. We develop fast multiscale algorithms having high accuracy to solve the regularized integral equations of the second kind. Numerical experiments show that the methods based on the continuous model perform much better than those based on discrete models, in terms of PSNR values and visual quality of the reconstructed images

  17. A boundary integral equation for boundary element applications in multigroup neutron diffusion theory

    International Nuclear Information System (INIS)

    Ozgener, B.

    1998-01-01

    A boundary integral equation (BIE) is developed for the application of the boundary element method to the multigroup neutron diffusion equations. The developed BIE contains no explicit scattering term; the scattering effects are taken into account by redefining the unknowns. Boundary elements of the linear and constant variety are utilised for validation of the developed boundary integral formulation

  18. Introduction to stochastic analysis integrals and differential equations

    CERN Document Server

    Mackevicius, Vigirdas

    2013-01-01

    This is an introduction to stochastic integration and stochastic differential equations written in an understandable way for a wide audience, from students of mathematics to practitioners in biology, chemistry, physics, and finances. The presentation is based on the naïve stochastic integration, rather than on abstract theories of measure and stochastic processes. The proofs are rather simple for practitioners and, at the same time, rather rigorous for mathematicians. Detailed application examples in natural sciences and finance are presented. Much attention is paid to simulation diffusion pro

  19. Local linearization methods for the numerical integration of ordinary differential equations: An overview

    International Nuclear Information System (INIS)

    Jimenez, J.C.

    2009-06-01

    Local Linearization (LL) methods conform a class of one-step explicit integrators for ODEs derived from the following primary and common strategy: the vector field of the differential equation is locally (piecewise) approximated through a first-order Taylor expansion at each time step, thus obtaining successive linear equations that are explicitly integrated. Hereafter, the LL approach may include some additional strategies to improve that basic affine approximation. Theoretical and practical results have shown that the LL integrators have a number of convenient properties. These include arbitrary order of convergence, A-stability, linearization preserving, regularity under quite general conditions, preservation of the dynamics of the exact solution around hyperbolic equilibrium points and periodic orbits, integration of stiff and high-dimensional equations, low computational cost, and others. In this paper, a review of the LL methods and their properties is presented. (author)

  20. A computational method for direct integration of motion equations of structural systems

    International Nuclear Information System (INIS)

    Brusa, L.; Ciacci, R.; Creco, A.; Rossi, F.

    1975-01-01

    The dynamic analysis of structural systems requires the solution of the matrix equations: Md 2 delta/dt(t) + Cddelta/dt(t) + Kdelta(t) = F(t). Many numerical methods are available for direct integration of this equation and their efficiency is due to the fulfillment of the following requirements: A reasonable order of accuracy must be obtained for the approximation of the response relevant to the first modes: the model contributions relevant to the eigenvalues with large real part must be essentially neglected. This paper presents a step-by-step numerical scheme for the integration of this equation which satisfies the requirements previously mentioned. (Auth.)

  1. Discrete integration of continuous Kalman filtering equations for time invariant second-order structural systems

    Science.gov (United States)

    Park, K. C.; Belvin, W. Keith

    1990-01-01

    A general form for the first-order representation of the continuous second-order linear structural-dynamics equations is introduced to derive a corresponding form of first-order continuous Kalman filtering equations. Time integration of the resulting equations is carried out via a set of linear multistep integration formulas. It is shown that a judicious combined selection of computational paths and the undetermined matrices introduced in the general form of the first-order linear structural systems leads to a class of second-order discrete Kalman filtering equations involving only symmetric sparse N x N solution matrices.

  2. The role of the commutator equations in integration methods in tetrad formalisms in general relativity

    International Nuclear Information System (INIS)

    Edgar, S.B.

    1990-01-01

    The structures of the N.P. and G.H.P formalisms are reviewed in order to understand and demonstrate the important role played by the commutator equations in the associated integration procedures. Particular attention is focused on how the commutator equations are to be satisfied, or checked for consistency. It is shown that Held's integration method will only guarantee genuine solutions of Einstein's equations when all the commutator equations are correctly and completely satisfied. (authors)

  3. Scattering integral equations and four nucleon problem. Four nucleon bound states and scattering

    International Nuclear Information System (INIS)

    Narodetskij, I.M.

    1981-01-01

    Existing results from the application of integral equation technique four-nucleon bound states and scattering are reviewed. The purpose of this review is to provide a clear and elementary introduction in the integral equation method and to demonstrate its usefulness in physical applications. Developments in the actual numerical solutions of Faddeev-Yakubovsky type equations are such that a detailed comparison can be made with experiment. Bound state calculations indicate that a nonrelativistic description with pairwise nuclear forces does not suffice and additional degrees of freedom are noted [ru

  4. Aitken extrapolation and epsilon algorithm for an accelerated solution of weakly singular nonlinear Volterra integral equations

    International Nuclear Information System (INIS)

    Mesgarani, H; Parmour, P; Aghazadeh, N

    2010-01-01

    In this paper, we apply Aitken extrapolation and epsilon algorithm as acceleration technique for the solution of a weakly singular nonlinear Volterra integral equation of the second kind. In this paper, based on Tao and Yong (2006 J. Math. Anal. Appl. 324 225-37.) the integral equation is solved by Navot's quadrature formula. Also, Tao and Yong (2006) for the first time applied Richardson extrapolation to accelerating convergence for the weakly singular nonlinear Volterra integral equations of the second kind. To our knowledge, this paper may be the first attempt to apply Aitken extrapolation and epsilon algorithm for the weakly singular nonlinear Volterra integral equations of the second kind.

  5. Low-frequency scaling of the standard and mixed magnetic field and Müller integral equations

    KAUST Repository

    Bogaert, Ignace; Cools, Kristof; Andriulli, Francesco P.; Bagci, Hakan

    2014-01-01

    The standard and mixed discretizations for the magnetic field integral equation (MFIE) and the Müller integral equation (MUIE) are investigated in the context of low-frequency (LF) scattering problems involving simply connected scatterers

  6. On the maximal cut of Feynman integrals and the solution of their differential equations

    Directory of Open Access Journals (Sweden)

    Amedeo Primo

    2017-03-01

    Full Text Available The standard procedure for computing scalar multi-loop Feynman integrals consists in reducing them to a basis of so-called master integrals, derive differential equations in the external invariants satisfied by the latter and, finally, try to solve them as a Laurent series in ϵ=(4−d/2, where d are the space–time dimensions. The differential equations are, in general, coupled and can be solved using Euler's variation of constants, provided that a set of homogeneous solutions is known. Given an arbitrary differential equation of order higher than one, there exists no general method for finding its homogeneous solutions. In this paper we show that the maximal cut of the integrals under consideration provides one set of homogeneous solutions, simplifying substantially the solution of the differential equations.

  7. Non-integrability of time-dependent spherically symmetric Yang-Mills equations

    International Nuclear Information System (INIS)

    Matinyan, S.G.; Prokhorenko, E.V.; Savvidy, G.K.

    1986-01-01

    The integrability of time-dependent spherically symmetric Yang-Mills equations is studied using the Fermi-Pasta-Ulam method. The phase space of this system is shown to have no quasi-periodic motion specific for integrable systems. In particular, the well-known Wu-Yang static solution is unstable, so its vicinity in phase is the stochasticity region

  8. Recovering an obstacle using integral equations

    KAUST Repository

    Rundell, William

    2009-05-01

    We consider the inverse problem of recovering the shape, location and surface properties of an object where the surrounding medium is both conductive and homogeneous and we measure Cauchy data on an accessible part of the exterior boundary. It is assumed that the physical situation is modelled by harmonic functions and the boundary condition on the obstacle is one of Dirichlet type. The purpose of this paper is to answer some of the questions raised in a recent paper that introduced a nonlinear integral equation approach for the solution of this type of problem.

  9. An Integrable Discrete Generalized Nonlinear Schrödinger Equation and Its Reductions

    International Nuclear Information System (INIS)

    Li Hong-Min; Li Yu-Qi; Chen Yong

    2014-01-01

    An integrable discrete system obtained by the algebraization of the difference operator is studied. The system is named discrete generalized nonlinear Schrödinger (GNLS) equation, which can be reduced to classical discrete nonlinear Schrödinger (NLS) equation. Furthermore, all of the linear reductions for the discrete GNLS equation are given through the theory of circulant matrices and the discrete NLS equation is obtained by one of the reductions. At the same time, the recursion operator and symmetries of continuous GNLS equation are successfully recovered by its corresponding discrete ones. (general)

  10. Oscillatory integrals on Hilbert spaces and Schroedinger equation with magnetic fields

    International Nuclear Information System (INIS)

    Albeverio, S.; Brzezniak, Z.

    1994-01-01

    We extend the theory of oscillatory integrals on Hilbert spaces (the mathematical version of ''Feynman path integrals'') to cover more general integrable functions, preserving the property of the integrals to have converging finite dimensional approximations. We give an application to the representation of solutions of the time dependent Schroedinger equation with a scalar and a magnetic potential by oscillatory integrals on Hilbert spaces. A relation with Ramer's functional in the corresponding probabilistic setting is found. (orig.)

  11. Integrable motion of curves in self-consistent potentials: Relation to spin systems and soliton equations

    Energy Technology Data Exchange (ETDEWEB)

    Myrzakulov, R.; Mamyrbekova, G.K.; Nugmanova, G.N.; Yesmakhanova, K.R. [Eurasian International Center for Theoretical Physics and Department of General and Theoretical Physics, Eurasian National University, Astana 010008 (Kazakhstan); Lakshmanan, M., E-mail: lakshman@cnld.bdu.ac.in [Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirapalli 620 024 (India)

    2014-06-13

    Motion of curves and surfaces in R{sup 3} lead to nonlinear evolution equations which are often integrable. They are also intimately connected to the dynamics of spin chains in the continuum limit and integrable soliton systems through geometric and gauge symmetric connections/equivalence. Here we point out the fact that a more general situation in which the curves evolve in the presence of additional self-consistent vector potentials can lead to interesting generalized spin systems with self-consistent potentials or soliton equations with self-consistent potentials. We obtain the general form of the evolution equations of underlying curves and report specific examples of generalized spin chains and soliton equations. These include principal chiral model and various Myrzakulov spin equations in (1+1) dimensions and their geometrically equivalent generalized nonlinear Schrödinger (NLS) family of equations, including Hirota–Maxwell–Bloch equations, all in the presence of self-consistent potential fields. The associated gauge equivalent Lax pairs are also presented to confirm their integrability. - Highlights: • Geometry of continuum spin chain with self-consistent potentials explored. • Mapping on moving space curves in R{sup 3} in the presence of potential fields carried out. • Equivalent generalized nonlinear Schrödinger (NLS) family of equations identified. • Integrability of identified nonlinear systems proved by deducing appropriate Lax pairs.

  12. Integrable equation of state for noisy cosmic string

    International Nuclear Information System (INIS)

    Carter, B.

    1990-01-01

    It is argued that, independently of the detailed (thermal or more general) noise spectrum of the microscopic extrinsic excitations that can be expected on an ordinary cosmic string, their effect can be taken into account at a macroscopic level by replacing the standard isotropic Goto-Nambu-type string model by the nondegenerate string model characterized by an equation of state of the nondispersive ''fixed determinant'' type, with the effective surface stress-energy tensor satisfying (T ν ν ) 2 -T μ ν T ν μ =2T 0 2 , where T 0 is a constant representing the null-state limit of the string tension T, whose product with the energy density U of the string is thereby held fixed: TU=T 0 2 . It is shown that this equation of state has the special property of giving rise (in a flat background) to explicitly integrable dynamical equations

  13. Heat transfer from the moving heat source of arbitrary shape

    International Nuclear Information System (INIS)

    Fomin, Sergei A.

    2000-01-01

    The present research is related to contact melting by a moving heat source of arbitrary shape. Heat conduction in the melting material is governed by 3D differential equation, where the thermal conductivity of the surrounding material is assumed to be strongly temperature dependent. By using the Green's formula, the boundary-value problem is converted to the boundary integral equation. This non-linear equation is solved numerically by interactions utilizing the boundary element method. Different shapes of heat sources are investigated. Since the obtained integral equation is the Fredholm type equation of the first kind and belongs to the family of so-called ill-posed problems, therefore, supplementary computations, that verify the stability of numerical algorithm, are provided. For the special cases associated with thermodrilling technology, some analytical estimations and solutions are obtained. Particularly, if the melting velocity is high (Pe>10), asymptotic solutions are found. In this case the integral equation is significantly reduced, that simplifies the computations. Numerical results are in good agreement with the closed-form solutions available for the elliptical shape of a solid-liquid interface. (author)

  14. A review of some basic aspects related to integration of airplane’s equations of motion

    Directory of Open Access Journals (Sweden)

    Dan TURCANU

    2017-09-01

    Full Text Available Numerical integration of the airplane’s equations of motion has long been considered among the most fundamental calculations in airplane’s analysis. Numerical algorithms have been implemented and experimentally validated. However, the need for superior speed and accuracy is still very topical, as, nowadays, various optimization algorithms rely heavily on data generated from the integration of the equations of motion and having access to larger amounts of data can increase the quality of the optimization. Now, for a number of decades, engineers have relied heavily on commercial codes based on automatically selected integration steps. However, optimally chosen constant integration steps can save time and allows for larger numbers of integrations to be performed. Yet, the basic papers that presented the fundamentals of numerical integration, as applied to airplane’s equations of motion are nowadays not easy to locate. Consequently, this paper presents a review of basic aspects related to the integration of airplane’s equation of motion. The discussion covers fundamentals of longitudinal and lateral-directional motion as well as the implementation of some numerical integration methods. The relation between numerical integration steps, accuracy, computational resource usage, numerical stability and their relation with the parameters describing the dynamic response of the airplane is considered and suggestions are presented for a faster yet accurate numerical integration.

  15. The ICVSIE: A General Purpose Integral Equation Method for Bio-Electromagnetic Analysis.

    Science.gov (United States)

    Gomez, Luis J; Yucel, Abdulkadir C; Michielssen, Eric

    2018-03-01

    An internally combined volume surface integral equation (ICVSIE) for analyzing electromagnetic (EM) interactions with biological tissue and wide ranging diagnostic, therapeutic, and research applications, is proposed. The ICVSIE is a system of integral equations in terms of volume and surface equivalent currents in biological tissue subject to fields produced by externally or internally positioned devices. The system is created by using equivalence principles and solved numerically; the resulting current values are used to evaluate scattered and total electric fields, specific absorption rates, and related quantities. The validity, applicability, and efficiency of the ICVSIE are demonstrated by EM analysis of transcranial magnetic stimulation, magnetic resonance imaging, and neuromuscular electrical stimulation. Unlike previous integral equations, the ICVSIE is stable regardless of the electric permittivities of the tissue or frequency of operation, providing an application-agnostic computational framework for EM-biomedical analysis. Use of the general purpose and robust ICVSIE permits streamlining the development, deployment, and safety analysis of EM-biomedical technologies.

  16. Integral equations of hadronic correlation functions a functional- bootstrap approach

    CERN Document Server

    Manesis, E K

    1974-01-01

    A reasonable 'microscopic' foundation of the Feynman hadron-liquid analogy is offered, based on a class of models for hadron production. In an external field formalism, the equivalence (complementarity) of the exclusive and inclusive descriptions of hadronic reactions is specifically expressed in a functional-bootstrap form, and integral equations between inclusive and exclusive correlation functions are derived. Using the latest CERN-ISR data on the two-pion inclusive correlation function, and assuming rapidity translational invariance for the exclusive one, the simplest integral equation is solved in the 'central region' and an exclusive correlation length in rapidity predicted. An explanation is also offered for the unexpected similarity observed between pi /sup +/ pi /sup -/ and pi /sup -/ pi /sup -/ inclusive correlations. (31 refs).

  17. Integral-equation formulation for drift eigenmodes in cylindrically symmetric systems

    International Nuclear Information System (INIS)

    Linsker, R.

    1980-12-01

    A method for solving the integral eigenmode equation for drift waves in cylindrical (or slab) geometry is presented. A leading-order kinematic effect that has been noted in the past, but incorrectly ignored in recent integral-equation calculations, is incorporated. The present method also allows electrons to be treated with a physical mass ratio (unlike earlier work that is restricted to artificially small m/sub i//m/sub e/ owing to resolution limitations). Results for the universal mode and for the ion-temperature-gradient driven mode are presented. The kinematic effect qualitatively changes the spectrum of the ion mode, and a new second region of instability for k/sub perpendicular to/rho/sub i/greater than or equal to 1 is found

  18. Local first integrals for systems of differential equations

    International Nuclear Information System (INIS)

    Zhang Xiang

    2003-01-01

    The main purpose of this paper is to provide some sufficient conditions for a system of differential equations to have local first integrals in a certain neighbourhood of a singularity. Our results generalize those given in Kwek et al (2003 Z. Angew. Math. Phys. 54 26) and Li et al (2003 Z. Angew. Math. Phys. 54 235)

  19. The Volterra's integral equation theory for accelerator single-freedom nonlinear components

    International Nuclear Information System (INIS)

    Wang Sheng; Xie Xi

    1996-01-01

    The Volterra's integral equation equivalent to the dynamic equation of accelerator single-freedom nonlinear components is given, starting from which the transport operator of accelerator single-freedom nonlinear components and its inverse transport operator are obtained. Therefore, another algorithm for the expert system of the beam transport operator of accelerator single-freedom nonlinear components is developed

  20. Expressing Solutions of the Dirac Equation in Terms of Feynman Path Integral

    CERN Document Server

    Hose, R D

    2006-01-01

    Using the separation of the variables technique, the free particle solutions of the Dirac equation in the momentum space are shown to be actually providing the definition of Delta function for the Schr dinger picture. Further, the said solution is shown to be derivable on the sole strength of geometrical argument that the Dirac equation for free particle is an equation of a plane in momentum space. During the evolution of time in the Schr dinger picture, the normal to the said Dirac equation plane is shown to be constantly changing in direction due to the uncertainty principle and thereby, leading to a zigzag path for the Dirac particle in the momentum space. Further, the time evolution of the said Delta function solutions of the Dirac equation is shown to provide Feynman integral of all such zigzag paths in the momentum space. Towards the end of the paper, Feynman path integral between two fixed spatial points in the co-ordinate space during a certain time interv! al is shown to be composed, in time sequence...

  1. A unified approach to deconvolution radiation spectra measured by radiochromic films

    CERN Document Server

    Stancic, V; Ljubenov, V

    2002-01-01

    A method for the evaluation of energy distribution of a radiation source on the basis of measured space distribution of deposited energy is proposed. The measured data were obtained by using radiochromic films. Mathematical modeling is defined as a Fredholm integral equation inversion problem. Negative solutions were treated as an additional condition expressed through undefined energy group boundaries, caused by virtue of the physical phenomenon of statistical uncertainty. Examples are given of the electron source and neutron radiation field.

  2. The deconvolution of Doppler-broadened positron annihilation measurements using fast Fourier transforms and power spectral analysis

    International Nuclear Information System (INIS)

    Schaffer, J.P.; Shaughnessy, E.J.; Jones, P.L.

    1984-01-01

    A deconvolution procedure which corrects Doppler-broadened positron annihilation spectra for instrument resolution is described. The method employs fast Fourier transforms, is model independent, and does not require iteration. The mathematical difficulties associated with the incorrectly posed first order Fredholm integral equation are overcome by using power spectral analysis to select a limited number of low frequency Fourier coefficients. The FFT/power spectrum method is then demonstrated for an irradiated high purity single crystal sapphire sample. (orig.)

  3. Integrated vehicle dynamics control using State Dependent Riccati Equations

    NARCIS (Netherlands)

    Bonsen, B.; Mansvelders, R.; Vermeer, E.

    2010-01-01

    In this paper we discuss a State Dependent Riccati Equations (SDRE) solution for Integrated Vehicle Dynamics Control (IVDC). The SDRE approach is a nonlinear variant of the well known Linear Quadratic Regulator (LQR) and implements a quadratic cost function optimization. A modified version of this

  4. Anti-symmetrically fused model and non-linear integral equations in the three-state Uimin-Sutherland model

    International Nuclear Information System (INIS)

    Fujii, Akira; Kluemper, Andreas

    1999-01-01

    We derive the non-linear integral equations determining the free energy of the three-state pure bosonic Uimin-Sutherland model. In order to find a complete set of auxiliary functions, the anti-symmetric fusion procedure is utilized. We solve the non-linear integral equations numerically and see that the low-temperature behavior coincides with that predicted by conformal field theory. The magnetization and magnetic susceptibility are also calculated by means of the non-linear integral equation

  5. On the Volterra integral equation relating creep and relaxation

    International Nuclear Information System (INIS)

    Anderssen, R S; De Hoog, F R; Davies, A R

    2008-01-01

    The evolving stress–strain response of a material to an applied deformation is causal. If the current response depends on the earlier history of the stress–strain dynamics of the material (i.e. the material has memory), then Volterra integral equations become the natural framework within which to model the response. For viscoelastic materials, when the response is linear, the dual linear Boltzmann causal integral equations are the appropriate model. The choice of one rather than the other depends on whether the applied deformation is a stress or a strain, and the associated response is, respectively, a creep or a relaxation. The duality between creep and relaxation is known explicitly and is referred to as the 'interconversion equation'. Rheologically, its importance relates to the fact that it allows the creep to be determined from knowledge of the relaxation and vice versa. Computationally, it has been known for some time that the recovery of the relaxation from the creep is more problematic than the creep from the relaxation. Recent research, using discrete models for the creep and relaxation, has confirmed that this is an essential feature of interconversion. In this paper, the corresponding result is generalized for continuous models of the creep and relaxation

  6. Transforming differential equations of multi-loop Feynman integrals into canonical form

    Energy Technology Data Exchange (ETDEWEB)

    Meyer, Christoph [Institut für Physik, Humboldt-Universität zu Berlin,12489 Berlin (Germany)

    2017-04-03

    The method of differential equations has been proven to be a powerful tool for the computation of multi-loop Feynman integrals appearing in quantum field theory. It has been observed that in many instances a canonical basis can be chosen, which drastically simplifies the solution of the differential equation. In this paper, an algorithm is presented that computes the transformation to a canonical basis, starting from some basis that is, for instance, obtained by the usual integration-by-parts reduction techniques. The algorithm requires the existence of a rational transformation to a canonical basis, but is otherwise completely agnostic about the differential equation. In particular, it is applicable to problems involving multiple scales and allows for a rational dependence on the dimensional regulator. It is demonstrated that the algorithm is suitable for current multi-loop calculations by presenting its successful application to a number of non-trivial examples.

  7. Transforming differential equations of multi-loop Feynman integrals into canonical form

    Science.gov (United States)

    Meyer, Christoph

    2017-04-01

    The method of differential equations has been proven to be a powerful tool for the computation of multi-loop Feynman integrals appearing in quantum field theory. It has been observed that in many instances a canonical basis can be chosen, which drastically simplifies the solution of the differential equation. In this paper, an algorithm is presented that computes the transformation to a canonical basis, starting from some basis that is, for instance, obtained by the usual integration-by-parts reduction techniques. The algorithm requires the existence of a rational transformation to a canonical basis, but is otherwise completely agnostic about the differential equation. In particular, it is applicable to problems involving multiple scales and allows for a rational dependence on the dimensional regulator. It is demonstrated that the algorithm is suitable for current multi-loop calculations by presenting its successful application to a number of non-trivial examples.

  8. Transforming differential equations of multi-loop Feynman integrals into canonical form

    International Nuclear Information System (INIS)

    Meyer, Christoph

    2017-01-01

    The method of differential equations has been proven to be a powerful tool for the computation of multi-loop Feynman integrals appearing in quantum field theory. It has been observed that in many instances a canonical basis can be chosen, which drastically simplifies the solution of the differential equation. In this paper, an algorithm is presented that computes the transformation to a canonical basis, starting from some basis that is, for instance, obtained by the usual integration-by-parts reduction techniques. The algorithm requires the existence of a rational transformation to a canonical basis, but is otherwise completely agnostic about the differential equation. In particular, it is applicable to problems involving multiple scales and allows for a rational dependence on the dimensional regulator. It is demonstrated that the algorithm is suitable for current multi-loop calculations by presenting its successful application to a number of non-trivial examples.

  9. Introduction to quantum mechanics Schrödinger equation and path integral

    CERN Document Server

    Müller-Kirsten, H J W

    2012-01-01

    This text on quantum mechanics begins by covering all the main topics of an introduction to the subject. It then concentrates on newer developments. In particular it continues with the perturbative solution of the Schrodinger equation for various potentials and thereafter with the introduction and evaluation of their path integral counterparts. Considerations of the large order behavior of the perturbation expansions show that in most applications these are asymptotic expansions. The parallel consideration of path integrals requires the evaluation of these around periodic classical configurations, the fluctuation equations about which lead back to specific wave equations. The period of the classical configurations is related to temperature, and permits transitions to the thermal domain to be classified as phase transitions. In this second edition of the text important applications and numerous examples have been added. In particular, the chapter on the Coulomb potential has been extended to include an introdu...

  10. Multi-soliton management by the integrable nonautonomous nonlinear integro-differential Schrödinger equation

    International Nuclear Information System (INIS)

    Zhang, Yu-Juan; Zhao, Dun; Luo, Hong-Gang

    2014-01-01

    We consider a wide class of integrable nonautonomous nonlinear integro-differential Schrödinger equation which contains the models for the soliton management in Bose–Einstein condensates, nonlinear optics, and inhomogeneous Heisenberg spin chain. With the help of the nonisospectral AKNS hierarchy, we obtain the N-fold Darboux transformation and the N-fold soliton-like solutions for the equation. The soliton management, especially the synchronized dispersive and nonlinear management in optical fibers is discussed. It is found that in the situation without external potential, the synchronized dispersive and nonlinear management can keep the integrability of the nonlinear Schrödinger equation; this suggests that in optical fibers, the synchronized dispersive and nonlinear management can control and maintain the propagation of a multi-soliton. - Highlights: • We consider a unified model for soliton management by an integrable integro-differential Schrödinger equation. • Using Lax pair, the N-fold Darboux transformation for the equation is presented. • The multi-soliton management is considered. • The synchronized dispersive and nonlinear management is suggested

  11. A calderón multiplicative preconditioner for the combined field integral equation

    KAUST Repository

    Bagci, Hakan

    2009-10-01

    A Calderón multiplicative preconditioner (CMP) for the combined field integral equation (CFIE) is developed. Just like with previously proposed Caldern-preconditioned CFIEs, a localization procedure is employed to ensure that the equation is resonance-free. The iterative solution of the linear system of equations obtained via the CMP-based discretization of the CFIE converges rapidly regardless of the discretization density and the frequency of excitation. © 2009 IEEE.

  12. Continuous limits for an integrable coupling system of Toda equation hierarchy

    International Nuclear Information System (INIS)

    Li Li; Yu Fajun

    2009-01-01

    In this Letter, we present an integrable coupling system of lattice hierarchy and its continuous limits by using of Lie algebra sl(4). By introducing a complex discrete spectral problem, the integrable coupling system of Toda lattice hierarchy is derived. It is shown that a new complex lattice spectral problem converges to the integrable couplings of discrete soliton equation hierarchy, which has the integrable coupling system of C-KdV hierarchy as a new kind of continuous limit.

  13. Continuous limits for an integrable coupling system of Toda equation hierarchy

    Energy Technology Data Exchange (ETDEWEB)

    Li Li [College of Maths and Systematic Science, Shenyang Normal University, Shenyang 110034 (China); Yu Fajun, E-mail: yfajun@163.co [College of Maths and Systematic Science, Shenyang Normal University, Shenyang 110034 (China)

    2009-09-21

    In this Letter, we present an integrable coupling system of lattice hierarchy and its continuous limits by using of Lie algebra sl(4). By introducing a complex discrete spectral problem, the integrable coupling system of Toda lattice hierarchy is derived. It is shown that a new complex lattice spectral problem converges to the integrable couplings of discrete soliton equation hierarchy, which has the integrable coupling system of C-KdV hierarchy as a new kind of continuous limit.

  14. Integral equation methods for vesicle electrohydrodynamics in three dimensions

    Science.gov (United States)

    Veerapaneni, Shravan

    2016-12-01

    In this paper, we develop a new boundary integral equation formulation that describes the coupled electro- and hydro-dynamics of a vesicle suspended in a viscous fluid and subjected to external flow and electric fields. The dynamics of the vesicle are characterized by a competition between the elastic, electric and viscous forces on its membrane. The classical Taylor-Melcher leaky-dielectric model is employed for the electric response of the vesicle and the Helfrich energy model combined with local inextensibility is employed for its elastic response. The coupled governing equations for the vesicle position and its transmembrane electric potential are solved using a numerical method that is spectrally accurate in space and first-order in time. The method uses a semi-implicit time-stepping scheme to overcome the numerical stiffness associated with the governing equations.

  15. A generalized Clebsch transformation leading to a first integral of Navier–Stokes equations

    Energy Technology Data Exchange (ETDEWEB)

    Scholle, M., E-mail: markus.scholle@hs-heilbronn.de; Marner, F., E-mail: florian.marner@hs-heilbronn.de

    2016-09-23

    In fluid dynamics, the Clebsch transformation allows for the construction of a first integral of the equations of motion leading to a self-adjoint form of the equations. A remarkable feature is the description of the vorticity by means of only two potential fields fulfilling simple transport equations. Despite useful applications in fluid dynamics and other physical disciplines as well, the classical Clebsch transformation has ever been restricted to inviscid flow. In the present paper a novel, generalized Clebsch transformation is developed which also covers the case of incompressible viscous flow. The resulting field equations are discussed briefly and solved for a flow example. Perspectives for a further extension of the method as well as perspectives towards the development of new solution strategies are presented. - Highlights: • A generalized Clebsch transformation is established applying to viscous flow. • The resulting 5 equations are a first integral of Navier–Stokes-equations. • An axisymmetric stagnation flow against a solid wall is considered as flow example. • Perspectives of the method for other problems, e.g. in solid mechanics are discussed.

  16. A generalized Clebsch transformation leading to a first integral of Navier–Stokes equations

    International Nuclear Information System (INIS)

    Scholle, M.; Marner, F.

    2016-01-01

    In fluid dynamics, the Clebsch transformation allows for the construction of a first integral of the equations of motion leading to a self-adjoint form of the equations. A remarkable feature is the description of the vorticity by means of only two potential fields fulfilling simple transport equations. Despite useful applications in fluid dynamics and other physical disciplines as well, the classical Clebsch transformation has ever been restricted to inviscid flow. In the present paper a novel, generalized Clebsch transformation is developed which also covers the case of incompressible viscous flow. The resulting field equations are discussed briefly and solved for a flow example. Perspectives for a further extension of the method as well as perspectives towards the development of new solution strategies are presented. - Highlights: • A generalized Clebsch transformation is established applying to viscous flow. • The resulting 5 equations are a first integral of Navier–Stokes-equations. • An axisymmetric stagnation flow against a solid wall is considered as flow example. • Perspectives of the method for other problems, e.g. in solid mechanics are discussed.

  17. Crossover integral equation theory for the liquid structure study

    International Nuclear Information System (INIS)

    Lai, S.K.; Chen, H.C.

    1994-08-01

    The main purpose of this work is to report on a calculation that describes the role of the long-range bridge function [H. Iyetomi and S. Ichimaru, Phys. Rev. A 25, 2434 (1982)] as applied to the study of structure of simple liquid metals. It was found here that this bridge function accounts pretty well for the major part of long-range interactions but is physically inadequate for describing the short-range part of liquid structure. To improve on the theory we have drawn attention to the crossover integral equation method which, in essence, amounts to adding to the above bridge function a short-range correction of bridge diagrams. The suggested crossover procedure has been tested for the case of liquid metal Cs. Remarkably good agreement with experiment was obtained confirming our conjecture that the crossover integral equation approach as stressed in this work is potentially an appropriate theory for an accurate study of liquid structure possibly for the supercooled liquid regime. (author). 21 refs, 3 figs

  18. Feynman path integral application on deriving black-scholes diffusion equation for european option pricing

    International Nuclear Information System (INIS)

    Utama, Briandhika; Purqon, Acep

    2016-01-01

    Path Integral is a method to transform a function from its initial condition to final condition through multiplying its initial condition with the transition probability function, known as propagator. At the early development, several studies focused to apply this method for solving problems only in Quantum Mechanics. Nevertheless, Path Integral could also apply to other subjects with some modifications in the propagator function. In this study, we investigate the application of Path Integral method in financial derivatives, stock options. Black-Scholes Model (Nobel 1997) was a beginning anchor in Option Pricing study. Though this model did not successfully predict option price perfectly, especially because its sensitivity for the major changing on market, Black-Scholes Model still is a legitimate equation in pricing an option. The derivation of Black-Scholes has a high difficulty level because it is a stochastic partial differential equation. Black-Scholes equation has a similar principle with Path Integral, where in Black-Scholes the share's initial price is transformed to its final price. The Black-Scholes propagator function then derived by introducing a modified Lagrange based on Black-Scholes equation. Furthermore, we study the correlation between path integral analytical solution and Monte-Carlo numeric solution to find the similarity between this two methods. (paper)

  19. Discretization of the induced-charge boundary integral equation.

    Science.gov (United States)

    Bardhan, Jaydeep P; Eisenberg, Robert S; Gillespie, Dirk

    2009-07-01

    Boundary-element methods (BEMs) for solving integral equations numerically have been used in many fields to compute the induced charges at dielectric boundaries. In this paper, we consider a more accurate implementation of BEM in the context of ions in aqueous solution near proteins, but our results are applicable more generally. The ions that modulate protein function are often within a few angstroms of the protein, which leads to the significant accumulation of polarization charge at the protein-solvent interface. Computing the induced charge accurately and quickly poses a numerical challenge in solving a popular integral equation using BEM. In particular, the accuracy of simulations can depend strongly on seemingly minor details of how the entries of the BEM matrix are calculated. We demonstrate that when the dielectric interface is discretized into flat tiles, the qualocation method of Tausch [IEEE Trans Comput.-Comput.-Aided Des. 20, 1398 (2001)] to compute the BEM matrix elements is always more accurate than the traditional centroid-collocation method. Qualocation is not more expensive to implement than collocation and can save significant computational time by reducing the number of boundary elements needed to discretize the dielectric interfaces.

  20. Discretization of the induced-charge boundary integral equation.

    Energy Technology Data Exchange (ETDEWEB)

    Bardhan, J. P.; Eisenberg, R. S.; Gillespie, D.; Rush Univ. Medical Center

    2009-07-01

    Boundary-element methods (BEMs) for solving integral equations numerically have been used in many fields to compute the induced charges at dielectric boundaries. In this paper, we consider a more accurate implementation of BEM in the context of ions in aqueous solution near proteins, but our results are applicable more generally. The ions that modulate protein function are often within a few angstroms of the protein, which leads to the significant accumulation of polarization charge at the protein-solvent interface. Computing the induced charge accurately and quickly poses a numerical challenge in solving a popular integral equation using BEM. In particular, the accuracy of simulations can depend strongly on seemingly minor details of how the entries of the BEM matrix are calculated. We demonstrate that when the dielectric interface is discretized into flat tiles, the qualocation method of Tausch et al. [IEEE Trans Comput.-Comput.-Aided Des. 20, 1398 (2001)] to compute the BEM matrix elements is always more accurate than the traditional centroid-collocation method. Qualocation is not more expensive to implement than collocation and can save significant computational time by reducing the number of boundary elements needed to discretize the dielectric interfaces.

  1. A predictor-corrector scheme for solving the Volterra integral equation

    KAUST Repository

    Al Jarro, Ahmed; Bagci, Hakan

    2011-01-01

    The occurrence of late time instabilities is a common problem of almost all time marching methods developed for solving time domain integral equations. Implicit marching algorithms are now considered stable with various efforts that have been

  2. Numerical Integration of the Transport Equation For Infinite Homogeneous Media

    Energy Technology Data Exchange (ETDEWEB)

    Haakansson, Rune

    1962-01-15

    The transport equation for neutrons in infinite homogeneous media is solved by direct numerical integration. Accounts are taken to the anisotropy and the inelastic scattering. The integration has been performed by means of the trapezoidal rule and the length of the energy intervals are constant in lethargy scale. The machine used is a Ferranti Mercury computer. Results are given for water, heavy water, aluminium water mixture and iron-aluminium-water mixture.

  3. On the initial condition problem of the time domain PMCHWT surface integral equation

    KAUST Repository

    Uysal, Ismail Enes

    2017-05-13

    Non-physical, linearly increasing and constant current components are induced in marching on-in-time solution of time domain surface integral equations when initial conditions on time derivatives of (unknown) equivalent currents are not enforced properly. This problem can be remedied by solving the time integral of the surface integral for auxiliary currents that are defined to be the time derivatives of the equivalent currents. Then the equivalent currents are obtained by numerically differentiating the auxiliary ones. In this work, this approach is applied to the marching on-in-time solution of the time domain Poggio-Miller-Chan-Harrington-Wu-Tsai surface integral equation enforced on dispersive/plasmonic scatterers. Accuracy of the proposed method is demonstrated by a numerical example.

  4. Integrable discretizations for the short-wave model of the Camassa-Holm equation

    International Nuclear Information System (INIS)

    Feng Baofeng; Maruno, Ken-ichi; Ohta, Yasuhiro

    2010-01-01

    The link between the short-wave model of the Camassa-Holm equation (SCHE) and bilinear equations of the two-dimensional Toda lattice equation is clarified. The parametric form of the N-cuspon solution of the SCHE in Casorati determinant is then given. Based on the above finding, integrable semi-discrete and full-discrete analogues of the SCHE are constructed. The determinant solutions of both semi-discrete and fully discrete analogues of the SCHE are also presented.

  5. Integral transform method for solving time fractional systems and fractional heat equation

    Directory of Open Access Journals (Sweden)

    Arman Aghili

    2014-01-01

    Full Text Available In the present paper, time fractional partial differential equation is considered, where the fractional derivative is defined in the Caputo sense. Laplace transform method has been applied to obtain an exact solution. The authors solved certain homogeneous and nonhomogeneous time fractional heat equations using integral transform. Transform method is a powerful tool for solving fractional singular Integro - differential equations and PDEs. The result reveals that the transform method is very convenient and effective.

  6. Electromagnetic wave propagation over an inhomogeneous flat earth (two-dimensional integral equation formulation)

    International Nuclear Information System (INIS)

    de Jong, G.

    1975-01-01

    With the aid of a two-dimensional integral equation formulation, the ground wave propagation of electromagnetic waves transmitted by a vertical electric dipole over an inhomogeneous flat earth is investigated. For the configuration in which a ground wave is propagating across an ''island'' on a flat earth, the modulus and argument of the attenuation function have been computed. The results for the two-dimensional treatment are significantly more accurate in detail than the calculations using a one-dimensional integral equation

  7. Efficiently and easily integrating differential equations with JiTCODE, JiTCDDE, and JiTCSDE

    Science.gov (United States)

    Ansmann, Gerrit

    2018-04-01

    We present a family of Python modules for the numerical integration of ordinary, delay, or stochastic differential equations. The key features are that the user enters the derivative symbolically and it is just-in-time-compiled, allowing the user to efficiently integrate differential equations from a higher-level interpreted language. The presented modules are particularly suited for large systems of differential equations such as those used to describe dynamics on complex networks. Through the selected method of input, the presented modules also allow almost complete automatization of the process of estimating regular as well as transversal Lyapunov exponents for ordinary and delay differential equations. We conceptually discuss the modules' design, analyze their performance, and demonstrate their capabilities by application to timely problems.

  8. Some thoughts on the pressure integration requirements of the Navier–Stokes equations

    International Nuclear Information System (INIS)

    Saad, Tony; Majdalani, Joseph

    2012-01-01

    The Navier–Stokes formulation represents a uniquely challenging system of partial differential equations that continues to influence modern applied science and engineering. In its simplest form, the system can be used to prescribe the motion of a viscous incompressible fluid with constant properties. It consists of four equations in three-dimensional space that account for both the kinematic and dynamic conditions that a fluid element senses. In this work, we investigate the pressure integration rules and restrictions that affect the resolution of the scalar pressure field. We begin our analysis by exploring the integration properties of Euler's equations in two dimensions while making use of Clairaut's theorem on the commutativity of mixed partial derivatives. We then extend our findings to three-dimensional space. This process gives rise to a theorem and four corollaries that help to clarify the conditions needed to obtain exact or asymptotic solutions for the pressure distribution. Consequently, we identify the fundamental conditions under which the Navier–Stokes equations can be properly integrated to arrive at an analytic expression for the pressure field, namely, one that is continuous and twice differentiable. In closing, several configurations are used to test the theorem and showcase its connection with the pressure formulation. These include potential flows for which the pressure can be obtained unconditionally, and inviscid rotational motions of the Taylor–Culick type with and without headwall injection. (paper)

  9. Nonperturbative time-convolutionless quantum master equation from the path integral approach

    International Nuclear Information System (INIS)

    Nan Guangjun; Shi Qiang; Shuai Zhigang

    2009-01-01

    The time-convolutionless quantum master equation is widely used to simulate reduced dynamics of a quantum system coupled to a bath. However, except for several special cases, applications of this equation are based on perturbative calculation of the dissipative tensor, and are limited to the weak system-bath coupling regime. In this paper, we derive an exact time-convolutionless quantum master equation from the path integral approach, which provides a new way to calculate the dissipative tensor nonperturbatively. Application of the new method is demonstrated in the case of an asymmetrical two-level system linearly coupled to a harmonic bath.

  10. A two-component generalization of the reduced Ostrovsky equation and its integrable semi-discrete analogue

    International Nuclear Information System (INIS)

    Feng, Bao-Feng; Maruno, Ken-ichi; Ohta, Yasuhiro

    2017-01-01

    In the present paper, we propose a two-component generalization of the reduced Ostrovsky (Vakhnenko) equation, whose differential form can be viewed as the short-wave limit of a two-component Degasperis–Procesi (DP) equation. They are integrable due to the existence of Lax pairs. Moreover, we have shown that the two-component reduced Ostrovsky equation can be reduced from an extended BKP hierarchy with negative flow through a pseudo 3-reduction and a hodograph (reciprocal) transform. As a by-product, its bilinear form and N -soliton solution in terms of pfaffians are presented. One- and two-soliton solutions are provided and analyzed. In the second part of the paper, we start with a modified BKP hierarchy, which is a Bäcklund transformation of the above extended BKP hierarchy, an integrable semi-discrete analogue of the two-component reduced Ostrovsky equation is constructed by defining an appropriate discrete hodograph transform and dependent variable transformations. In particular, the backward difference form of above semi-discrete two-component reduced Ostrovsky equation gives rise to the integrable semi-discretization of the short wave limit of a two-component DP equation. Their N -soliton solutions in terms of pffafians are also provided. (paper)

  11. The integral equation method applied to eddy currents

    International Nuclear Information System (INIS)

    Biddlecombe, C.S.; Collie, C.J.; Simkin, J.; Trowbridge, C.W.

    1976-04-01

    An algorithm for the numerical solution of eddy current problems is described, based on the direct solution of the integral equation for the potentials. In this method only the conducting and iron regions need to be divided into elements, and there are no boundary conditions. Results from two computer programs using this method for iron free problems for various two-dimensional geometries are presented and compared with analytic solutions. (author)

  12. Rational first integrals of geodesic equations and generalised hidden symmetries

    International Nuclear Information System (INIS)

    Aoki, Arata; Houri, Tsuyoshi; Tomoda, Kentaro

    2016-01-01

    We discuss novel generalisations of Killing tensors, which are introduced by considering rational first integrals of geodesic equations. We introduce the notion of inconstructible generalised Killing tensors, which cannot be constructed from ordinary Killing tensors. Moreover, we introduce inconstructible rational first integrals, which are constructed from inconstructible generalised Killing tensors, and provide a method for checking the inconstructibility of a rational first integral. Using the method, we show that the rational first integral of the Collinson–O’Donnell solution is not inconstructible. We also provide several examples of metrics admitting an inconstructible rational first integral in two and four-dimensions, by using the Maciejewski–Przybylska system. Furthermore, we attempt to generalise other hidden symmetries such as Killing–Yano tensors. (paper)

  13. Time-independent integral equation for Maxwell's system. Application of radar cross section computation

    International Nuclear Information System (INIS)

    Pujols, Agnes

    1991-01-01

    We prove that the scattering operator for the wave equation in the exterior of an non-homogeneous obstacle exists. Its distribution kernel is represented by a time-dependent boundary integral equation. A space-time integral variational formulation is developed for determining the current induced by the scattering of an electromagnetic wave by an homogeneous object. The discrete approximation of the variational problem using a finite element method in both space and time leads to stable convergent schemes, giving a numerical code for perfectly conducting cylinders. (author) [fr

  14. Nystro¨m Method in transport theory

    Energy Technology Data Exchange (ETDEWEB)

    Dalmolin, Débora; Azevedo, Fabio Souto de; Sauter, Esequia, E-mail: mtmdalmolin@gmail.com, E-mail: fabio.azevedo@ufrgs.br, E-mail: esequia.sauter@ufrgs.br [Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, RS (Brazil). Departamento de Matemática Pura e Aplicada

    2017-07-01

    We consider a system of equations modelling the steady-state transport equation in a participative medium with internal sources and semi-reflective boundaries. Based on this model, we discuss the implementation of the Nystro¨m method to solve the integral formulation of this transport equation. The analytical problems of existence and uniqueness of solution as well as numerical results for these equations have already been established in the literature. To obtain a numerical solution for the scalar flux for this problem, we will write the equation as a Fredholm equation of the second type and analyze quadrature schemes such as the Boole and Gauss-Legendre rules. Analytical and computational techniques were implemented to deal with singularities. We show the efficiency of the proposed method through some numerical tests and compare our results with those that can be found in the literature. (author)

  15. Continuous properties of the data-to-solution map for a generalized μ-Camassa-Holm integrable equation

    Science.gov (United States)

    Yu, Shengqi

    2018-05-01

    This work studies a generalized μ-type integrable equation with both quadratic and cubic nonlinearities; the μ-Camassa-Holm and modified μ-Camassa-Holm equations are members of this family of equations. It has been shown that the Cauchy problem for this generalized μ-Camassa-Holm integrable equation is locally well-posed for initial data u0 ∈ Hs, s > 5/2. In this work, we further investigate the continuity properties to this equation. It is proved in this work that the data-to-solution map of the proposed equation is not uniformly continuous. It is also found that the solution map is Hölder continuous in the Hr-topology when 0 ≤ r < s with Hölder exponent α depending on both s and r.

  16. On a Painleve test for the complete integrability of Bogomolny's monopole equation

    International Nuclear Information System (INIS)

    Roy Chowdhury, A.; Chanda, P.K.

    1984-09-01

    We have made an analysis of the monopole equation of Bogomolny from the stand point of Painleve test. The idea that any non-linear partial differential equation admitting a Lax representation should conform to the criterion of the Painleve analysis seems to hold well in case of Bogomolny equation. We have determined the position for resonances and have proved that at each of these the coefficients in the Forbenius type expansion of the gauge potentials do become arbitrary signalling the complete integrability of the system. (author)

  17. Two-dimensional nonlinear string-type equations and their exact integration

    International Nuclear Information System (INIS)

    Leznov, A.N.; Saveliev, M.V.

    1982-01-01

    On the base of group-theoretical formulation for exactly integrable two-dimensional non-linear dynamical systems associated with a local part of an arbitrary graded Lie algebra we study a string-type subclass of the equations. Explicit expressions have been obtained for their general solutions

  18. Fringe integral equation method for a truncated grounded dielectric slab

    DEFF Research Database (Denmark)

    Jørgensen, Erik; Maci, S.; Toccafondi, A.

    2001-01-01

    The problem of scattering by a semi-infinite grounded dielectric slab illuminated by an arbitrary incident TMz polarized electric field is studied by solving a new set of “fringe” integral equations (F-IEs), whose functional unknowns are physically associated to the wave diffraction processes...

  19. Quadratic algebras and noncommutative integration of Klein-Gordon equations in non-steckel Riemann spaces

    International Nuclear Information System (INIS)

    Varaksin, O.L.; Firstov, V.V.; Shapovalov, A.V.; Shirokov, I.V.

    1995-01-01

    The method of noncommutative integration of linear partial differential equations is used to solve the Klein-Gordon equations in Riemann space, in the case when the set of noncommutating symmetry operators of this equation for a quadratic algebra consists of one second-order operator and several first-order operators. Solutions that do not permit variable separation are presented

  20. A trick loop algebra and a corresponding Liouville integrable hierarchy of evolution equations

    International Nuclear Information System (INIS)

    Zhang Yufeng; Xu Xixiang

    2004-01-01

    A subalgebra of loop algebra A-bar 2 is first constructed, which has its own special feature. It follows that a new Liouville integrable hierarchy of evolution equations is obtained, possessing a tri-Hamiltonian structure, which is proved by us in this paper. Especially, three symplectic operators are constructed directly from recurrence relations. The conjugate operator of a recurrence operator is a hereditary symmetry. As reduction cases of the hierarchy presented in this paper, the celebrated MKdV equation and heat-conduction equation are engendered, respectively. Therefore, we call the hierarchy a generalized MKdV-H system. At last, a high-dimension loop algebra G-bar is constructed by making use of a proper scalar transformation. As a result, a type expanding integrable model of the MKdV-H system is given

  1. The First-Integral Method and Abundant Explicit Exact Solutions to the Zakharov Equations

    Directory of Open Access Journals (Sweden)

    Yadong Shang

    2012-01-01

    Full Text Available This paper is concerned with the system of Zakharov equations which involves the interactions between Langmuir and ion-acoustic waves in plasma. Abundant explicit and exact solutions of the system of Zakharov equations are derived uniformly by using the first integral method. These exact solutions are include that of the solitary wave solutions of bell-type for n and E, the solitary wave solutions of kink-type for E and bell-type for n, the singular traveling wave solutions, periodic wave solutions of triangle functions, Jacobi elliptic function doubly periodic solutions, and Weierstrass elliptic function doubly periodic wave solutions. The results obtained confirm that the first integral method is an efficient technique for analytic treatment of a wide variety of nonlinear systems of partial differential equations.

  2. Particle connectedness and cluster formation in sequential depositions of particles: integral-equation theory.

    Science.gov (United States)

    Danwanichakul, Panu; Glandt, Eduardo D

    2004-11-15

    We applied the integral-equation theory to the connectedness problem. The method originally applied to the study of continuum percolation in various equilibrium systems was modified for our sequential quenching model, a particular limit of an irreversible adsorption. The development of the theory based on the (quenched-annealed) binary-mixture approximation includes the Ornstein-Zernike equation, the Percus-Yevick closure, and an additional term involving the three-body connectedness function. This function is simplified by introducing a Kirkwood-like superposition approximation. We studied the three-dimensional (3D) system of randomly placed spheres and 2D systems of square-well particles, both with a narrow and with a wide well. The results from our integral-equation theory are in good accordance with simulation results within a certain range of densities.

  3. On the Space of Functions with Growths Tempered by a Modulus of Continuity and Its Applications

    Directory of Open Access Journals (Sweden)

    Józef Banaś

    2013-01-01

    Full Text Available We are going to study the space of real functions defined on a bounded metric space and having growths tempered by a modulus of continuity. We prove also a sufficient condition for the relative compactness in the mentioned function space. Using that condition and the classical Schauder fixed point theorem, we show the existence theorem for some quadratic integral equations of Fredholm type in the space of functions satisfying the Hölder condition. An example illustrating the mentioned existence result is also included.

  4. Towards multi-exponential analysis in optically stimulated luminescence

    DEFF Research Database (Denmark)

    Ankjærgaard, Christina; Jain, Mayank; Hansen, Per Christian

    2010-01-01

    analysis of the performance and stability of two models, the decay and peak form, and we consider different transformation methods for obtaining the peak form. For the numerical computations we use a nonlinear least squares (NLS) method and a method based on a first-kind Fredholm integral equation (FIE......). Our analysis uses artificial data with three components (seven parameters including the background) and ten different levels of background, both the signal and the background contain Poisson distributed noise. Parameters derived using both models are acceptable (statistically consistent...

  5. Nodal integral method for the neutron diffusion equation in cylindrical geometry

    International Nuclear Information System (INIS)

    Azmy, Y.Y.

    1987-01-01

    The nodal methodology is based on retaining a higher a higher degree of analyticity in the process of deriving the discrete-variable equations compared to conventional numerical methods. As a result, extensive numerical testing of nodal methods developed for a wide variety of partial differential equations and comparison of the results to conventional methods have established the superior accuracy of nodal methods on coarse meshes. Moreover, these tests have shown that nodal methods are more computationally efficient than finite difference and finite-element methods in the sense that they require shorter CPU times to achieve comparable accuracy in the solutions. However, nodal formalisms and the final discrete-variable equations they produce are, in general, more complicated than their conventional counterparts. This, together with anticipated difficulties in applying the transverse-averaging procedure in curvilinear coordinates, has limited the applications of nodal methods, so far, to Cartesian geometry, and with additional approximations to hexagonal geometry. In this paper the authors report recent progress in deriving and numerically implementing a nodal integral method (NIM) for solving the neutron diffusion equation in cylindrical r-z geometry. Also, presented are comparisons of numerical solutions to two test problems with those obtained by the Exterminator-2 code, which indicate the superior accuracy of the nodal integral method solutions on much coarser meshes

  6. Exact Mathisson-Papapetrou equations in the Schwarzschild metric with integrals of motion

    International Nuclear Information System (INIS)

    Plyatsko, R.M.; Stefanishin, O.B.

    2011-01-01

    A new representation for exact Mathisson-Papapetrou equations under the Mathisson-Pirani condition in the Schwarzschild gravitational field, which does not contain third-order derivatives with respect to spinning particle coordinates, has been obtained. For this purpose, the integrals of energy and angular momentum of a spinning particle, as well as a differential relation following from the Mathisson-Papapetrou equations for an arbitrary metric, are used.

  7. Reinforcement of a plate weakened by multiple holes with several patches for different types of plate-patch attachment

    KAUST Repository

    Zemlyanova, A.

    2014-01-24

    The most general situation of the reinforcement of a plate with multiple holes by several patches is considered. There is no restriction on the number and the location of the patches. Two types of patch attachment are considered: only along the boundary of the patch or both along the boundary of the patch and the boundaries of the holes which this patch covers. The unattached boundaries of the holes may be loaded with given in-plane stresses. The mechanical problem is reduced to a system of singular integral equations which can be further reduced to a system of Fredholm equations. A new numerical procedure for the solution of the system of singular integral equations is proposed in this paper. It is demonstrated on numerical examples that this procedure has advantages in the case of multiple patches and holes and allows achievement of better numerical convergence with less computational effort.

  8. An approximation method for nonlinear integral equations of Hammerstein type

    International Nuclear Information System (INIS)

    Chidume, C.E.; Moore, C.

    1989-05-01

    The solution of a nonlinear integral equation of Hammerstein type in Hilbert spaces is approximated by means of a fixed point iteration method. Explicit error estimates are given and, in some cases, convergence is shown to be at least as fast as a geometric progression. (author). 25 refs

  9. Magnetostatic fields computed using an integral equation derived from Green's theorems

    International Nuclear Information System (INIS)

    Simkin, J.; Trowbridge, C.W.

    1976-04-01

    A method of computing magnetostatic fields is described that is based on a numerical solution of the integral equation obtained from Green's Theorems. The magnetic scalar potential and its normal derivative on the surfaces of volumes are found by solving a set of linear equations. These are obtained from Green's Second Theorem and the continuity conditions at interfaces between volumes. Results from a two-dimensional computer program are presented and these show the method to be accurate and efficient. (author)

  10. Time-Dependent Heat Conduction Problems Solved by an Integral-Equation Approach

    International Nuclear Information System (INIS)

    Oberaigner, E.R.; Leindl, M.; Antretter, T.

    2010-01-01

    Full text: A classical task of mathematical physics is the formulation and solution of a time dependent thermoelastic problem. In this work we develop an algorithm for solving the time-dependent heat conduction equation c p ρ∂ t T-kT, ii =0 in an analytical, exact fashion for a two-component domain. By the Green's function approach the formal solution of the problem is obtained. As an intermediate result an integral-equation for the temperature history at the domain interface is formulated which can be solved analytically. This method is applied to a classical engineering problem, i.e. to a special case of a Stefan-Problem. The Green's function approach in conjunction with the integral-equation method is very useful in cases were strong discontinuities or jumps occur. The initial conditions and the system parameters of the investigated problem give rise to two jumps in the temperature field. Purely numerical solutions are obtained by using the FEM (finite element method) and the FDM (finite difference method) and compared with the analytical approach. At the domain boundary the analytical solution and the FEM-solution are in good agreement, but the FDM results show a signicant smearing effect. (author)

  11. Direct linearizing transform for three-dimensional discrete integrable systems: the lattice AKP, BKP and CKP equations.

    Science.gov (United States)

    Fu, Wei; Nijhoff, Frank W

    2017-07-01

    A unified framework is presented for the solution structure of three-dimensional discrete integrable systems, including the lattice AKP, BKP and CKP equations. This is done through the so-called direct linearizing transform, which establishes a general class of integral transforms between solutions. As a particular application, novel soliton-type solutions for the lattice CKP equation are obtained.

  12. Dissolution process analysis using model-free Noyes-Whitney integral equation.

    Science.gov (United States)

    Hattori, Yusuke; Haruna, Yoshimasa; Otsuka, Makoto

    2013-02-01

    Drug dissolution process of solid dosages is theoretically described by Noyes-Whitney-Nernst equation. However, the analysis of the process is demonstrated assuming some models. Normally, the model-dependent methods are idealized and require some limitations. In this study, Noyes-Whitney integral equation was proposed and applied to represent the drug dissolution profiles of a solid formulation via the non-linear least squares (NLLS) method. The integral equation is a model-free formula involving the dissolution rate constant as a parameter. In the present study, several solid formulations were prepared via changing the blending time of magnesium stearate (MgSt) with theophylline monohydrate, α-lactose monohydrate, and crystalline cellulose. The formula could excellently represent the dissolution profile, and thereby the rate constant and specific surface area could be obtained by NLLS method. Since the long time blending coated the particle surface with MgSt, it was found that the water permeation was disturbed by its layer dissociating into disintegrant particles. In the end, the solid formulations were not disintegrated; however, the specific surface area gradually increased during the process of dissolution. The X-ray CT observation supported this result and demonstrated that the rough surface was dominant as compared to dissolution, and thus, specific surface area of the solid formulation gradually increased. Copyright © 2012 Elsevier B.V. All rights reserved.

  13. Singular integral equations boundary problems of function theory and their application to mathematical physics

    CERN Document Server

    Muskhelishvili, N I

    2011-01-01

    Singular integral equations play important roles in physics and theoretical mechanics, particularly in the areas of elasticity, aerodynamics, and unsteady aerofoil theory. They are highly effective in solving boundary problems occurring in the theory of functions of a complex variable, potential theory, the theory of elasticity, and the theory of fluid mechanics.This high-level treatment by a noted mathematician considers one-dimensional singular integral equations involving Cauchy principal values. Its coverage includes such topics as the Hölder condition, Hilbert and Riemann-Hilbert problem

  14. RBSDE's with jumps and the related obstacle problems for integral-partial differential equations

    Institute of Scientific and Technical Information of China (English)

    FAN; Yulian

    2006-01-01

    The author proves, when the noise is driven by a Brownian motion and an independent Poisson random measure, the one-dimensional reflected backward stochastic differential equation with a stopping time terminal has a unique solution. And in a Markovian framework, the solution can provide a probabilistic interpretation for the obstacle problem for the integral-partial differential equation.

  15. Choquet Integral of Fuzzy-Number-Valued Functions: The Differentiability of the Primitive with respect to Fuzzy Measures and Choquet Integral Equations

    Directory of Open Access Journals (Sweden)

    Zengtai Gong

    2014-01-01

    Full Text Available This paper deals with the Choquet integral of fuzzy-number-valued functions based on the nonnegative real line. We firstly give the definitions and the characterizations of the Choquet integrals of interval-valued functions and fuzzy-number-valued functions based on the nonadditive measure. Furthermore, the operational schemes of above several classes of integrals on a discrete set are investigated which enable us to calculate Choquet integrals in some applications. Secondly, we give a representation of the Choquet integral of a nonnegative, continuous, and increasing fuzzy-number-valued function with respect to a fuzzy measure. In addition, in order to solve Choquet integral equations of fuzzy-number-valued functions, a concept of the Laplace transformation for the fuzzy-number-valued functions in the sense of Choquet integral is introduced. For distorted Lebesgue measures, it is shown that Choquet integral equations of fuzzy-number-valued functions can be solved by the Laplace transformation. Finally, an example is given to illustrate the main results at the end of the paper.

  16. A Fibonacci collocation method for solving a class of Fredholm–Volterra integral equations in two-dimensional spaces

    Directory of Open Access Journals (Sweden)

    Farshid Mirzaee

    2014-06-01

    Full Text Available In this paper, we present a numerical method for solving two-dimensional Fredholm–Volterra integral equations (F-VIE. The method reduces the solution of these integral equations to the solution of a linear system of algebraic equations. The existence and uniqueness of the solution and error analysis of proposed method are discussed. The method is computationally very simple and attractive. Finally, numerical examples illustrate the efficiency and accuracy of the method.

  17. Numerical solution of integral equations, describing mass spectrum of vector mesons

    International Nuclear Information System (INIS)

    Zhidkov, E.P.; Nikonov, E.G.; Sidorov, A.V.; Skachkov, N.B.; Khoromskij, B.N.

    1988-01-01

    The description of the numerical algorithm for solving quasipotential integral equation in impulse space is presented. The results of numerical computations of the vector meson mass spectrum and the leptonic decay width are given in comparison with the experimental data

  18. A higher order space-time Galerkin scheme for time domain integral equations

    KAUST Repository

    Pray, Andrew J.

    2014-12-01

    Stability of time domain integral equation (TDIE) solvers has remained an elusive goal formany years. Advancement of this research has largely progressed on four fronts: 1) Exact integration, 2) Lubich quadrature, 3) smooth temporal basis functions, and 4) space-time separation of convolutions with the retarded potential. The latter method\\'s efficacy in stabilizing solutions to the time domain electric field integral equation (TD-EFIE) was previously reported for first-order surface descriptions (flat elements) and zeroth-order functions as the temporal basis. In this work, we develop the methodology necessary to extend the scheme to higher order surface descriptions as well as to enable its use with higher order basis functions in both space and time. These basis functions are then used in a space-time Galerkin framework. A number of results are presented that demonstrate convergence in time. The viability of the space-time separation method in producing stable results is demonstrated experimentally for these examples.

  19. A higher order space-time Galerkin scheme for time domain integral equations

    KAUST Repository

    Pray, Andrew J.; Beghein, Yves; Nair, Naveen V.; Cools, Kristof; Bagci, Hakan; Shanker, Balasubramaniam

    2014-01-01

    Stability of time domain integral equation (TDIE) solvers has remained an elusive goal formany years. Advancement of this research has largely progressed on four fronts: 1) Exact integration, 2) Lubich quadrature, 3) smooth temporal basis functions, and 4) space-time separation of convolutions with the retarded potential. The latter method's efficacy in stabilizing solutions to the time domain electric field integral equation (TD-EFIE) was previously reported for first-order surface descriptions (flat elements) and zeroth-order functions as the temporal basis. In this work, we develop the methodology necessary to extend the scheme to higher order surface descriptions as well as to enable its use with higher order basis functions in both space and time. These basis functions are then used in a space-time Galerkin framework. A number of results are presented that demonstrate convergence in time. The viability of the space-time separation method in producing stable results is demonstrated experimentally for these examples.

  20. The ε-form of the differential equations for Feynman integrals in the elliptic case

    Science.gov (United States)

    Adams, Luise; Weinzierl, Stefan

    2018-06-01

    Feynman integrals are easily solved if their system of differential equations is in ε-form. In this letter we show by the explicit example of the kite integral family that an ε-form can even be achieved, if the Feynman integrals do not evaluate to multiple polylogarithms. The ε-form is obtained by a (non-algebraic) change of basis for the master integrals.

  1. Path integral solutions of the master equation. [Lagrangian function, Ehrenfest-type theorem, Cauchy method, inverse functions

    Energy Technology Data Exchange (ETDEWEB)

    Etim, E; Basili, C [Rome Univ. (Italy). Ist. di Matematica

    1978-08-21

    The lagrangian in the path integral solution of the master equation of a stationary Markov process is derived by application of the Ehrenfest-type theorem of quantum mechanics and the Cauchy method of finding inverse functions. Applied to the non-linear Fokker-Planck equation the authors reproduce the result obtained by integrating over Fourier series coefficients and by other methods.

  2. Functional analysis in the study of differential and integral equations

    International Nuclear Information System (INIS)

    Sell, G.R.

    1976-01-01

    This paper illustrates the use of functional analysis in the study of differential equations. Our particular starting point, the theory of flows or dynamical systems, originated with the work of H. Poincare, who is the founder of the qualitative theory of ordinary differential equations. In the qualitative theory one tries to describe the behaviour of a solution, or a collection of solutions, without ''solving'' the differential equation. As a starting point one assumes the existence, and sometimes the uniqueness, of solutions and then one tries to describe the asymptotic behaviour, as time t→+infinity, of these solutions. We compare the notion of a flow with that of a C 0 -group of bounded linear operators on a Banach space. We shall show how the concept C 0 -group, or more generally a C 0 -semigroup, can be used to study the behaviour of solutions of certain differential and integral equations. Our main objective is to show how the concept of a C 0 -group and especially the notion of weak-compactness can be used to prove the existence of an invariant measure for a flow on a compact Hausdorff space. Applications to the theory of ordinary differential equations are included. (author)

  3. High-Order Calderón Preconditioned Time Domain Integral Equation Solvers

    KAUST Repository

    Valdes, Felipe; Ghaffari-Miab, Mohsen; Andriulli, Francesco P.; Cools, Kristof; Michielssen,

    2013-01-01

    Two high-order accurate Calderón preconditioned time domain electric field integral equation (TDEFIE) solvers are presented. In contrast to existing Calderón preconditioned time domain solvers, the proposed preconditioner allows for high-order surface representations and current expansions by using a novel set of fully-localized high-order div-and quasi curl-conforming (DQCC) basis functions. Numerical results demonstrate that the linear systems of equations obtained using the proposed basis functions converge rapidly, regardless of the mesh density and of the order of the current expansion. © 1963-2012 IEEE.

  4. High-Order Calderón Preconditioned Time Domain Integral Equation Solvers

    KAUST Repository

    Valdes, Felipe

    2013-05-01

    Two high-order accurate Calderón preconditioned time domain electric field integral equation (TDEFIE) solvers are presented. In contrast to existing Calderón preconditioned time domain solvers, the proposed preconditioner allows for high-order surface representations and current expansions by using a novel set of fully-localized high-order div-and quasi curl-conforming (DQCC) basis functions. Numerical results demonstrate that the linear systems of equations obtained using the proposed basis functions converge rapidly, regardless of the mesh density and of the order of the current expansion. © 1963-2012 IEEE.

  5. Optimum biasing of integral equations in Monte Carlo calculations

    International Nuclear Information System (INIS)

    Hoogenboom, J.E.

    1979-01-01

    In solving integral equations and estimating average values with the Monte Carlo method, biasing functions may be used to reduce the variancee of the estimates. A simple derivation was used to prove the existence of a zero-variance collision estimator if a specific biasing function and survival probability are applied. This optimum biasing function is the same as that used for the well known zero-variance last-event estimator

  6. Channel surface plasmons in a continuous and flat graphene sheet

    Science.gov (United States)

    Chaves, A. J.; Peres, N. M. R.; da Costa, D. R.; Farias, G. A.

    2018-05-01

    We derive an integral equation describing surface-plasmon polaritons in graphene deposited on a substrate with a planar surface and a dielectric protrusion in the opposite surface of the dielectric slab. We show that the problem is mathematically equivalent to the solution of a Fredholm equation, which we solve exactly. In addition, we show that the dispersion relation of the channel surface plasmons is determined by the geometric parameters of the protrusion alone. We also show that such a system supports both even and odd modes. We give the electrostatic potential and the intensity plot of the electrostatic field, which clearly show the transverse localized nature of the surface plasmons in a continuous and flat graphene sheet.

  7. Explicit solution of Calderon preconditioned time domain integral equations

    KAUST Repository

    Ulku, Huseyin Arda

    2013-07-01

    An explicit marching on-in-time (MOT) scheme for solving Calderon-preconditioned time domain integral equations is proposed. The scheme uses Rao-Wilton-Glisson and Buffa-Christiansen functions to discretize the domain and range of the integral operators and a PE(CE)m type linear multistep to march on in time. Unlike its implicit counterpart, the proposed explicit solver requires the solution of an MOT system with a Gram matrix that is sparse and well-conditioned independent of the time step size. Numerical results demonstrate that the explicit solver maintains its accuracy and stability even when the time step size is chosen as large as that typically used by an implicit solver. © 2013 IEEE.

  8. On the mixed discretization of the time domain magnetic field integral equation

    KAUST Repository

    Ulku, Huseyin Arda; Bogaert, Ignace; Cools, Kristof; Andriulli, Francesco P.; Bagci, Hakan

    2012-01-01

    Time domain magnetic field integral equation (MFIE) is discretized using divergence-conforming Rao-Wilton-Glisson (RWG) and curl-conforming Buffa-Christiansen (BC) functions as spatial basis and testing functions, respectively. The resulting mixed

  9. Cut cancellation in the planar integral equation for the Reggeon

    International Nuclear Information System (INIS)

    Bishari, M.; Veneziano, G.

    1975-01-01

    Planar unitarity for the Reggeon, analyticity and the multi-Regge assumption with cluster production lead to integral equations of the Chew-Goldberger-Low type with separable self-consistent kernel. Contrary to common prejudice, the authors show the existence of solutions exhibiting moving poles and exact, non-perturbative cancellation of the cut. Previously studied consistency conditions are rederived. (Auth.)

  10. The Green's matrix and the boundary integral equations for analysis of time-harmonic dynamics of elastic helical springs.

    Science.gov (United States)

    Sorokin, Sergey V

    2011-03-01

    Helical springs serve as vibration isolators in virtually any suspension system. Various exact and approximate methods may be employed to determine the eigenfrequencies of vibrations of these structural elements and their dynamic transfer functions. The method of boundary integral equations is a meaningful alternative to obtain exact solutions of problems of the time-harmonic dynamics of elastic springs in the framework of Bernoulli-Euler beam theory. In this paper, the derivations of the Green's matrix, of the Somigliana's identities, and of the boundary integral equations are presented. The vibrational power transmission in an infinitely long spring is analyzed by means of the Green's matrix. The eigenfrequencies and the dynamic transfer functions are found by solving the boundary integral equations. In the course of analysis, the essential features and advantages of the method of boundary integral equations are highlighted. The reported analytical results may be used to study the time-harmonic motion in any wave guide governed by a system of linear differential equations in a single spatial coordinate along its axis. © 2011 Acoustical Society of America

  11. Spheroidal Integral Equations for Geodetic Inversion of Geopotential Gradients

    Science.gov (United States)

    Novák, Pavel; Šprlák, Michal

    2018-03-01

    The static Earth's gravitational field has traditionally been described in geodesy and geophysics by the gravitational potential (geopotential for short), a scalar function of 3-D position. Although not directly observable, geopotential functionals such as its first- and second-order gradients are routinely measured by ground, airborne and/or satellite sensors. In geodesy, these observables are often used for recovery of the static geopotential at some simple reference surface approximating the actual Earth's surface. A generalized mathematical model is represented by a surface integral equation which originates in solving Dirichlet's boundary-value problem of the potential theory defined for the harmonic geopotential, spheroidal boundary and globally distributed gradient data. The mathematical model can be used for combining various geopotential gradients without necessity of their re-sampling or prior continuation in space. The model extends the apparatus of integral equations which results from solving boundary-value problems of the potential theory to all geopotential gradients observed by current ground, airborne and satellite sensors. Differences between spherical and spheroidal formulations of integral kernel functions of Green's kind are investigated. Estimated differences reach relative values at the level of 3% which demonstrates the significance of spheroidal approximation for flattened bodies such as the Earth. The observation model can be used for combined inversion of currently available geopotential gradients while exploring their spectral and stochastic characteristics. The model would be even more relevant to gravitational field modelling of other bodies in space with more pronounced spheroidal geometry than that of the Earth.

  12. Integrable Equations of the Form qt=L1(x,t,q,qx,qxx)qxxx+L2(x,t,q,qx,qxx)

    International Nuclear Information System (INIS)

    Satir, Ahmet

    2003-01-01

    Integrable equations of the form q t =L 1 (x,t,q,q x ,q xx )q xxx +L 2 (x,t,q,q x ,q xx ) are considered using linearization. A new type of integrable equations which are the generalization of the integrable equations of Fokas and Ibragimov and Shabat are given

  13. Acidity in DMSO from the embedded cluster integral equation quantum solvation model.

    Science.gov (United States)

    Heil, Jochen; Tomazic, Daniel; Egbers, Simon; Kast, Stefan M

    2014-04-01

    The embedded cluster reference interaction site model (EC-RISM) is applied to the prediction of acidity constants of organic molecules in dimethyl sulfoxide (DMSO) solution. EC-RISM is based on a self-consistent treatment of the solute's electronic structure and the solvent's structure by coupling quantum-chemical calculations with three-dimensional (3D) RISM integral equation theory. We compare available DMSO force fields with reference calculations obtained using the polarizable continuum model (PCM). The results are evaluated statistically using two different approaches to eliminating the proton contribution: a linear regression model and an analysis of pK(a) shifts for compound pairs. Suitable levels of theory for the integral equation methodology are benchmarked. The results are further analyzed and illustrated by visualizing solvent site distribution functions and comparing them with an aqueous environment.

  14. Quadratic algebras applied to noncommutative integration of the Klein-Gordon equation: Four-dimensional quadratic algebras containing three-dimensional nilpotent lie algebras

    International Nuclear Information System (INIS)

    Varaksin, O.L.; Firstov, V.V.; Shapovalov, A.V.

    1995-01-01

    The study is continued on noncommutative integration of linear partial differential equations in application to the exact integration of quantum-mechanical equations in a Riemann space. That method gives solutions to the Klein-Gordon equation when the set of noncommutative symmetry operations for that equation forms a quadratic algebra consisting of one second-order operator and of first-order operators forming a Lie algebra. The paper is a continuation of, where a single nontrivial example is used to demonstrate noncommutative integration of the Klein-Gordon equation in a Riemann space not permitting variable separation

  15. Time-domain single-source integral equations for analyzing scattering from homogeneous penetrable objects

    KAUST Repository

    Valdés, Felipe

    2013-03-01

    Single-source time-domain electric-and magnetic-field integral equations for analyzing scattering from homogeneous penetrable objects are presented. Their temporal discretization is effected by using shifted piecewise polynomial temporal basis functions and a collocation testing procedure, thus allowing for a marching-on-in-time (MOT) solution scheme. Unlike dual-source formulations, single-source equations involve space-time domain operator products, for which spatial discretization techniques developed for standalone operators do not apply. Here, the spatial discretization of the single-source time-domain integral equations is achieved by using the high-order divergence-conforming basis functions developed by Graglia alongside the high-order divergence-and quasi curl-conforming (DQCC) basis functions of Valdés The combination of these two sets allows for a well-conditioned mapping from div-to curl-conforming function spaces that fully respects the space-mapping properties of the space-time operators involved. Numerical results corroborate the fact that the proposed procedure guarantees accuracy and stability of the MOT scheme. © 2012 IEEE.

  16. A calderón-preconditioned single source combined field integral equation for analyzing scattering from homogeneous penetrable objects

    KAUST Repository

    Valdés, Felipe

    2011-06-01

    A new regularized single source equation for analyzing scattering from homogeneous penetrable objects is presented. The proposed equation is a linear combination of a Calderón-preconditioned single source electric field integral equation and a single source magnetic field integral equation. The equation is immune to low-frequency and dense-mesh breakdown, and free from spurious resonances. Unlike dual source formulations, this equation involves operator products that cannot be discretized using standard procedures for discretizing standalone electric, magnetic, and combined field operators. Instead, the single source equation proposed here is discretized using a recently developed technique that achieves a well-conditioned mapping from div- to curl-conforming function spaces, thereby fully respecting the space mapping properties of the operators involved, and guaranteeing accuracy and stability. Numerical results show that the proposed equation and discretization technique give rise to rapidly convergent solutions. They also validate the equation\\'s resonant free character. © 2006 IEEE.

  17. A novel hierarchy of differential—integral equations and their generalized bi-Hamiltonian structures

    International Nuclear Information System (INIS)

    Zhai Yun-Yun; Geng Xian-Guo; He Guo-Liang

    2014-01-01

    With the aid of the zero-curvature equation, a novel integrable hierarchy of nonlinear evolution equations associated with a 3 × 3 matrix spectral problem is proposed. By using the trace identity, the bi-Hamiltonian structures of the hierarchy are established with two skew-symmetric operators. Based on two linear spectral problems, we obtain the infinite many conservation laws of the first member in the hierarchy

  18. Nonlinear integral equations for thermodynamics of the sl(r + 1) Uimin-Sutherland model

    International Nuclear Information System (INIS)

    Tsuboi, Zengo

    2003-01-01

    We derive traditional thermodynamic Bethe ansatz (TBA) equations for the sl(r+1) Uimin-Sutherland model from the T-system of the quantum transfer matrix. These TBA equations are identical to the those from the string hypothesis. Next we derive a new family of nonlinear integral equations (NLIEs). In particular, a subset of these NLIEs forms a system of NLIEs which contains only a finite number of unknown functions. For r=1, this subset of NLIEs reduces to Takahashi's NLIE for the XXX spin chain. A relation between the traditional TBA equations and our new NLIEs is clarified. Based on our new NLIEs, we also calculate the high-temperature expansion of the free energy

  19. An integrable (2+1)-dimensional Toda equation with two discrete variables

    International Nuclear Information System (INIS)

    Cao Cewen; Cao Jianli

    2007-01-01

    An integrable (2+1)-dimensional Toda equation with two discrete variables is presented from the compatible condition of a Lax triad composed of the ZS-AKNS (Zakharov, Shabat; Ablowitz, Kaup, Newell, Segur) eigenvalue problem and two discrete spectral problems. Through the nonlinearization technique, the Lax triad is transformed into a Hamiltonian system and two symplectic maps, respectively, which are integrable in the Liouville sense, sharing the same set of integrals, functionally independent and involutive with each other. In the Jacobi variety of the associated algebraic curve, both the continuous and the discrete flows are straightened out by the Abel-Jacobi coordinates, and are integrated by quadratures. An explicit algebraic-geometric solution in the original variable is obtained by the Riemann-Jacobi inversion

  20. Conservation properties of numerical integration methods for systems of ordinary differential equations

    Science.gov (United States)

    Rosenbaum, J. S.

    1976-01-01

    If a system of ordinary differential equations represents a property conserving system that can be expressed linearly (e.g., conservation of mass), it is then desirable that the numerical integration method used conserve the same quantity. It is shown that both linear multistep methods and Runge-Kutta methods are 'conservative' and that Newton-type methods used to solve the implicit equations preserve the inherent conservation of the numerical method. It is further shown that a method used by several authors is not conservative.

  1. New lumps of Veselov-Novikov integrable nonlinear equation and new exact rational potentials of two-dimensional stationary Schroedinger equation via ∂-macron-dressing method

    International Nuclear Information System (INIS)

    Dubrovsky, V.G.; Formusatik, I.B.

    2003-01-01

    The scheme for calculating via Zakharov-Manakov ∂-macron-dressing method of new rational solutions with constant asymptotic values at infinity of the famous two-dimensional Veselov-Novikov (VN) integrable nonlinear evolution equation and new exact rational potentials of two-dimensional stationary Schroedinger (2DSchr) equation with multiple pole wave functions is developed. As examples new lumps of VN nonlinear equation and new exact rational potentials of 2DSchr equation with multiple pole of order two wave functions are calculated. Among the constructed rational solutions are as nonsingular and also singular

  2. Solution of the Stokes system by boundary integral equations and fixed point iterative schemes

    International Nuclear Information System (INIS)

    Chidume, C.E.; Lubuma, M.S.

    1990-01-01

    The solution to the exterior three dimensional Stokes problem is sought in the form of a single layer potential of unknown density. This reduces the problem to a boundary integral equation of the first kind whose operator is the velocity component of the single layer potential. It is shown that this component is an isomorphism between two appropriate Sobolev spaces containing the unknown densities and the data respectively. The isomorphism corresponds to a variational problem with coercive bilinear form. The latter property allows us to consider various fixed point iterative schemes that converge to the unique solution of the integral equation. Explicit error estimates are also obtained. The successive approximations are also considered in a more computable form by using the product integration method of Atkinson. (author). 47 refs

  3. Stability and square integrability of solutions of nonlinear fourth order differential equations

    Directory of Open Access Journals (Sweden)

    Moussadek Remili

    2016-05-01

    Full Text Available The aim of the present paper is to establish a new result, which guarantees the asymptotic stability of zero solution and square integrability of solutions and their derivatives to nonlinear differential equations of fourth order.

  4. 3-D electromagnetic modeling for very early time sounding of shallow targets using integral equations

    International Nuclear Information System (INIS)

    Xiong, Z.; Tripp, A.C.

    1994-01-01

    This paper presents an integral equation algorithm for 3D EM modeling at high frequencies for applications in engineering an environmental studies. The integral equation method remains the same for low and high frequencies, but the dominant roles of the displacements currents complicate both numerical treatments and interpretations. With singularity extraction technique they successively extended the application of the Hankel filtering technique to the computation of Hankel integrals occurring in high frequency EM modeling. Time domain results are calculated from frequency domain results via Fourier transforms. While frequency domain data are not obvious for interpretations, time domain data show wave-like pictures that resemble seismograms. Both 1D and 3D numerical results show clearly the layer interfaces

  5. Hierarchical Matrices Method and Its Application in Electromagnetic Integral Equations

    Directory of Open Access Journals (Sweden)

    Han Guo

    2012-01-01

    Full Text Available Hierarchical (H- matrices method is a general mathematical framework providing a highly compact representation and efficient numerical arithmetic. When applied in integral-equation- (IE- based computational electromagnetics, H-matrices can be regarded as a fast algorithm; therefore, both the CPU time and memory requirement are reduced significantly. Its kernel independent feature also makes it suitable for any kind of integral equation. To solve H-matrices system, Krylov iteration methods can be employed with appropriate preconditioners, and direct solvers based on the hierarchical structure of H-matrices are also available along with high efficiency and accuracy, which is a unique advantage compared to other fast algorithms. In this paper, a novel sparse approximate inverse (SAI preconditioner in multilevel fashion is proposed to accelerate the convergence rate of Krylov iterations for solving H-matrices system in electromagnetic applications, and a group of parallel fast direct solvers are developed for dealing with multiple right-hand-side cases. Finally, numerical experiments are given to demonstrate the advantages of the proposed multilevel preconditioner compared to conventional “single level” preconditioners and the practicability of the fast direct solvers for arbitrary complex structures.

  6. N-fold Darboux Transformation for Integrable Couplings of AKNS Equations

    Science.gov (United States)

    Yu, Jing; Chen, Shou-Ting; Han, Jing-Wei; Ma, Wen-Xiu

    2018-04-01

    For the integrable couplings of Ablowitz-Kaup-Newell-Segur (ICAKNS) equations, N-fold Darboux transformation (DT) TN, which is a 4 × 4 matrix, is constructed in this paper. Each element of this matrix is expressed by a ratio of the (4N + 1)-order determinant and 4N-order determinant of eigenfunctions. By making use of these formulae, the determinant expressions of N-transformed new solutions p [N], q [N], r [N] and s [N] are generated by this N-fold DT. Furthermore, when the reduced conditions q = ‑p* and s = ‑r* are chosen, we obtain determinant representations of N-fold DT and N-transformed solutions for the integrable couplings of nonlinear Schrödinger (ICNLS) equations. Starting from the zero seed solutions, one-soliton solutions are explicitly given as an example. Supported by the National Natural Science Foundation of China under Grant Nos. 61771174, 11371326, 11371361, 11301454, and 11271168, Natural Science Fund for Colleges and Universities of Jiangsu Province of China under Grant No. 17KJB110020, and General Research Project of Department of Education of Zhejiang Province (Y201636538)

  7. WKB: an interactive code for solving differential equations using phase integral methods

    International Nuclear Information System (INIS)

    White, R.B.

    1978-01-01

    A small code for the analysis of ordinary differential equations interactively through the use of Phase Integral Methods (WKB) has been written for use on the DEC 10. This note is a descriptive manual for those interested in using the code

  8. Approximations to the Probability of Failure in Random Vibration by Integral Equation Methods

    DEFF Research Database (Denmark)

    Nielsen, Søren R.K.; Sørensen, John Dalsgaard

    Close approximations to the first passage probability of failure in random vibration can be obtained by integral equation methods. A simple relation exists between the first passage probability density function and the distribution function for the time interval spent below a barrier before...... passage probability density. The results of the theory agree well with simulation results for narrow banded processes dominated by a single frequency, as well as for bimodal processes with 2 dominating frequencies in the structural response....... outcrossing. An integral equation for the probability density function of the time interval is formulated, and adequate approximations for the kernel are suggested. The kernel approximation results in approximate solutions for the probability density function of the time interval, and hence for the first...

  9. Deterministic methods to solve the integral transport equation in neutronic

    International Nuclear Information System (INIS)

    Warin, X.

    1993-11-01

    We present a synthesis of the methods used to solve the integral transport equation in neutronic. This formulation is above all used to compute solutions in 2D in heterogeneous assemblies. Three kinds of methods are described: - the collision probability method; - the interface current method; - the current coupling collision probability method. These methods don't seem to be the most effective in 3D. (author). 9 figs

  10. On Models with Uncountable Set of Spin Values on a Cayley Tree: Integral Equations

    International Nuclear Information System (INIS)

    Rozikov, Utkir A.; Eshkobilov, Yusup Kh.

    2010-01-01

    We consider models with nearest-neighbor interactions and with the set [0, 1] of spin values, on a Cayley tree of order k ≥ 1. We reduce the problem of describing the 'splitting Gibbs measures' of the model to the description of the solutions of some nonlinear integral equation. For k = 1 we show that the integral equation has a unique solution. In case k ≥ 2 some models (with the set [0, 1] of spin values) which have a unique splitting Gibbs measure are constructed. Also for the Potts model with uncountable set of spin values it is proven that there is unique splitting Gibbs measure.

  11. Time-domain single-source integral equations for analyzing scattering from homogeneous penetrable objects

    KAUST Repository

    Valdé s, Felipe; Andriulli, Francesco P.; Bagci, Hakan; Michielssen, Eric

    2013-01-01

    Single-source time-domain electric-and magnetic-field integral equations for analyzing scattering from homogeneous penetrable objects are presented. Their temporal discretization is effected by using shifted piecewise polynomial temporal basis

  12. A calderón-preconditioned single source combined field integral equation for analyzing scattering from homogeneous penetrable objects

    KAUST Repository

    Valdé s, Felipe; Andriulli, Francesco P.; Bagci, Hakan; Michielssen, Eric

    2011-01-01

    A new regularized single source equation for analyzing scattering from homogeneous penetrable objects is presented. The proposed equation is a linear combination of a Calderón-preconditioned single source electric field integral equation and a

  13. Fast Near-Field Calculation for Volume Integral Equations for Layered Media

    DEFF Research Database (Denmark)

    Kim, Oleksiy S.; Meincke, Peter; Breinbjerg, Olav

    2005-01-01

    . Afterwards, the scattered electric field can be easily computed at a regular rectangular grid on any horizontal plane us-ing a 2-dimensional FFT. This approach provides significant speedup in the near-field calculation in comparison to a straightforward numerical evaluation of the ra-diation integral since......An efficient technique based on the Fast Fourier Transform (FFT) for calculating near-field scattering by dielectric objects in layered media is presented. A higher or-der method of moments technique is employed to solve the volume integral equation for the unknown induced volume current density...

  14. Assessment of available integration algorithms for initial value ordinary differential equations

    International Nuclear Information System (INIS)

    Carver, M.B.; Stewart, D.G.

    1979-11-01

    There exists an extremely large number of algorithms designed for the ordinary differential equation initial value problem. The integration is normally done by a finite sum at time intervals which are chosen dynamically to satisfy an imposed error tolerance. This report describes the basic logistics of the integration process, identifies common areas of difficulty, and establishes a comprehensive test profile for integration algorithms. A number of algorithms are described, and selected published subroutines are evaluated using the test profile. It concludes that an effective library for general use need have only two such routines. The two selected are versions of the well-known Gear and Runge-Kutta-Fehlberg algorithms. Full documentation and listings are included. (auth)

  15. Integral solution for the spherically symmetric Fokker-Planck equation

    International Nuclear Information System (INIS)

    Donoso, J.M.; Soler, M.

    1993-01-01

    We propose an integral method to deal with the spherically symmetric non-linear Fokker-Planck equation appearing in plasma physics. A probability transition expression is obtained, which takes into account the proper domain for the radial velocity component. The analytical and computational results are new, and the time evolution is completely satisfactory. The main achievement of the method is conservation of both the initial norm and energy for unlimited times, which has not been attained in the differential approach to the problem. (orig.)

  16. The Abel symposium 2008 on differential equations: geometry, symmetries and integrability

    CERN Document Server

    Lychagin, Valentin; Straume, Eldar; Abel symposium 2008; Differential equations; Geometry, symmetries and integrability

    2008-01-01

    The Abel Symposium 2008 focused on the modern theory of differential equations and their applications in geometry, mechanics, and mathematical physics. Following the tradition of Monge, Abel and Lie, the scientific program emphasized the role of algebro-geometric methods, which nowadays permeate all mathematical models in natural and engineering sciences. The ideas of invariance and symmetry are of fundamental importance in the geometric approach to differential equations, with a serious impact coming from the area of integrable systems and field theories. This volume consists of original contributions and broad overview lectures of the participants of the Symposium. The papers in this volume present the modern approach to this classical subject.

  17. A purely Lagrangian method for the numerical integration of Fokker-Planck equations

    International Nuclear Information System (INIS)

    Combis, P.; Fronteau, J.

    1986-01-01

    A new numerical approach to Fokker-Planck equations is presented, in which the integration grid moves according to the solution of a differential system. The method is purely Lagrangian, the mean effect of the diffusion being inserted into the differential system itself

  18. Mechanical quadrature method as applied to singular integral equations with logarithmic singularity on the right-hand side

    Science.gov (United States)

    Amirjanyan, A. A.; Sahakyan, A. V.

    2017-08-01

    A singular integral equation with a Cauchy kernel and a logarithmic singularity on its righthand side is considered on a finite interval. An algorithm is proposed for the numerical solution of this equation. The contact elasticity problem of a П-shaped rigid punch indented into a half-plane is solved in the case of a uniform hydrostatic pressure occurring under the punch, which leads to a logarithmic singularity at an endpoint of the integration interval. The numerical solution of this problem shows the efficiency of the proposed approach and suggests that the singularity has to be taken into account in solving the equation.

  19. Convergence Analysis of Generalized Jacobi-Galerkin Methods for Second Kind Volterra Integral Equations with Weakly Singular Kernels

    Directory of Open Access Journals (Sweden)

    Haotao Cai

    2017-01-01

    Full Text Available We develop a generalized Jacobi-Galerkin method for second kind Volterra integral equations with weakly singular kernels. In this method, we first introduce some known singular nonpolynomial functions in the approximation space of the conventional Jacobi-Galerkin method. Secondly, we use the Gauss-Jacobi quadrature rules to approximate the integral term in the resulting equation so as to obtain high-order accuracy for the approximation. Then, we establish that the approximate equation has a unique solution and the approximate solution arrives at an optimal convergence order. One numerical example is presented to demonstrate the effectiveness of the proposed method.

  20. Stochastic integration of the Bethe-Salpeter equation for two bound fermions

    International Nuclear Information System (INIS)

    Salomon, M.

    1988-09-01

    A non-perturbative method using a Monte Carlo algorithm is used to integrate the Bethe-Salpeter equation in momentum space. Solutions for two scalars and two fermions with an arbitrary coupling constant are calculated for bound states in the ladder approximation. The results are compared with other numerical methods. (Author) (13 refs., 2 figs.)

  1. Applications of integral equation methods for the numerical solution of magnetostatic and eddy current problems

    Energy Technology Data Exchange (ETDEWEB)

    Trowbridge, C W

    1976-06-01

    Various integral equation methods are described. For magnetostatic problems three formulations are considered in detail, (a) the direct solution method for the magnetisation distribution in permeable materials, (b) a method based on a scalar potential, and (c) the use of an integral equation derived from Green's Theorem, i.e. the so-called Boundary Integral Method (BIM). In the case of (a) results are given for two-and three-dimensional non-linear problems with comparisons against measurement. For methods (b) and (c), which both lead to a more economical use of the computer than (a), some preliminary results are given for simple cases. For eddy current problems various methods are discussed and some results are given from a computer program based on a vector potential formulation.

  2. Applications of integral equation methods for the numerical solution of magnetostatic and eddy current problems

    International Nuclear Information System (INIS)

    Trowbridge, C.W.

    1976-06-01

    Various integral equation methods are described. For magnetostatic problems three formulations are considered in detail, (a) the direct solution method for the magnetisation distribution in permeable materials, (b) a method based on a scalar potential and (c) the use of an integral equation derived from Green's Theorem, i.e. the so-called Boundary Integral Method (BIM). In the case of (a) results are given for two-and three-dimensional non-linear problems with comparisons against measurement. For methods (b) and (c) which both lead to a more economic use of the computer than (a) some preliminary results are given for simple cases. For eddy current problems various methods are discussed and some results are given from a computer program based on a vector potential formulation. (author)

  3. Numerical Integration of a Class of Singularly Perturbed Delay Differential Equations with Small Shift

    Directory of Open Access Journals (Sweden)

    Gemechis File

    2012-01-01

    Full Text Available We have presented a numerical integration method to solve a class of singularly perturbed delay differential equations with small shift. First, we have replaced the second-order singularly perturbed delay differential equation by an asymptotically equivalent first-order delay differential equation. Then, Simpson’s rule and linear interpolation are employed to get the three-term recurrence relation which is solved easily by discrete invariant imbedding algorithm. The method is demonstrated by implementing it on several linear and nonlinear model examples by taking various values for the delay parameter and the perturbation parameter .

  4. Solution of fractional kinetic equation by a class of integral transform of pathway type

    Science.gov (United States)

    Kumar, Dilip

    2013-04-01

    Solutions of fractional kinetic equations are obtained through an integral transform named Pα-transform introduced in this paper. The Pα-transform is a binomial type transform containing many class of transforms including the well known Laplace transform. The paper is motivated by the idea of pathway model introduced by Mathai [Linear Algebra Appl. 396, 317-328 (2005), 10.1016/j.laa.2004.09.022]. The composition of the transform with differential and integral operators are proved along with convolution theorem. As an illustration of applications to the general theory of differential equations, a simple differential equation is solved by the new transform. Being a new transform, the Pα-transform of some elementary functions as well as some generalized special functions such as H-function, G-function, Wright generalized hypergeometric function, generalized hypergeometric function, and Mittag-Leffler function are also obtained. The results for the classical Laplace transform is retrieved by letting α → 1.

  5. Differential-algebraic integrability analysis of the generalized Riemann type and Korteweg-de Vries hydrodynamical equations

    Energy Technology Data Exchange (ETDEWEB)

    Prykarpatsky, Anatoliy K [Department of Mining Geodesy, AGH University of Science and Technology, Cracow 30059 (Poland); Artemovych, Orest D [Department of Algebra and Topology, Faculty of Mathematics and Informatics of the Vasyl Stefanyk Pre-Carpathian National University, Ivano-Frankivsk (Ukraine); Popowicz, Ziemowit [Institute of Theoretical Physics, University of Wroclaw (Poland); Pavlov, Maxim V, E-mail: pryk.anat@ua.f, E-mail: artemo@usk.pk.edu.p, E-mail: ziemek@ift.uni.wroc.p, E-mail: M.V.Pavlov@lboro.ac.u [Department of Mathematical Physics, P.N. Lebedev Physical Institute, 53 Leninskij Prospekt, Moscow 119991 (Russian Federation)

    2010-07-23

    A differential-algebraic approach to studying the Lax-type integrability of the generalized Riemann-type hydrodynamic equations at N = 3, 4 is devised. The approach is also applied to studying the Lax-type integrability of the well-known Korteweg-de Vries dynamical system.

  6. Differential-algebraic integrability analysis of the generalized Riemann type and Korteweg-de Vries hydrodynamical equations

    International Nuclear Information System (INIS)

    Prykarpatsky, Anatoliy K; Artemovych, Orest D; Popowicz, Ziemowit; Pavlov, Maxim V

    2010-01-01

    A differential-algebraic approach to studying the Lax-type integrability of the generalized Riemann-type hydrodynamic equations at N = 3, 4 is devised. The approach is also applied to studying the Lax-type integrability of the well-known Korteweg-de Vries dynamical system.

  7. REFLECT: a program to integrate the wave equation through a plane stratified plasma

    International Nuclear Information System (INIS)

    Greene, J.W.

    1975-01-01

    A program was developed to integrate the wave equation through a plane stratified plasma with a general density distribution. The reflection and transmission of a plane wave are computed as a function of the angle of incidence. The polarization of the electric vector is assumed to be perpendicular to the plane of incidence. The model for absorption by classical inverse bremsstrahlung avoids the improper extrapolation of underdense formulae that are singular at the plasma critical surface. Surprisingly good agreement with the geometric-optics analysis of a linear layer was found. The system of ordinary differential equations is integrated by the variable-step, variable-order Adams method in the Lawrence Livermore Laboratory Gear package. Parametric studies of the absorption are summarized, and some possibilities for further development of the code are discussed. (auth)

  8. Determination of the Geometric Form of a Plane of a Tectonic Gap as the Inverse III-posed Problem of Mathematical Physics

    Science.gov (United States)

    Sirota, Dmitry; Ivanov, Vadim

    2017-11-01

    Any mining operations influence stability of natural and technogenic massifs are the reason of emergence of the sources of differences of mechanical tension. These sources generate a quasistationary electric field with a Newtonian potential. The paper reviews the method of determining the shape and size of a flat source field with this kind of potential. This common problem meets in many fields of mining: geological exploration mineral resources, ore deposits, control of mining by underground method, determining coal self-heating source, localization of the rock crack's sources and other applied problems of practical physics. This problems are ill-posed and inverse and solved by converting to Fredholm-Uryson integral equation of the first kind. This equation will be solved by A.N. Tikhonov regularization method.

  9. Consistency of direct integral estimator for partially observed systems of ordinary differential equations

    NARCIS (Netherlands)

    Vujačić, Ivan; Dattner, Itai

    In this paper we use the sieve framework to prove consistency of the ‘direct integral estimator’ of parameters for partially observed systems of ordinary differential equations, which are commonly used for modeling dynamic processes.

  10. On the initial condition problem of the time domain PMCHWT surface integral equation

    KAUST Repository

    Uysal, Ismail Enes; Bagci, Hakan; Ergin, A. Arif; Ulku, H. Arda

    2017-01-01

    Non-physical, linearly increasing and constant current components are induced in marching on-in-time solution of time domain surface integral equations when initial conditions on time derivatives of (unknown) equivalent currents are not enforced

  11. Numerical Solution of Nonlinear Volterra Integral Equations System Using Simpson’s 3/8 Rule

    Directory of Open Access Journals (Sweden)

    Adem Kılıçman

    2012-01-01

    Full Text Available The Simpson’s 3/8 rule is used to solve the nonlinear Volterra integral equations system. Using this rule the system is converted to a nonlinear block system and then by solving this nonlinear system we find approximate solution of nonlinear Volterra integral equations system. One of the advantages of the proposed method is its simplicity in application. Further, we investigate the convergence of the proposed method and it is shown that its convergence is of order O(h4. Numerical examples are given to show abilities of the proposed method for solving linear as well as nonlinear systems. Our results show that the proposed method is simple and effective.

  12. Three semi-direct sum Lie algebras and three discrete integrable couplings associated with the modified K dV lattice equation

    International Nuclear Information System (INIS)

    Yu Zhang; Zhang Yufeng

    2009-01-01

    Three semi-direct sum Lie algebras are constructed, which is an efficient and new way to obtain discrete integrable couplings. As its applications, three discrete integrable couplings associated with the modified K dV lattice equation are worked out. The approach can be used to produce other discrete integrable couplings of the discrete hierarchies of soliton equations.

  13. An exterior Poisson solver using fast direct methods and boundary integral equations with applications to nonlinear potential flow

    Science.gov (United States)

    Young, D. P.; Woo, A. C.; Bussoletti, J. E.; Johnson, F. T.

    1986-01-01

    A general method is developed combining fast direct methods and boundary integral equation methods to solve Poisson's equation on irregular exterior regions. The method requires O(N log N) operations where N is the number of grid points. Error estimates are given that hold for regions with corners and other boundary irregularities. Computational results are given in the context of computational aerodynamics for a two-dimensional lifting airfoil. Solutions of boundary integral equations for lifting and nonlifting aerodynamic configurations using preconditioned conjugate gradient are examined for varying degrees of thinness.

  14. Magnetic field integral equation analysis of surface plasmon scattering by rectangular dielectric channel discontinuities.

    Science.gov (United States)

    Chremmos, Ioannis

    2010-01-01

    The scattering of a surface plasmon polariton (SPP) by a rectangular dielectric channel discontinuity is analyzed through a rigorous magnetic field integral equation method. The scattering phenomenon is formulated by means of the magnetic-type scalar integral equation, which is subsequently treated through an entire-domain Galerkin method of moments (MoM), based on a Fourier-series plane wave expansion of the magnetic field inside the discontinuity. The use of Green's function Fourier transform allows all integrations over the area and along the boundary of the discontinuity to be performed analytically, resulting in a MoM matrix with entries that are expressed as spectral integrals of closed-form expressions. Complex analysis techniques, such as Cauchy's residue theorem and the saddle-point method, are applied to obtain the amplitudes of the transmitted and reflected SPP modes and the radiated field pattern. Through numerical results, we examine the wavelength selectivity of transmission and reflection against the channel dimensions as well as the sensitivity to changes in the refractive index of the discontinuity, which is useful for sensing applications.

  15. On a method for constructing the Lax pairs for nonlinear integrable equations

    International Nuclear Information System (INIS)

    Habibullin, I T; Poptsova, M N; Khakimova, A R

    2016-01-01

    We suggest a direct algorithm for searching the Lax pairs for nonlinear integrable equations. It is effective for both continuous and discrete models. The first operator of the Lax pair corresponding to a given nonlinear equation is found immediately, coinciding with the linearization of the considered nonlinear equation. The second one is obtained as an invariant manifold to the linearized equation. A surprisingly simple relation between the second operator of the Lax pair and the recursion operator is discussed: the recursion operator can immediately be found from the Lax pair. Examples considered in the article are convincing evidence that the found Lax pairs differ from the classical ones. The examples also show that the suggested objects are true Lax pairs which allow the construction of infinite series of conservation laws and hierarchies of higher symmetries. In the case of the hyperbolic type partial differential equation our algorithm is slightly modified; in order to construct the Lax pairs from the invariant manifolds we use the cutting off conditions for the corresponding infinite Laplace sequence. The efficiency of the method is illustrated by application to some equations given in the Svinolupov–Sokolov classification list for which the Lax pairs and the recursion operators have not been found earlier. (paper)

  16. Some applications of perturbation theory to numerical integration methods for the Schroedinger equation

    International Nuclear Information System (INIS)

    Killingbeck, J.

    1979-01-01

    By using the methods of perturbation theory it is possible to construct simple formulae for the numerical integration of the Schroedinger equation, and also to calculate expectation values solely by means of simple eigenvalue calculations. (Auth.)

  17. Variance analysis of the Monte-Carlo perturbation source method in inhomogeneous linear particle transport problems

    International Nuclear Information System (INIS)

    Noack, K.

    1982-01-01

    The perturbation source method may be a powerful Monte-Carlo means to calculate small effects in a particle field. In a preceding paper we have formulated this methos in inhomogeneous linear particle transport problems describing the particle fields by solutions of Fredholm integral equations and have derived formulae for the second moment of the difference event point estimator. In the present paper we analyse the general structure of its variance, point out the variance peculiarities, discuss the dependence on certain transport games and on generation procedures of the auxiliary particles and draw conclusions to improve this method

  18. Development of steel head joints with fiberglass sucker rod on the base of contact stresses investigation

    Energy Technology Data Exchange (ETDEWEB)

    Kopey, B.V.; Kopey, L.B. [Ivano-Frankivsk State Technical Oil and Gas University (Ukraine); Maksymuk, A.V.; Shcherbyna, N.M. [National Ukrainian Academy of Sciences (Ukraine)

    1998-12-31

    The methods of calculation of contact stresses during cylinder shell tube - steel bandage interaction are presented. Tymoshenko`s generalized theory of shells serves as a basis for investigating steel head to fiberglass sucker rod joint strength. This theory allows to consider mechanical performance of composite materials. The problem is reduced to solving Fredholm integral equation of second degree. The numeric analysis is performed. Several joints of composite body with steel head are proposed. The full-size sucker rod fatigue tests are performed to determine the fatigue limit under the bending and axial cyclic loads in the medium of oil well fluids. (orig.)

  19. Reduction of infinite dimensional equations

    Directory of Open Access Journals (Sweden)

    Zhongding Li

    2006-02-01

    Full Text Available In this paper, we use the general Legendre transformation to show the infinite dimensional integrable equations can be reduced to a finite dimensional integrable Hamiltonian system on an invariant set under the flow of the integrable equations. Then we obtain the periodic or quasi-periodic solution of the equation. This generalizes the results of Lax and Novikov regarding the periodic or quasi-periodic solution of the KdV equation to the general case of isospectral Hamiltonian integrable equation. And finally, we discuss the AKNS hierarchy as a special example.

  20. Hamiltonian structures and integrability for a discrete coupled KdV-type equation hierarchy

    International Nuclear Information System (INIS)

    Zhao Haiqiong; Zhu Zuonong; Zhang Jingli

    2011-01-01

    Coupled Korteweg-de Vries (KdV) systems have many important physical applications. By considering a 4 × 4 spectral problem, we derive a discrete coupled KdV-type equation hierarchy. Our hierarchy includes the coupled Volterra system proposed by Lou et al. (e-print arXiv: 0711.0420) as the first member which is a discrete version of the coupled KdV equation. We also investigate the integrability in the Liouville sense and the multi-Hamiltonian structures for the obtained hierarchy. (authors)

  1. Applied Mathematical Methods in Theoretical Physics

    Science.gov (United States)

    Masujima, Michio

    2005-04-01

    All there is to know about functional analysis, integral equations and calculus of variations in a single volume. This advanced textbook is divided into two parts: The first on integral equations and the second on the calculus of variations. It begins with a short introduction to functional analysis, including a short review of complex analysis, before continuing a systematic discussion of different types of equations, such as Volterra integral equations, singular integral equations of Cauchy type, integral equations of the Fredholm type, with a special emphasis on Wiener-Hopf integral equations and Wiener-Hopf sum equations. After a few remarks on the historical development, the second part starts with an introduction to the calculus of variations and the relationship between integral equations and applications of the calculus of variations. It further covers applications of the calculus of variations developed in the second half of the 20th century in the fields of quantum mechanics, quantum statistical mechanics and quantum field theory. Throughout the book, the author presents over 150 problems and exercises -- many from such branches of physics as quantum mechanics, quantum statistical mechanics, and quantum field theory -- together with outlines of the solutions in each case. Detailed solutions are given, supplementing the materials discussed in the main text, allowing problems to be solved making direct use of the method illustrated. The original references are given for difficult problems. The result is complete coverage of the mathematical tools and techniques used by physicists and applied mathematicians Intended for senior undergraduates and first-year graduates in science and engineering, this is equally useful as a reference and self-study guide.

  2. Integrability of systems of two second-order ordinary differential equations admitting four-dimensional Lie algebras.

    Science.gov (United States)

    Gainetdinova, A A; Gazizov, R K

    2017-01-01

    We suggest an algorithm for integrating systems of two second-order ordinary differential equations with four symmetries. In particular, if the admitted transformation group has two second-order differential invariants, the corresponding system can be integrated by quadratures using invariant representation and the operator of invariant differentiation. Otherwise, the systems reduce to partially uncoupled forms and can also be integrated by quadratures.

  3. An integrated approach to determine phenomenological equations in metallic systems

    Science.gov (United States)

    Ghamarian, Iman

    It is highly desirable to be able to make predictions of properties in metallic materials based upon the composition of the material and the microstructure. Unfortunately, the complexity of real, multi-component, multi-phase engineering alloys makes the provision of constituent-based (i.e., composition or microstructure) phenomenological equations extremely difficult. Due to these difficulties, qualitative predictions are frequently used to study the influence of microstructure or composition on the properties. Neural networks were used as a tool to get a quantitative model from a database. However, the developed model is not a phenomenological model. In this study, a new method based upon the integration of three separate modeling approaches, specifically artificial neural networks, genetic algorithms, and monte carlo was proposed. These three methods, when coupled in the manner described in this study, allows for the extraction of phenomenological equations with a concurrent analysis of uncertainty. This approach has been applied to a multi-component, multi-phase microstructure exhibiting phases with varying spatial and morphological distributions. Specifically, this approach has been applied to derive a phenomenological equation for the prediction of yield strength in alpha+beta processed Ti-6-4. The equation is consistent with not only the current dataset but also, where available, the limited information regarding certain parameters such as intrinsic yield strength of pure hexagonal close-packed alpha titanium.

  4. The effect of a curvature-dependent surface tension on the singularities at the tips of a straight interface crack

    KAUST Repository

    Zemlyanova, A. Y.

    2013-03-08

    A problem of an interface crack between two semi-planes made out of different materials under an action of an in-plane loading of general tensile-shear type is treated in a semi-analytical manner with the help of Dirichlet-to-Neumann mappings. The boundaries of the crack and the interface between semi-planes are subjected to a curvature-dependent surface tension. The resulting system of six singular integro-differential equations is reduced to the system of three Fredholm equations. It is shown that the introduction of the curvature-dependent surface tension eliminates both classical integrable power singularity of the order 1/2 and an oscillating singularity present in a classical linear elasticity solutions. The numerical results are obtained by solving the original system of singular integro-differential equations by approximating unknown functions with Taylor polynomials. © 2013 The Author.

  5. Spectral Properties of Homogeneous and Nonhomogeneous Radar Images

    DEFF Research Database (Denmark)

    Madsen, Søren Nørvang

    1987-01-01

    On the basis of a two-dimensional, nonstationary white noisemodel for the complex radar backscatter, the spectral properties ofa one-look synthetic-aperture radar (SAR) system is derived. It isshown that the power spectrum of the complex SAR image is sceneindependent. It is also shown that the sp......On the basis of a two-dimensional, nonstationary white noisemodel for the complex radar backscatter, the spectral properties ofa one-look synthetic-aperture radar (SAR) system is derived. It isshown that the power spectrum of the complex SAR image is sceneindependent. It is also shown...... that the spectrum of the intensityimage is in general related to the radar scene spectrum by a linearintegral equation, a Fredholm's integral equation of the third kind.Under simplifying assumptions, a closed-form equation giving theradar scene spectrum as a function of the SAR image spectrum canbe derived....

  6. Asymptotically Stable Solutions of a Generalized Fractional Quadratic Functional-Integral Equation of Erdélyi-Kober Type

    Directory of Open Access Journals (Sweden)

    Mohamed Abdalla Darwish

    2014-01-01

    Full Text Available We study a generalized fractional quadratic functional-integral equation of Erdélyi-Kober type in the Banach space BC(ℝ+. We show that this equation has at least one asymptotically stable solution.

  7. On a numereeical method for solving the Faddv integral equation without deformation of contour

    International Nuclear Information System (INIS)

    Belyaev, V.O.; Moller, K.

    1976-01-01

    A numerical method is proposed for solving the Faddeev equation for separable potentials at positive total energy. The method is based on the fact that after applying a simple interpolation procedure the logarithmic singularities in the kernel of the integral equation can be extracted in the same way as usually the pole singularity is extracted. The method has been applied to calculate the eigenvalues of the Faddeev kernel

  8. Numerical evaluation of path-integral solutions to Fokker-Planck equations. II. Restricted stochastic processes

    International Nuclear Information System (INIS)

    Wehner, M.F.

    1983-01-01

    A path-integral solution is derived for processes described by nonlinear Fokker-Plank equations together with externally imposed boundary conditions. This path-integral solution is written in the form of a path sum for small time steps and contains, in addition to the conventional volume integral, a surface integral which incorporates the boundary conditions. A previously developed numerical method, based on a histogram representation of the probability distribution, is extended to a trapezoidal representation. This improved numerical approach is combined with the present path-integral formalism for restricted processes and is show t give accurate results. 35 refs., 5 figs

  9. On a functional equation related to the intermediate long wave equation

    International Nuclear Information System (INIS)

    Hone, A N W; Novikov, V S

    2004-01-01

    We resolve an open problem stated by Ablowitz et al (1982 J. Phys. A: Math. Gen. 15 781) concerning the integral operator appearing in the intermediate long wave equation. We explain how this is resolved using the perturbative symmetry approach introduced by one of us with Mikhailov. By solving a certain functional equation, we prove that the intermediate long wave equation and the Benjamin-Ono equation are the unique integrable cases within a particular class of integro-differential equations. Furthermore, we explain how the perturbative symmetry approach is naturally extended to treat equations on a periodic domain. (letter to the editor)

  10. Symmetries, integrals, and three-dimensional reductions of Plebanski's second heavenly equation

    International Nuclear Information System (INIS)

    Neyzi, F.; Sheftel, M. B.; Yazici, D.

    2007-01-01

    We study symmetries and conservation laws for Plebanski's second heavenly equation written as a first-order nonlinear evolutionary system which admits a multi-Hamiltonian structure. We construct an optimal system of one-dimensional subalgebras and all inequivalent three-dimensional symmetry reductions of the original four-dimensional system. We consider these two-component evolutionary systems in three dimensions as natural candidates for integrable systems

  11. To the complete integrability of long-wave short-wave interaction equations

    International Nuclear Information System (INIS)

    Roy Chowdhury, A.; Chanda, P.K.

    1984-10-01

    We show that the non-linear partial differential equations governing the interaction of long and short waves are completely integrable. The methodology we use is that of Ablowitz et al. though in the last section of our paper we have discussed the problem also in the light of the procedure due to Weiss et al. and have obtained a Baecklund transformation. (author)

  12. A family of integrable differential–difference equations, its bi-Hamiltonian structure and binary nonlinearization of the Lax pairs and adjoint Lax pairs

    International Nuclear Information System (INIS)

    Xu Xixiang

    2012-01-01

    Highlights: ► We deduce a family of integrable differential–difference equations. ► We present a discrete Hamiltonian operator involving two arbitrary real parameters. ► We establish the bi-Hamiltonian structure for obtained integrable family. ► Liouvolle integrability of the obtained family is demonstrated. ► Every equation in obtained family is factored through the binary nonlinearization. - Abstract: A family of integrable differential–difference equations is derived by the method of Lax pairs. A discrete Hamiltonian operator involving two arbitrary real parameters is introduced. When the parameters are suitably selected, a pair of discrete Hamiltonian operators is presented. Bi-Hamiltonian structure of obtained family is established by discrete trace identity. Then, Liouville integrability for the obtained family is proved. Ultimately, through the binary nonlinearization of the Lax pairs and adjoint Lax pairs, every differential–difference equation in obtained family is factored by an integrable symplectic map and a finite-dimensional integrable system in Liouville sense.

  13. Analytical solutions of linear diffusion and wave equations in semi-infinite domains by using a new integral transform

    Directory of Open Access Journals (Sweden)

    Gao Lin

    2017-01-01

    Full Text Available Recently, a new integral transform similar to Sumudu transform has been proposed by Yang [1]. Some of the properties of the integral transform are expanded in the present article. Meanwhile, new applications to the linear wave and diffusion equations in semi-infinite domains are discussed in detail. The proposed method provides an alternative approach to solve the partial differential equations in mathematical physics.

  14. Local instant conservation equations

    International Nuclear Information System (INIS)

    Delaje, Dzh.

    1984-01-01

    Local instant conservation equations for two-phase flow are derived. Derivation of the equation starts from the recording of integral laws of conservation for a fixed reference volume, containing both phases. Transformation of the laws, using the Leibniz rule and Gauss theory permits to obtain the sum of two integrals as to the volume and integral as to the surface. Integrals as to the volume result in local instant differential equations, in particular derivatives for each phase, and integrals as to the surface reflect local instant conditions of a jump on interface surface

  15. Combined Helmholtz Integral Equation - Fourier series formulation of acoustical radiation and scattering problems

    CSIR Research Space (South Africa)

    Fedotov, I

    2006-07-01

    Full Text Available The Combined Helmholtz Integral Equation – Fourier series Formulation (CHIEFF) is based on representation of a velocity potential in terms of Fourier series and finding the Fourier coefficients of this expansion. The solution could be substantially...

  16. Bound states of quarks calculated with stochastic integration of the Bethe-Salpeter equation

    International Nuclear Information System (INIS)

    Salomon, M.

    1992-07-01

    We have computed the masses, wave functions and sea quark content of mesons in their ground state by integrating the Bethe-Salpeter equation with a stochastic algorithm. This method allows the inclusion of a large set of diagrams. Inspection of the kernel of the equation shows that q-q-bar pairs with similar constituent masses in a singlet spin state exhibit a high bound state which is not present in other pairs. The pion, kaon and eta belongs to this category. 19 refs., 2 figs., 2 tabs

  17. Transient analysis of scattering from ferromagnetic objects using Landau-Lifshitz-Gilbert and volume integral equations

    KAUST Repository

    Sayed, Sadeed Bin

    2016-11-02

    An explicit marching on-in-time scheme for analyzing transient electromagnetic wave interactions on ferromagnetic scatterers is described. The proposed method solves a coupled system of time domain magnetic field volume integral and Landau-Lifshitz-Gilbert (LLG) equations. The unknown fluxes and fields are discretized using full and half Schaubert-Wilton-Glisson functions in space and bandlimited temporal interpolation functions in time. The coupled system is cast in the form of an ordinary differential equation and integrated in time using a PE(CE)m type linear multistep method to obtain the unknown expansion coefficients. Numerical results demonstrating the stability and accuracy of the proposed scheme are presented.

  18. Transient analysis of scattering from ferromagnetic objects using Landau-Lifshitz-Gilbert and volume integral equations

    KAUST Repository

    Sayed, Sadeed Bin; Ulku, Huseyin Arda; Bagci, Hakan

    2016-01-01

    An explicit marching on-in-time scheme for analyzing transient electromagnetic wave interactions on ferromagnetic scatterers is described. The proposed method solves a coupled system of time domain magnetic field volume integral and Landau-Lifshitz-Gilbert (LLG) equations. The unknown fluxes and fields are discretized using full and half Schaubert-Wilton-Glisson functions in space and bandlimited temporal interpolation functions in time. The coupled system is cast in the form of an ordinary differential equation and integrated in time using a PE(CE)m type linear multistep method to obtain the unknown expansion coefficients. Numerical results demonstrating the stability and accuracy of the proposed scheme are presented.

  19. Two hierarchies of multi-component Kaup-Newell equations and theirs integrable couplings

    International Nuclear Information System (INIS)

    Zhu Fubo; Ji Jie; Zhang Jianbin

    2008-01-01

    Two hierarchies of multi-component Kaup-Newell equations are derived from an arbitrary order matrix spectral problem, including positive non-isospectral Kaup-Newell hierarchy and negative non-isospectral Kaup-Newell hierarchy. Moreover, new integrable couplings of the resulting Kaup-Newell soliton hierarchies are constructed by enlarging the associated matrix spectral problem

  20. Phase integral approximation for coupled ordinary differential equations of the Schroedinger type

    International Nuclear Information System (INIS)

    Skorupski, Andrzej A.

    2008-01-01

    Four generalizations of the phase integral approximation (PIA) to sets of ordinary differential equations of Schroedinger type [u j '' (x)+Σ k=1 N R jk (x)u k (x)=0, j=1,2,...,N] are described. The recurrence relations for higher order corrections are given in a form valid to arbitrary order and for the matrix R(x)[≡(R jk (x))] either Hermitian or non-Hermitian. For Hermitian and negative definite R(x) matrices, a Wronskian conserving PIA theory is formulated, which generalizes Fulling's current conserving theory pertinent to positive definite R(x) matrices. The idea of a modification of the PIA, which is well known for one equation [u '' (x)+R(x)u(x)=0], is generalized to sets. A simplification of Wronskian or current conserving theories is proposed which in each order eliminates one integration from the formulas for higher order corrections. If the PIA is generated by a nondegenerate eigenvalue of the R(x) matrix, the eliminated integration is the only one present. In that case, the simplified theory becomes fully algorithmic and is generalized to non-Hermitian R(x) matrices. The general theory is illustrated by a few examples automatically generated by using the author's program in MATHEMATICA published in e-print arXiv:0710.5406 [math-ph

  1. Time-integration methods for finite element discretisations of the second-order Maxwell equation

    NARCIS (Netherlands)

    Sarmany, D.; Bochev, Mikhail A.; van der Vegt, Jacobus J.W.

    This article deals with time integration for the second-order Maxwell equations with possibly non-zero conductivity in the context of the discontinuous Galerkin finite element method DG-FEM) and the $H(\\mathrm{curl})$-conforming FEM. For the spatial discretisation, hierarchic

  2. On the use of a variational technique based on integral equations for plane acoustic and vibro-acoustic problems

    DEFF Research Database (Denmark)

    Brunskog, Jonas; Richard, Antoine Philippe André

    2016-01-01

    Problems such as sound insulation and absorption of plane structures in laboratory conditions can theoretically be described as an integral or integral-differential equation. This equation contains the Green’s function integrated over the surface, which describes the radiation from the surface....... A variational technique, well described by Morse and Ingard, has successfully been used for both absorption and sound insulation for a plane incident wave. The resulting formulas are surprisingly simple, accurate and robust. Moreover, they capture the physics of sound radiation of a finite surface well. However...

  3. Soliton solutions of an integrable nonlinear Schrödinger equation with quintic terms.

    Science.gov (United States)

    Chowdury, A; Kedziora, D J; Ankiewicz, A; Akhmediev, N

    2014-09-01

    We present the fifth-order equation of the nonlinear Schrödinger hierarchy. This integrable partial differential equation contains fifth-order dispersion and nonlinear terms related to it. We present the Lax pair and use Darboux transformations to derive exact expressions for the most representative soliton solutions. This set includes two-soliton collisions and the degenerate case of the two-soliton solution, as well as beating structures composed of two or three solitons. Ultimately, the new quintic operator and the terms it adds to the standard nonlinear Schrödinger equation (NLSE) are found to primarily affect the velocity of solutions, with complicated flow-on effects. Furthermore, we present a new structure, composed of coincident equal-amplitude solitons, which cannot exist for the standard NLSE.

  4. On the integration of equations of motion for particle-in-cell codes

    Czech Academy of Sciences Publication Activity Database

    Fuchs, Vladimír; Gunn, J. P.

    2006-01-01

    Roč. 214, - (2006), s. 299-315 ISSN 0021-9991 R&D Projects: GA ČR GA202/04/0360 Institutional research plan: CEZ:AV0Z20430508 Keywords : Equations of motion * 2nd order integration methods * nonlinear oscillations Subject RIV: BM - Solid Matter Physics ; Magnetism Impact factor: 2.328, year: 2006

  5. Killing spinor equations in dimension 7 and geometry of integrable G2-manifolds

    International Nuclear Information System (INIS)

    Friedrich, Thomas; Ivanov, Stefan

    2001-12-01

    We compute the scalar curvature of 7-dimensional G 2 -manifolds admitting a connection with totally skew-symmetric torsion. We prove the formula for the general solution of the Killing spinor equation and express the Riemannian scalar curvature of the solution in terms of the dilation function and the NS 3-form field. In dimension n=7 the dilation function involved in the second fermionic string equation has an interpretation as a conformal change of the underlying integrable G 2 -structure into a cocalibrated one of pure type W 3 . (author)

  6. Wavelet-like bases for thin-wire integral equations in electromagnetics

    Science.gov (United States)

    Francomano, E.; Tortorici, A.; Toscano, E.; Ala, G.; Viola, F.

    2005-03-01

    In this paper, wavelets are used in solving, by the method of moments, a modified version of the thin-wire electric field integral equation, in frequency domain. The time domain electromagnetic quantities, are obtained by using the inverse discrete fast Fourier transform. The retarded scalar electric and vector magnetic potentials are employed in order to obtain the integral formulation. The discretized model generated by applying the direct method of moments via point-matching procedure, results in a linear system with a dense matrix which have to be solved for each frequency of the Fourier spectrum of the time domain impressed source. Therefore, orthogonal wavelet-like basis transform is used to sparsify the moment matrix. In particular, dyadic and M-band wavelet transforms have been adopted, so generating different sparse matrix structures. This leads to an efficient solution in solving the resulting sparse matrix equation. Moreover, a wavelet preconditioner is used to accelerate the convergence rate of the iterative solver employed. These numerical features are used in analyzing the transient behavior of a lightning protection system. In particular, the transient performance of the earth termination system of a lightning protection system or of the earth electrode of an electric power substation, during its operation is focused. The numerical results, obtained by running a complex structure, are discussed and the features of the used method are underlined.

  7. New Positive and Negative Hierarchies of Integrable Differential-Difference Equations and Conservation Laws

    International Nuclear Information System (INIS)

    Li Xinyue; Zhao Qiulan

    2009-01-01

    Two hierarchies of nonlinear integrable positive and negative lattice equations are derived from a discrete spectral problem. The two lattice hierarchies are proved to have discrete zero curvature representations associated with a discrete spectral problem, which also shows that the positive and negative hierarchies correspond to positive and negative power expansions of Lax operators with respect to the spectral parameter, respectively. Moreover, the integrable lattice models in the positive hierarchy are of polynomial type, and the integrable lattice models in the negative hierarchy are of rational type. Further, we construct infinite conservation laws about the positive hierarchy.

  8. Low-frequency scaling of the standard and mixed magnetic field and Müller integral equations

    KAUST Repository

    Bogaert, Ignace

    2014-02-01

    The standard and mixed discretizations for the magnetic field integral equation (MFIE) and the Müller integral equation (MUIE) are investigated in the context of low-frequency (LF) scattering problems involving simply connected scatterers. It is proved that, at low frequencies, the frequency scaling of the nonsolenoidal part of the solution current can be incorrect for the standard discretization. In addition, it is proved that the frequency scaling obtained with the mixed discretization is correct. The reason for this problem in the standard discretization scheme is the absence of exact solenoidal currents in the rotated RWG finite element space. The adoption of the mixed discretization scheme eliminates this problem and leads to a well-conditioned system of linear equations that remains accurate at low frequencies. Numerical results confirm these theoretical predictions and also show that, when the frequency is lowered, a finer and finer mesh is required to keep the accuracy constant with the standard discretization. © 1963-2012 IEEE.

  9. An integral geometry lemma and its applications: The nonlocality of the Pavlov equation and a tomographic problem with opaque parabolic objects

    Science.gov (United States)

    Grinevich, P. G.; Santini, P. M.

    2016-10-01

    Written in the evolutionary form, the multidimensional integrable dispersionless equations, exactly like the soliton equations in 2+1 dimensions, become nonlocal. In particular, the Pavlov equation is brought to the form v t = v x v y - ∂ x -1 ∂ y [ v y + v x 2], where the formal integral ∂ x -1 becomes the asymmetric integral - int_x^∞ {dx'} . We show that this result could be guessed using an apparently new integral geometry lemma. It states that the integral of a sufficiently general smooth function f( X, Y) over a parabola in the plane ( X, Y) can be expressed in terms of the integrals of f( X, Y) over straight lines not intersecting the parabola. We expect that this result can have applications in two-dimensional linear tomography problems with an opaque parabolic obstacle.

  10. An efficient explicit marching on in time solver for magnetic field volume integral equation

    KAUST Repository

    Sayed, Sadeed Bin

    2015-07-25

    An efficient explicit marching on in time (MOT) scheme for solving the magnetic field volume integral equation is proposed. The MOT system is cast in the form of an ordinary differential equation and is integrated in time using a PE(CE)m multistep scheme. At each time step, a system with a Gram matrix is solved for the predicted/corrected field expansion coefficients. Depending on the type of spatial testing scheme Gram matrix is sparse or consists of blocks with only diagonal entries regardless of the time step size. Consequently, the resulting MOT scheme is more efficient than its implicit counterparts, which call for inversion of fuller matrix system at lower frequencies. Numerical results, which demonstrate the efficiency, accuracy, and stability of the proposed MOT scheme, are presented.

  11. Properties of linear integral equations related to the six-vertex model with disorder parameter II

    International Nuclear Information System (INIS)

    Boos, Hermann; Göhmann, Frank

    2012-01-01

    We study certain functions arising in the context of the calculation of correlation functions of the XXZ spin chain and of integrable field theories related to various scaling limits of the underlying six-vertex model. We show that several of these functions that are related to linear integral equations can be obtained by acting with (deformed) difference operators on a master function Φ. The latter is defined in terms of a functional equation and of its asymptotic behavior. Concentrating on the so-called temperature case, we show that these conditions uniquely determine the high-temperature series expansions of the master function. This provides an efficient calculation scheme for the high-temperature expansions of the derived functions as well. (paper)

  12. Integration of the time-dependent heat equation in the fuel rod performance program IAMBUS

    International Nuclear Information System (INIS)

    West, G.

    1982-01-01

    An iterative numerical method for integration of the time-dependent heat equation is described. No presuppositions are made for the dependency of the thermal conductivity and heat capacity on space, time and temperature. (orig.) [de

  13. Quantitative emission tomography by coded aperture imaging in nuclear medicine

    International Nuclear Information System (INIS)

    Guilhem, J.B.

    1982-06-01

    The coded aperture imaging is applied to nuclear medicine, since ten years. However no satisfactory clinical results have been obtained thus for. The reason is that digital reconstruction methods which have been implemented, in particular the method which use deconvolution filtering are not appropriate for quantification. Indeed these methods which all based on the assumption of shift invariance of the coding procedure, which is contradictory to the geometrical recording conditions giving the best depth resolution, do not take into account gamma rays attenuation by tissues and in most cases give tomograms with artefacts from blurred structures. A method is proposed which has not these limitations and considers the reconstruction problem as the ill-conditioned problem of solving a Fredholm integral equation. The main advantage of this method lies in fact that the transmission kernel of the integral equation is obtained experimentally, and the approximate solution of this equation, close enough to the original 3-D radioactive object, can be obtained in spite of the ill-conditioned nature of the problem, by use of singular values decomposition (S. V. D.) of the kernel [fr

  14. Comment on the consistency of truncated nonlinear integral equation based theories of freezing

    International Nuclear Information System (INIS)

    Cerjan, C.; Bagchi, B.; Rice, S.A.

    1985-01-01

    We report the results of two studies of aspects of the consistency of truncated nonlinear integral equation based theories of freezing: (i) We show that the self-consistent solutions to these nonlinear equations are unfortunately sensitive to the level of truncation. For the hard sphere system, if the Wertheim--Thiele representation of the pair direct correlation function is used, the inclusion of part but not all of the triplet direct correlation function contribution, as has been common, worsens the predictions considerably. We also show that the convergence of the solutions found, with respect to number of reciprocal lattice vectors kept in the Fourier expansion of the crystal singlet density, is slow. These conclusions imply great sensitivity to the quality of the pair direct correlation function employed in the theory. (ii) We show the direct correlation function based and the pair correlation function based theories of freezing can be cast into a form which requires solution of isomorphous nonlinear integral equations. However, in the pair correlation function theory the usual neglect of the influence of inhomogeneity of the density distribution on the pair correlation function is shown to be inconsistent to the lowest order in the change of density on freezing, and to lead to erroneous predictions

  15. Modern integral equation techniques for quantum reactive scattering theory

    International Nuclear Information System (INIS)

    Auerbach, S.M.

    1993-11-01

    Rigorous calculations of cross sections and rate constants for elementary gas phase chemical reactions are performed for comparison with experiment, to ensure that our picture of the chemical reaction is complete. We focus on the H/D+H 2 → H 2 /DH + H reaction, and use the time independent integral equation technique in quantum reactive scattering theory. We examine the sensitivity of H+H 2 state resolved integral cross sections σ v'j',vj (E) for the transitions (v = 0,j = 0) to (v' = 1,j' = 1,3), to the difference between the Liu-Siegbahn-Truhlar-Horowitz (LSTH) and double many body expansion (DMBE) ab initio potential energy surfaces (PES). This sensitivity analysis is performed to determine the origin of a large discrepancy between experimental cross sections with sharply peaked energy dependence and theoretical ones with smooth energy dependence. We find that the LSTH and DMBE PESs give virtually identical cross sections, which lends credence to the theoretical energy dependence

  16. Hierarchies of multi-component mKP equations and theirs integrable couplings

    International Nuclear Information System (INIS)

    Ji Jie; Yao Yuqin; Zhu Fubo; Chen Dengyuan

    2008-01-01

    First, a new multi-component modified Kadomtsev-Petviashvill (mKP) spectral problem is constructed by k-constraint imposed on a general pseudo-differential operator. Then, two hierarchies of multi-component mKP equations are derived, including positive non-isospectral mKP hierarchy and negative non-isospectral mKP hierarchy. Moreover, new integrable couplings of the resulting mKP soliton hierarchies are constructed by enlarging the associated matrix spectral problem

  17. Spatial symmetry, local integrability and tetrahedron equations in the Baxter-Bazhanov model

    International Nuclear Information System (INIS)

    Kashaev, R.M.; Mangazeev, V.V.; Stroganov, Yu.G.

    1992-01-01

    It is shown that the Baxter-Bazhanov model is invariant under the action of the cube symmetry group. The three-dimensional star-star relations, proposed by Baxter and Bazhanov as local integrability conditions, correspond to a particular transformation from this group. Invariant Boltzmann weights, parameterized in terms of the Zamolodchikov's angle variables, apparently satisfy the tetrahedron equations. 12 refs

  18. Parareal algorithms with local time-integrators for time fractional differential equations

    Science.gov (United States)

    Wu, Shu-Lin; Zhou, Tao

    2018-04-01

    It is challenge work to design parareal algorithms for time-fractional differential equations due to the historical effect of the fractional operator. A direct extension of the classical parareal method to such equations will lead to unbalance computational time in each process. In this work, we present an efficient parareal iteration scheme to overcome this issue, by adopting two recently developed local time-integrators for time fractional operators. In both approaches, one introduces auxiliary variables to localized the fractional operator. To this end, we propose a new strategy to perform the coarse grid correction so that the auxiliary variables and the solution variable are corrected separately in a mixed pattern. It is shown that the proposed parareal algorithm admits robust rate of convergence. Numerical examples are presented to support our conclusions.

  19. Asymptotic integration of a linear fourth order differential equation of Poincaré type

    Directory of Open Access Journals (Sweden)

    Anibal Coronel

    2015-11-01

    Full Text Available This article deals with the asymptotic behavior of nonoscillatory solutions of fourth order linear differential equation where the coefficients are perturbations of constants. We define a change of variable and deduce that the new variable satisfies a third order nonlinear differential equation. We assume three hypotheses. The first hypothesis is related to the constant coefficients and set up that the characteristic polynomial associated with the fourth order linear equation has simple and real roots. The other two hypotheses are related to the behavior of the perturbation functions and establish asymptotic integral smallness conditions of the perturbations. Under these general hypotheses, we obtain four main results. The first two results are related to the application of a fixed point argument to prove that the nonlinear third order equation has a unique solution. The next result concerns with the asymptotic behavior of the solutions of the nonlinear third order equation. The fourth main theorem is introduced to establish the existence of a fundamental system of solutions and to precise the formulas for the asymptotic behavior of the linear fourth order differential equation. In addition, we present an example to show that the results introduced in this paper can be applied in situations where the assumptions of some classical theorems are not satisfied.

  20. Integral Boundary Value Problems for Fractional Impulsive Integro Differential Equations in Banach Spaces

    Directory of Open Access Journals (Sweden)

    A. Anguraj

    2014-02-01

    Full Text Available We study in this paper,the existence of solutions for fractional integro differential equations with impulsive and integral conditions by using fixed point method. We establish the Sufficient conditions and unique solution for given problem. An Example is also explained to the main results.

  1. The anisotropic Ising correlations as elliptic integrals: duality and differential equations

    International Nuclear Information System (INIS)

    McCoy, B M; Maillard, J-M

    2016-01-01

    We present the reduction of the correlation functions of the Ising model on the anisotropic square lattice to complete elliptic integrals of the first, second and third kind, the extension of Kramers–Wannier duality to anisotropic correlation functions, and the linear differential equations for these anisotropic correlations. More precisely, we show that the anisotropic correlation functions are homogeneous polynomials of the complete elliptic integrals of the first, second and third kind. We give the exact dual transformation matching the correlation functions and the dual correlation functions. We show that the linear differential operators annihilating the general two-point correlation functions are factorized in a very simple way, in operators of decreasing orders. (paper)

  2. Analysis of electromagnetic wave interactions on nonlinear scatterers using time domain volume integral equations

    KAUST Repository

    Ulku, Huseyin Arda

    2014-07-06

    Effects of material nonlinearities on electromagnetic field interactions become dominant as field amplitudes increase. A typical example is observed in plasmonics, where highly localized fields “activate” Kerr nonlinearities. Naturally, time domain solvers are the method of choice when it comes simulating these nonlinear effects. Oftentimes, finite difference time domain (FDTD) method is used for this purpose. This is simply due to the fact that explicitness of the FDTD renders the implementation easier and the material nonlinearity can be easily accounted for using an auxiliary differential equation (J.H. Green and A. Taflove, Opt. Express, 14(18), 8305-8310, 2006). On the other hand, explicit marching on-in-time (MOT)-based time domain integral equation (TDIE) solvers have never been used for the same purpose even though they offer several advantages over FDTD (E. Michielssen, et al., ECCOMAS CFD, The Netherlands, Sep. 5-8, 2006). This is because explicit MOT solvers have never been stabilized until not so long ago. Recently an explicit but stable MOT scheme has been proposed for solving the time domain surface magnetic field integral equation (H.A. Ulku, et al., IEEE Trans. Antennas Propag., 61(8), 4120-4131, 2013) and later it has been extended for the time domain volume electric field integral equation (TDVEFIE) (S. B. Sayed, et al., Pr. Electromagn. Res. S., 378, Stockholm, 2013). This explicit MOT scheme uses predictor-corrector updates together with successive over relaxation during time marching to stabilize the solution even when time step is as large as in the implicit counterpart. In this work, an explicit MOT-TDVEFIE solver is proposed for analyzing electromagnetic wave interactions on scatterers exhibiting Kerr nonlinearity. Nonlinearity is accounted for using the constitutive relation between the electric field intensity and flux density. Then, this relation and the TDVEFIE are discretized together by expanding the intensity and flux - sing half

  3. Simulation of Logging-while-drilling Tool Response Using Integral Equation Fast Fourier Transform

    Directory of Open Access Journals (Sweden)

    Sun Xiang-Yang

    2017-01-01

    Full Text Available We rely on the volume integral equation (VIE method for the simulation of loggingwhile- drilling (LWG tool response using the integral equation fast Fourier transform (IE-FFT algorithm to accelerate the computation of the matrix-vector product in the iterative solver. Depending on the virtue of the Toeplitz structure of the interpolation of the Green’s function on the uniform Cartesian grids, this method uses FFT to calculate the matrix-vector multiplication. At the same time, this method reduce the memory requirement and CPU time. In this paper, IEFFT method is first used in the simulation of LWG. Numerical results are presented to demonstrate the accuracy and efficiency of this method. Compared with the Moment of Method (MOM and other fast algorithms, IE-FFT have distinct advantages in the fact of memory requirement and CPU time. In addition, this paper study the truncation, mesh elements, the size of the interpolation grids of IE-FFT and dip formation, and give some conclusion with wide applicability.

  4. Tokamak plasma shape identification based on the boundary integral equations

    International Nuclear Information System (INIS)

    Kurihara, Kenichi; Kimura, Toyoaki

    1992-05-01

    A necessary condition for tokamak plasma shape identification is discussed and a new identification method is proposed in this article. This method is based on the boundary integral equations governing a vacuum region around a plasma with only the measurement of either magnetic fluxes or magnetic flux intensities. It can identify various plasmas with low to high ellipticities with the precision determined by the number of the magnetic sensors. This method is applicable to real-time control and visualization using a 'table-look-up' procedure. (author)

  5. Scattering in quantum field theory: the M.P.S.A. approach in complex momentum space

    International Nuclear Information System (INIS)

    Bros, J.

    1981-02-01

    In this course, we intend to show how 'Many-Particle Structure Analysis' (M.P.S.A.) can be worked out in the standard field-theoretical framework, by using integral relations in complex momentum space involving 'l-particle irreducible kernels'. The ultimate purpose of this approach is to obtain the best possible knowledge of the singularities (location, nature, type of ramification) and of the ambient holomorphy (or meromorphy) domains of the n-point Green functions and scattering amplitudes, and at the same time to derive analytic structural equations for them which display the global organization of these singularities. The generation of Landau singularities for integrals and Fredholm resolvents, taken on cycles in complex space, will be explained on the basis of the Picard-Lefschetz formula (presented and used in simple situations). Among various results described, we present and analyse a structural equation for the six-point function (and for the 3 → 3 particle scattering function), valid in a domain containing the three-particle normal threshold

  6. A Calderón multiplicative preconditioner for coupled surface-volume electric field integral equations

    KAUST Repository

    Bagci, Hakan; Andriulli, Francesco P.; Cools, Kristof; Olyslager, Femke; Michielssen, Eric

    2010-01-01

    A well-conditioned coupled set of surface (S) and volume (V) electric field integral equations (S-EFIE and V-EFIE) for analyzing wave interactions with densely discretized composite structures is presented. Whereas the V-EFIE operator is well

  7. Method of interior boundaries in a mixed problem of acoustic scattering

    Directory of Open Access Journals (Sweden)

    P. A. Krutitskii

    1999-01-01

    Full Text Available The mixed problem for the Helmholtz equation in the exterior of several bodies (obstacles is studied in 2 and 3 dimensions. The Dirichlet boundary condition is given on some obstacles and the impedance boundary condition is specified on the rest. The problem is investigated by a special modification of the boundary integral equation method. This modification can be called ‘Method of interior boundaries’, because additional boundaries are introduced inside scattering bodies, where impedance boundary condition is given. The solution of the problem is obtained in the form of potentials on the whole boundary. The density in the potentials satisfies the uniquely solvable Fredholm equation of the second kind and can be computed by standard codes. In fact our method holds for any positive wave numbers. The Neumann, Dirichlet, impedance problems and mixed Dirichlet–Neumann problem are particular cases of our problem.

  8. Time-dependent integral equations of neutron transport for calculating the kinetics of nuclear reactors by the Monte Carlo method

    Energy Technology Data Exchange (ETDEWEB)

    Davidenko, V. D., E-mail: Davidenko-VD@nrcki.ru; Zinchenko, A. S., E-mail: zin-sn@mail.ru; Harchenko, I. K. [National Research Centre Kurchatov Institute (Russian Federation)

    2016-12-15

    Integral equations for the shape functions in the adiabatic, quasi-static, and improved quasi-static approximations are presented. The approach to solving these equations by the Monte Carlo method is described.

  9. Group theoretical and Hamiltonian structures of integrable evolution equations in 1x1 and 2x1 dimensions

    International Nuclear Information System (INIS)

    Konopel'chenko, B.G.

    1983-01-01

    New results in investigation of the group-theoretical and hamiltonian structure of the integrable evolution equations in 1+1 and 2+1 dimensions are briefly reviewed. Main general results, such as the form of integrable equations, Baecklund transfomations, symmetry groups, are turned out to have the same form for different spectral problems. The used generalized AKNS-method (the Ablowitz Kaup, Newell and Segur method) permits to prove that all nonlinear evolution equations considered are hamiltonians. The general condition of effective application of the ACNS mehtod to the concrete spectral problem is the possibility to calculate a recursion operator explicitly. The embedded representation is shown to be a fundamental object connected with different aspects of the inverse scattering problem

  10. A wavelet-based PWTD algorithm-accelerated time domain surface integral equation solver

    KAUST Repository

    Liu, Yang

    2015-10-26

    © 2015 IEEE. The multilevel plane-wave time-domain (PWTD) algorithm allows for fast and accurate analysis of transient scattering from, and radiation by, electrically large and complex structures. When used in tandem with marching-on-in-time (MOT)-based surface integral equation (SIE) solvers, it reduces the computational and memory costs of transient analysis from equation and equation to equation and equation, respectively, where Nt and Ns denote the number of temporal and spatial unknowns (Ergin et al., IEEE Trans. Antennas Mag., 41, 39-52, 1999). In the past, PWTD-accelerated MOT-SIE solvers have been applied to transient problems involving half million spatial unknowns (Shanker et al., IEEE Trans. Antennas Propag., 51, 628-641, 2003). Recently, a scalable parallel PWTD-accelerated MOT-SIE solver that leverages a hiearchical parallelization strategy has been developed and successfully applied to the transient problems involving ten million spatial unknowns (Liu et. al., in URSI Digest, 2013). We further enhanced the capabilities of this solver by implementing a compression scheme based on local cosine wavelet bases (LCBs) that exploits the sparsity in the temporal dimension (Liu et. al., in URSI Digest, 2014). Specifically, the LCB compression scheme was used to reduce the memory requirement of the PWTD ray data and computational cost of operations in the PWTD translation stage.

  11. Explicit Solutions for the (2 + 1-Dimensional Jaulent–Miodek Equation Using the Integrating Factors Method in an Unbounded Domain

    Directory of Open Access Journals (Sweden)

    Rahma Sadat

    2018-03-01

    Full Text Available In this work, we prove that the integrating factors can be used as a reduction method. Analytical solutions of the Jaulent–Miodek (JM equation are obtained using integrating factors as an extension of a recent work where, through hidden symmetries, the JM was reduced to ordinary differential equations (ODEs. Some of these ODEs had no quadrature. We here derive several new solutions for these non-solvable ODEs.

  12. Integral equations with contrasting kernels

    Directory of Open Access Journals (Sweden)

    Theodore Burton

    2008-01-01

    Full Text Available In this paper we study integral equations of the form $x(t=a(t-\\int^t_0 C(t,sx(sds$ with sharply contrasting kernels typified by $C^*(t,s=\\ln (e+(t-s$ and $D^*(t,s=[1+(t-s]^{-1}$. The kernel assigns a weight to $x(s$ and these kernels have exactly opposite effects of weighting. Each type is well represented in the literature. Our first project is to show that for $a\\in L^2[0,\\infty$, then solutions are largely indistinguishable regardless of which kernel is used. This is a surprise and it leads us to study the essential differences. In fact, those differences become large as the magnitude of $a(t$ increases. The form of the kernel alone projects necessary conditions concerning the magnitude of $a(t$ which could result in bounded solutions. Thus, the next project is to determine how close we can come to proving that the necessary conditions are also sufficient. The third project is to show that solutions will be bounded for given conditions on $C$ regardless of whether $a$ is chosen large or small; this is important in real-world problems since we would like to have $a(t$ as the sum of a bounded, but badly behaved function, and a large well behaved function.

  13. Remark on the solution of the Schroedinger equation for anharmonic oscillators via the Feynman path integral

    International Nuclear Information System (INIS)

    Rezende, J.

    1983-01-01

    We give a simple proof of Feynman's formula for the Green's function of the n-dimensional harmonic oscillator valid for every time t with Im t<=0. As a consequence the Schroedinger equation for the anharmonic oscillator is integrated and expressed by the Feynman path integral on Hilbert space. (orig.)

  14. The Wertheim integral equation theory with the ideal chain approximation and a dimer equation of state: Generalization to mixtures of hard-sphere chain fluids

    International Nuclear Information System (INIS)

    Chang, J.; Sandler, S.I.

    1995-01-01

    We have extended the Wertheim integral equation theory to mixtures of hard spheres with two attraction sites in order to model homonuclear hard-sphere chain fluids, and then solved these equations with the polymer-Percus--Yevick closure and the ideal chain approximation to obtain the average intermolecular and overall radial distribution functions. We obtain explicit expressions for the contact values of these distribution functions and a set of one-dimensional integral equations from which the distribution functions can be calculated without iteration or numerical Fourier transformation. We compare the resulting predictions for the distribution functions with Monte Carlo simulation results we report here for five selected binary mixtures. It is found that the accuracy of the prediction of the structure is the best for dimer mixtures and declines with increasing chain length and chain-length asymmetry. For the equation of state, we have extended the dimer version of the thermodynamic perturbation theory to the hard-sphere chain mixture by introducing the dimer mixture as an intermediate reference system. The Helmholtz free energy of chain fluids is then expressed in terms of the free energy of the hard-sphere mixture and the contact values of the correlation functions of monomer and dimer mixtures. We compared with the simulation results, the resulting equation of state is found to be the most accurate among existing theories with a relative average error of 1.79% for 4-mer/8-mer mixtures, which is the worst case studied in this work. copyright 1995 American Institute of Physics

  15. Dynamic propagation of a weak-discontinuous interface crack between two dissimilar functionally graded layers under anti-plane shear

    International Nuclear Information System (INIS)

    Shin, Jeong Woo; Lee, Young Shin

    2011-01-01

    The dynamic propagation of an interface crack between two functionally graded material (FGM) layers under anti-plane shear is analyzed using the integral transform method. The properties of the FGM layers vary continuously along their thicknesses. The properties of the two FGM layers vary and the two layers are connected weak-discontinuously. A constant velocity Yoffe-type moving crack is considered. The Fourier transform is used to reduce the problem to a dual integral equation, which is then expressed to a Fredholm integral equation of the second kind. Numerical values on the dynamic energy release rate (DERR) are presented for the FGM to show the effect of the gradient of material properties, crack moving velocity, and thickness of FGM layers. The following are helpful to increase resistance to interface crack propagation in FGMs: a) increasing the gradient of material properties, b) an increase of shear modulus and density from the interface to the upper and lower free surface, and c) increasing the thickness of the FGM layer. The DERR increases or decreases with increase of the crack moving velocity

  16. Exact integration of the unsteady incompressible Navier-Stokes equations, gauge criteria, and applications

    Science.gov (United States)

    Scholle, M.; Gaskell, P. H.; Marner, F.

    2018-04-01

    An exact first integral of the full, unsteady, incompressible Navier-Stokes equations is achieved in its most general form via the introduction of a tensor potential and parallels drawn with Maxwell's theory. Subsequent to this gauge freedoms are explored, showing that when used astutely they lead to a favourable reduction in the complexity of the associated equation set and number of unknowns, following which the inviscid limit case is discussed. Finally, it is shown how a change in gauge criteria enables a variational principle for steady viscous flow to be constructed having a self-adjoint form. Use of the new formulation is demonstrated, for different gauge variants of the first integral as the starting point, through the solution of a hierarchy of classical three-dimensional flow problems, two of which are tractable analytically, the third being solved numerically. In all cases the results obtained are found to be in excellent accord with corresponding solutions available in the open literature. Concurrently, the prescription of appropriate commonly occurring physical and necessary auxiliary boundary conditions, incorporating for completeness the derivation of a first integral of the dynamic boundary condition at a free surface, is established, together with how the general approach can be advantageously reformulated for application in solving unsteady flow problems with periodic boundaries.

  17. On the numerical evaluation of algebro-geometric solutions to integrable equations

    International Nuclear Information System (INIS)

    Kalla, C; Klein, C

    2012-01-01

    Physically meaningful periodic solutions to certain integrable partial differential equations are given in terms of multi-dimensional theta functions associated with real Riemann surfaces. Typical analytical problems in the numerical evaluation of these solutions are studied. In the case of hyperelliptic surfaces efficient algorithms exist even for almost degenerate surfaces. This allows the numerical study of solitonic limits. For general real Riemann surfaces, the choice of a homology basis adapted to the anti-holomorphic involution is important for a convenient formulation of the solutions and smoothness conditions. Since existing algorithms for algebraic curves produce a homology basis not related to automorphisms of the curve, we study symplectic transformations to an adapted basis and give explicit formulae for M-curves. As examples we discuss solutions of the Davey–Stewartson and the multi-component nonlinear Schrödinger equations

  18. Fuchsia : A tool for reducing differential equations for Feynman master integrals to epsilon form

    Science.gov (United States)

    Gituliar, Oleksandr; Magerya, Vitaly

    2017-10-01

    We present Fuchsia - an implementation of the Lee algorithm, which for a given system of ordinary differential equations with rational coefficients ∂x J(x , ɛ) = A(x , ɛ) J(x , ɛ) finds a basis transformation T(x , ɛ) , i.e., J(x , ɛ) = T(x , ɛ) J‧(x , ɛ) , such that the system turns into the epsilon form : ∂xJ‧(x , ɛ) = ɛ S(x) J‧(x , ɛ) , where S(x) is a Fuchsian matrix. A system of this form can be trivially solved in terms of polylogarithms as a Laurent series in the dimensional regulator ɛ. That makes the construction of the transformation T(x , ɛ) crucial for obtaining solutions of the initial system. In principle, Fuchsia can deal with any regular systems, however its primary task is to reduce differential equations for Feynman master integrals. It ensures that solutions contain only regular singularities due to the properties of Feynman integrals. Program Files doi:http://dx.doi.org/10.17632/zj6zn9vfkh.1 Licensing provisions: MIT Programming language:Python 2.7 Nature of problem: Feynman master integrals may be calculated from solutions of a linear system of differential equations with rational coefficients. Such a system can be easily solved as an ɛ-series when its epsilon form is known. Hence, a tool which is able to find the epsilon form transformations can be used to evaluate Feynman master integrals. Solution method: The solution method is based on the Lee algorithm (Lee, 2015) which consists of three main steps: fuchsification, normalization, and factorization. During the fuchsification step a given system of differential equations is transformed into the Fuchsian form with the help of the Moser method (Moser, 1959). Next, during the normalization step the system is transformed to the form where eigenvalues of all residues are proportional to the dimensional regulator ɛ. Finally, the system is factorized to the epsilon form by finding an unknown transformation which satisfies a system of linear equations. Additional comments

  19. A comparative analysis of Painleve, Lax pair, and similarity transformation methods in obtaining the integrability conditions of nonlinear Schroedinger equations

    International Nuclear Information System (INIS)

    Al Khawaja, U.

    2010-01-01

    We derive the integrability conditions of nonautonomous nonlinear Schroedinger equations using the Lax pair and similarity transformation methods. We present a comparative analysis of these integrability conditions with those of the Painleve method. We show that while the Painleve integrability conditions restrict the dispersion, nonlinearity, and dissipation/gain coefficients to be space independent and the external potential to be only a quadratic function of position, the Lax Pair and the similarity transformation methods allow for space-dependent coefficients and an external potential that is not restricted to the quadratic form. The integrability conditions of the Painleve method are retrieved as a special case of our general integrability conditions. We also derive the integrability conditions of nonautonomous nonlinear Schroedinger equations for two- and three-spacial dimensions.

  20. Pure radiation in space-time models that admit integration of the eikonal equation by the separation of variables method

    Science.gov (United States)

    Osetrin, Evgeny; Osetrin, Konstantin

    2017-11-01

    We consider space-time models with pure radiation, which admit integration of the eikonal equation by the method of separation of variables. For all types of these models, the equations of the energy-momentum conservation law are integrated. The resulting form of metric, energy density, and wave vectors of radiation as functions of metric for all types of spaces under consideration is presented. The solutions obtained can be used for any metric theories of gravitation.

  1. A surface-integral-equation approach to the propagation of waves in EBG-based devices

    NARCIS (Netherlands)

    Lancellotti, V.; Tijhuis, A.G.

    2012-01-01

    We combine surface integral equations with domain decomposition to formulate and (numerically) solve the problem of electromagnetic (EM) wave propagation inside finite-sized structures. The approach is of interest for (but not limited to) the analysis of devices based on the phenomenon of

  2. Symmetric and arbitrarily high-order Birkhoff-Hermite time integrators and their long-time behaviour for solving nonlinear Klein-Gordon equations

    Science.gov (United States)

    Liu, Changying; Iserles, Arieh; Wu, Xinyuan

    2018-03-01

    The Klein-Gordon equation with nonlinear potential occurs in a wide range of application areas in science and engineering. Its computation represents a major challenge. The main theme of this paper is the construction of symmetric and arbitrarily high-order time integrators for the nonlinear Klein-Gordon equation by integrating Birkhoff-Hermite interpolation polynomials. To this end, under the assumption of periodic boundary conditions, we begin with the formulation of the nonlinear Klein-Gordon equation as an abstract second-order ordinary differential equation (ODE) and its operator-variation-of-constants formula. We then derive a symmetric and arbitrarily high-order Birkhoff-Hermite time integration formula for the nonlinear abstract ODE. Accordingly, the stability, convergence and long-time behaviour are rigorously analysed once the spatial differential operator is approximated by an appropriate positive semi-definite matrix, subject to suitable temporal and spatial smoothness. A remarkable characteristic of this new approach is that the requirement of temporal smoothness is reduced compared with the traditional numerical methods for PDEs in the literature. Numerical results demonstrate the advantage and efficiency of our time integrators in comparison with the existing numerical approaches.

  3. Direct Yaw Control of Vehicle using State Dependent Riccati Equation with Integral Terms

    Directory of Open Access Journals (Sweden)

    SANDHU, F.

    2016-05-01

    Full Text Available Direct yaw control of four-wheel vehicles using optimal controllers such as the linear quadratic regulator (LQR and the sliding mode controller (SMC either considers only certain parameters constant in the nonlinear equations of vehicle model or totally neglect their effects to obtain simplified models, resulting in loss of states for the system. In this paper, a modified state-dependent Ricatti equation method obtained by the simplification of the vehicle model is proposed. This method overcomes the problem of the lost states by including state integrals. The results of the proposed system are compared with the sliding mode slip controller and state-dependent Ricatti equation method using high fidelity vehicle model in the vehicle simulation software package, Carsim. Results show 38% reduction in the lateral velocity, 34% reduction in roll and 16% reduction in excessive yaw by only increasing the fuel consumption by 6.07%.

  4. A parallel algorithm for solving the integral form of the discrete ordinates equations

    International Nuclear Information System (INIS)

    Zerr, R. J.; Azmy, Y. Y.

    2009-01-01

    The integral form of the discrete ordinates equations involves a system of equations that has a large, dense coefficient matrix. The serial construction methodology is presented and properties that affect the execution times to construct and solve the system are evaluated. Two approaches for massively parallel implementation of the solution algorithm are proposed and the current results of one of these are presented. The system of equations May be solved using two parallel solvers-block Jacobi and conjugate gradient. Results indicate that both methods can reduce overall wall-clock time for execution. The conjugate gradient solver exhibits better performance to compete with the traditional source iteration technique in terms of execution time and scalability. The parallel conjugate gradient method is synchronous, hence it does not increase the number of iterations for convergence compared to serial execution, and the efficiency of the algorithm demonstrates an apparent asymptotic decline. (authors)

  5. High Weak Order Methods for Stochastic Differential Equations Based on Modified Equations

    KAUST Repository

    Abdulle, Assyr

    2012-01-01

    © 2012 Society for Industrial and Applied Mathematics. Inspired by recent advances in the theory of modified differential equations, we propose a new methodology for constructing numerical integrators with high weak order for the time integration of stochastic differential equations. This approach is illustrated with the constructions of new methods of weak order two, in particular, semi-implicit integrators well suited for stiff (meansquare stable) stochastic problems, and implicit integrators that exactly conserve all quadratic first integrals of a stochastic dynamical system. Numerical examples confirm the theoretical results and show the versatility of our methodology.

  6. Solvability of a class of systems of infinite-dimensional integral equations and their application in statistical mechanics

    International Nuclear Information System (INIS)

    Gonchar, N.S.

    1986-01-01

    This paper presents a mathematical method developed for investigating a class of systems of infinite-dimensional integral equations which have application in statistical mechanics. Necessary and sufficient conditions are obtained for the uniqueness and bifurcation of the solution of this class of systems of equations. Problems of equilibrium statistical mechanics are considered on the basis of this method

  7. Integrable systems of partial differential equations determined by structure equations and Lax pair

    International Nuclear Information System (INIS)

    Bracken, Paul

    2010-01-01

    It is shown how a system of evolution equations can be developed both from the structure equations of a submanifold embedded in three-space as well as from a matrix SO(6) Lax pair. The two systems obtained this way correspond exactly when a constraint equation is selected and imposed on the system of equations. This allows for the possibility of selecting the coefficients in the second fundamental form in a general way.

  8. An Operational Matrix Technique for Solving Variable Order Fractional Differential-Integral Equation Based on the Second Kind of Chebyshev Polynomials

    Directory of Open Access Journals (Sweden)

    Jianping Liu

    2016-01-01

    Full Text Available An operational matrix technique is proposed to solve variable order fractional differential-integral equation based on the second kind of Chebyshev polynomials in this paper. The differential operational matrix and integral operational matrix are derived based on the second kind of Chebyshev polynomials. Using two types of operational matrixes, the original equation is transformed into the arithmetic product of several dependent matrixes, which can be viewed as an algebraic system after adopting the collocation points. Further, numerical solution of original equation is obtained by solving the algebraic system. Finally, several examples show that the numerical algorithm is computationally efficient.

  9. Algebraic entropy for differential-delay equations

    OpenAIRE

    Viallet, Claude M.

    2014-01-01

    We extend the definition of algebraic entropy to a class of differential-delay equations. The vanishing of the entropy, as a structural property of an equation, signals its integrability. We suggest a simple way to produce differential-delay equations with vanishing entropy from known integrable differential-difference equations.

  10. Integrable equations, addition theorems, and the Riemann-Schottky problem

    International Nuclear Information System (INIS)

    Buchstaber, Viktor M; Krichever, I M

    2006-01-01

    The classical Weierstrass theorem claims that, among the analytic functions, the only functions admitting an algebraic addition theorem are the elliptic functions and their degenerations. This survey is devoted to far-reaching generalizations of this result that are motivated by the theory of integrable systems. The authors discovered a strong form of the addition theorem for theta functions of Jacobian varieties, and this form led to new approaches to known problems in the geometry of Abelian varieties. It is shown that strong forms of addition theorems arise naturally in the theory of the so-called trilinear functional equations. Diverse aspects of the approaches suggested here are discussed, and some important open problems are formulated.

  11. Parallel, explicit, and PWTD-enhanced time domain volume integral equation solver

    KAUST Repository

    Liu, Yang

    2013-07-01

    Time domain volume integral equations (TDVIEs) are useful for analyzing transient scattering from inhomogeneous dielectric objects in applications as varied as photonics, optoelectronics, and bioelectromagnetics. TDVIEs typically are solved by implicit marching-on-in-time (MOT) schemes [N. T. Gres et al., Radio Sci., 36, 379-386, 2001], requiring the solution of a system of equations at each and every time step. To reduce the computational cost associated with such schemes, [A. Al-Jarro et al., IEEE Trans. Antennas Propagat., 60, 5203-5215, 2012] introduced an explicit MOT-TDVIE method that uses a predictor-corrector technique to stably update field values throughout the scatterer. By leveraging memory-efficient nodal spatial discretization and scalable parallelization schemes [A. Al-Jarro et al., in 28th Int. Rev. Progress Appl. Computat. Electromagn., 2012], this solver has been successfully applied to the analysis of scattering phenomena involving 0.5 million spatial unknowns. © 2013 IEEE.

  12. Four-dimensional integral equations for the MHD diffraction waves in plasma

    International Nuclear Information System (INIS)

    Alexandrova, A.A.; Khizhnyak, N.A.

    2000-01-01

    The superficial analysis of the boundary-value nonstationary problem for Alfven wave has shown the principal possibility of using the method of evolutionary integral equations of non-stationary macroscopic electrodynamical in a case of MHD description of waves in plasma. With the importance of strict mathematical solutions obtained for simple model problems that is the diffraction of one separately taken Alfven wave is that it can be the basis for construction of the approximate solutions of more complex boundary-value problems

  13. Lagrangian structures, integrability and chaos for 3D dynamical equations[45.20.Jj Lagrangian and Hamiltonian mechanics; 02.30.Ik Integrable systems; 05.45.Ac Low-dimensional chaos;

    Energy Technology Data Exchange (ETDEWEB)

    Bustamante, Miguel D [Departamento de Fisica, Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago, Chile (Chile); Hojman, Sergio A [Departamento de Fisica, Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago, Chile (Chile)

    2003-01-10

    In this paper, we consider the general setting for constructing action principles for three-dimensional first-order autonomous equations. We present the results for some integrable and non-integrable cases of the Lotka-Volterra equation, and show Lagrangian descriptions which are valid for systems satisfying Shil'nikov criteria on the existence of strange attractors, though chaotic behaviour has not been verified up to now. The Euler-Lagrange equations we get for these systems usually present 'time reparametrization' invariance, though other kinds of invariance may be found according to the kernel of the associated symplectic 2-form. The formulation of a Hamiltonian structure (Poisson brackets and Hamiltonians) for these systems from the Lagrangian viewpoint leads to a method of finding new constants of the motion starting from known ones, which is applied to some systems found in the literature known to possess a constant of the motion, to find the other and thus showing their integrability. In particular, we show that the so-called ABC system is completely integrable if it possesses one constant of the motion.

  14. On preconditioning techniques for dense linear systems arising from singular boundary integral equations

    Energy Technology Data Exchange (ETDEWEB)

    Chen, Ke [Univ. of Liverpool (United Kingdom)

    1996-12-31

    We study various preconditioning techniques for the iterative solution of boundary integral equations, and aim to provide a theory for a class of sparse preconditioners. Two related ideas are explored here: singularity separation and inverse approximation. Our preliminary conclusion is that singularity separation based preconditioners perform better than approximate inverse based while it is desirable to have both features.

  15. Positive nondecreasing solutions for a multi-term fractional-order functional differential equation with integral conditions

    OpenAIRE

    Ahmed M. A. El-Sayed; Ebtisam O. Bin-Taher

    2011-01-01

    In this article, we prove the existence of positive nondecreasing solutions for a multi-term fractional-order functional differential equations. We consider Cauchy boundary problems with: nonlocal conditions, two-point boundary conditions, integral conditions, and deviated arguments.

  16. Variational integration for ideal magnetohydrodynamics with built-in advection equations

    Energy Technology Data Exchange (ETDEWEB)

    Zhou, Yao; Burby, J. W.; Bhattacharjee, A. [Plasma Physics Laboratory, Princeton University, Princeton, New Jersey 08543 (United States); Qin, Hong [Plasma Physics Laboratory, Princeton University, Princeton, New Jersey 08543 (United States); Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026 (China)

    2014-10-15

    Newcomb's Lagrangian for ideal magnetohydrodynamics (MHD) in Lagrangian labeling is discretized using discrete exterior calculus. Variational integrators for ideal MHD are derived thereafter. Besides being symplectic and momentum-preserving, the schemes inherit built-in advection equations from Newcomb's formulation, and therefore avoid solving them and the accompanying error and dissipation. We implement the method in 2D and show that numerical reconnection does not take place when singular current sheets are present. We then apply it to studying the dynamics of the ideal coalescence instability with multiple islands. The relaxed equilibrium state with embedded current sheets is obtained numerically.

  17. Variational Integration for Ideal MHD with Built-in Advection Equations

    Energy Technology Data Exchange (ETDEWEB)

    Zhou, Yao [Princeton Plasma Physics Lab. (PPPL), Princeton, NJ (United States); Qin, Hong [Princeton Plasma Physics Lab. (PPPL), Princeton, NJ (United States); Burby, J. W. [Princeton Plasma Physics Lab. (PPPL), Princeton, NJ (United States); Bhattacharjee, A. [Princeton Plasma Physics Lab. (PPPL), Princeton, NJ (United States)

    2014-08-05

    Newcomb's Lagrangian for ideal MHD in Lagrangian labeling is discretized using discrete exterior calculus. Variational integrators for ideal MHD are derived thereafter. Besides being symplectic and momentum preserving, the schemes inherit built-in advection equations from Newcomb's formulation, and therefore avoid solving them and the accompanying error and dissipation. We implement the method in 2D and show that numerical reconnection does not take place when singular current sheets are present. We then apply it to studying the dynamics of the ideal coalescence instability with multiple islands. The relaxed equilibrium state with embedded current sheets is obtained numerically.

  18. Solving Ordinary Differential Equations

    Science.gov (United States)

    Krogh, F. T.

    1987-01-01

    Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.

  19. Fuchsia. A tool for reducing differential equations for Feynman master integral to epsilon form

    Energy Technology Data Exchange (ETDEWEB)

    Gituliar, Oleksandr [Hamburg Univ. (Germany). 2. Inst. fuer Theoretische Physik; Magerya, Vitaly

    2017-01-15

    We present Fuchsia - an implementation of the Lee algorithm, which for a given system of ordinary differential equations with rational coefficients ∂{sub x}f(x,ε)=A(x,ε)f(x,ε) finds a basis transformation T(x,ε), i.e., f(x,ε)=T(x,ε)g(x,ε), such that the system turns into the epsilon form: ∂{sub x}g(x,ε)=εS(x)g(x,ε), where S(x) is a Fuchsian matrix. A system of this form can be trivially solved in terms of polylogarithms as a Laurent series in the dimensional regulator ε. That makes the construction of the transformation T(x,ε) crucial for obtaining solutions of the initial system. In principle, Fuchsia can deal with any regular systems, however its primary task is to reduce differential equations for Feynman master integrals. It ensures that solutions contain only regular singularities due to the properties of Feynman integrals.

  20. Fuchsia. A tool for reducing differential equations for Feynman master integral to epsilon form

    International Nuclear Information System (INIS)

    Gituliar, Oleksandr; Magerya, Vitaly

    2017-01-01

    We present Fuchsia - an implementation of the Lee algorithm, which for a given system of ordinary differential equations with rational coefficients ∂ x f(x,ε)=A(x,ε)f(x,ε) finds a basis transformation T(x,ε), i.e., f(x,ε)=T(x,ε)g(x,ε), such that the system turns into the epsilon form: ∂ x g(x,ε)=εS(x)g(x,ε), where S(x) is a Fuchsian matrix. A system of this form can be trivially solved in terms of polylogarithms as a Laurent series in the dimensional regulator ε. That makes the construction of the transformation T(x,ε) crucial for obtaining solutions of the initial system. In principle, Fuchsia can deal with any regular systems, however its primary task is to reduce differential equations for Feynman master integrals. It ensures that solutions contain only regular singularities due to the properties of Feynman integrals.