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.

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)

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

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.

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

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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.

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

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

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)

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.

Transport equation solving methods

International Nuclear Information System (INIS)

Granjean, P.M.

1984-06-01

This work is mainly devoted to Csub(N) and Fsub(N) methods. CN method: starting from a lemma stated by Placzek, an equivalence is established between two problems: the first one is defined in a finite medium bounded by a surface S, the second one is defined in the whole space. In the first problem the angular flux on the surface S is shown to be the solution of an integral equation. This equation is solved by Galerkin's method. The Csub(N) method is applied here to one-velocity problems: in plane geometry, slab albedo and transmission with Rayleigh scattering, calculation of the extrapolation length; in cylindrical geometry, albedo and extrapolation length calculation with linear scattering. Fsub(N) method: the basic integral transport equation of the Csub(N) method is integrated on Case's elementary distributions; another integral transport equation is obtained: this equation is solved by a collocation method. The plane problems solved by the Csub(N) method are also solved by the Fsub(N) method. The Fsub(N) method is extended to any polynomial scattering law. Some simple spherical problems are also studied. Chandrasekhar's method, collision probability method, Case's method are presented for comparison with Csub(N) and Fsub(N) methods. This comparison shows the respective advantages of the two methods: a) fast convergence and possible extension to various geometries for Csub(N) method; b) easy calculations and easy extension to polynomial scattering for Fsub(N) method [fr

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

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.

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.)

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...

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.

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

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

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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.

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

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)

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)

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

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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.

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.

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)

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

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.

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

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

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

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...

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.)

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

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

Hierarchical Matrices Method and Its Application in Electromagnetic Integral Equations

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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.

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

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

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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.

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

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

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

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

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.

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

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.)

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...

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

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.

Acoustic 3D modeling by the method of integral equations

Science.gov (United States)

Malovichko, M.; Khokhlov, N.; Yavich, N.; Zhdanov, M.

2018-02-01

This paper presents a parallel algorithm for frequency-domain acoustic modeling by the method of integral equations (IE). The algorithm is applied to seismic simulation. The IE method reduces the size of the problem but leads to a dense system matrix. A tolerable memory consumption and numerical complexity were achieved by applying an iterative solver, accompanied by an effective matrix-vector multiplication operation, based on the fast Fourier transform (FFT). We demonstrate that, the IE system matrix is better conditioned than that of the finite-difference (FD) method, and discuss its relation to a specially preconditioned FD matrix. We considered several methods of matrix-vector multiplication for the free-space and layered host models. The developed algorithm and computer code were benchmarked against the FD time-domain solution. It was demonstrated that, the method could accurately calculate the seismic field for the models with sharp material boundaries and a point source and receiver located close to the free surface. We used OpenMP to speed up the matrix-vector multiplication, while MPI was used to speed up the solution of the system equations, and also for parallelizing across multiple sources. The practical examples and efficiency tests are presented as well.

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.

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

Numerical method for solving integral equations of neutron transport. II

International Nuclear Information System (INIS)

Loyalka, S.K.; Tsai, R.W.

1975-01-01

In a recent paper it was pointed out that the weakly singular integral equations of neutron transport can be quite conveniently solved by a method based on subtraction of singularity. This previous paper was devoted entirely to the consideration of simple one-dimensional isotropic-scattering and one-group problems. The present paper constitutes interesting extensions of the previous work in that in addition to a typical two-group anisotropic-scattering albedo problem in the slab geometry, the method is also applied to an isotropic-scattering problem in the x-y geometry. These results are compared with discrete S/sub N/ (ANISN or TWOTRAN-II) results, and for the problems considered here, the proposed method is found to be quite effective. Thus, the method appears to hold considerable potential for future applications. (auth)

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...

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

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.

Solving Hammerstein Type Integral Equation by New Discrete Adomian Decomposition Methods

Directory of Open Access Journals (Sweden)

Huda O. Bakodah

2013-01-01

Full Text Available New discrete Adomian decomposition methods are presented by using some identified Clenshaw-Curtis quadrature rules. We investigate two mixed quadrature rules one of precision five and the other of precision seven. The first rule is formed by using the Fejér second rule of precision three and Simpson rule of precision three, while the second rule is formed by using the Fejér second rule of precision five and the Boole rule of precision five. Our methods were applied to a nonlinear integral equation of the Hammerstein type and some examples are given to illustrate the validity of our methods.

Runge-Kutta Integration of the Equal Width Wave Equation Using the Method of Lines

Directory of Open Access Journals (Sweden)

M. A. Banaja

2015-01-01

Full Text Available The equal width (EW equation governs nonlinear wave phenomena like waves in shallow water. Numerical solution of the (EW equation is obtained by using the method of lines (MOL based on Runge-Kutta integration. Using von Neumann stability analysis, the scheme is found to be unconditionally stable. Solitary wave motion and interaction of two solitary waves are studied using the proposed method. The three invariants of the motion are evaluated to determine the conservation properties of the generated scheme. Accuracy of the proposed method is discussed by computing the L2 and L∞ error norms. The results are found in good agreement with exact solution.

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.

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

He's homotopy perturbation method for solving systems of Volterra integral equations of the second kind

International Nuclear Information System (INIS)

Biazar, J.; Ghazvini, H.

2009-01-01

In this paper, the He's homotopy perturbation method is applied to solve systems of Volterra integral equations of the second kind. Some examples are presented to illustrate the ability of the method for linear and non-linear such systems. The results reveal that the method is very effective and simple.

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.

Numerical treatments for solving nonlinear mixed integral equation

Directory of Open Access Journals (Sweden)

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.

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.

The reduced basis method for the electric field integral equation

International Nuclear Information System (INIS)

Fares, M.; Hesthaven, J.S.; Maday, Y.; Stamm, B.

2011-01-01

We introduce the reduced basis method (RBM) as an efficient tool for parametrized scattering problems in computational electromagnetics for problems where field solutions are computed using a standard Boundary Element Method (BEM) for the parametrized electric field integral equation (EFIE). This combination enables an algorithmic cooperation which results in a two step procedure. The first step consists of a computationally intense assembling of the reduced basis, that needs to be effected only once. In the second step, we compute output functionals of the solution, such as the Radar Cross Section (RCS), independently of the dimension of the discretization space, for many different parameter values in a many-query context at very little cost. Parameters include the wavenumber, the angle of the incident plane wave and its polarization.

Application of homotopy perturbation method for systems of Volterra integral equations of the first kind

International Nuclear Information System (INIS)

Biazar, J.; Eslami, M.; Aminikhah, H.

2009-01-01

In this article, an application of He's homotopy perturbation method is applied to solve systems of Volterra integral equations of the first kind. Some non-linear examples are prepared to illustrate the efficiency and simplicity of the method. Applying the method for linear systems is so easily that it does not worth to have any example.

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)

Boundary integral equation methods in eigenvalue problems of elastodynamics and thin plates

CERN Document Server

Kitahara, M

1985-01-01

The boundary integral equation (BIE) method has been used more and more in the last 20 years for solving various engineering problems. It has important advantages over other techniques for numerical treatment of a wide class of boundary value problems and is now regarded as an indispensable tool for potential problems, electromagnetism problems, heat transfer, fluid flow, elastostatics, stress concentration and fracture problems, geomechanical problems, and steady-state and transient electrodynamics.In this book, the author gives a complete, thorough and detailed survey of the method. It pro

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.

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

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.

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.

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.

Application of the method of integral equations to calculating the electrodynamic characteristics of periodically corrugated waveguides

International Nuclear Information System (INIS)

Belov, V.E.; Rodygin, L.V.; Fil'chenko, S.E.; Yunakovskii, A.D.

1988-01-01

A method is described for calculating the electrodynamic characteristics of periodically corrugated waveguide systems. This method is based on representing the field as the solution of the Helmholtz vector equation in the form of a simple layer potential, transformed with the use of the Floquet conditions. Systems of compound integral equations based on a weighted vector function of the simple layer potential are derived for waveguides with azimuthally symmetric and helical corrugations. A numerical realization of the Fourier method is cited for seeking the dispersion relation of azimuthally symmetric waves of a circular corrugated waveguide

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

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 –

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.

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

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

An irrational trial equation method and its applications

Indian Academy of Sciences (India)

equation method which is different from those direct methods. Liu's key idea is that exact solution to a differential equation can be given by solving an integration. For example, consider a differential equation of u. We always assume that its exact solution satisfies a solvable equation u = F(u). Therefore, our task is just to find.

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

Integral-equation based methods for parameter estimation in output pulses of radiation detectors: Application in nuclear medicine and spectroscopy

Science.gov (United States)

Mohammadian-Behbahani, Mohammad-Reza; Saramad, Shahyar

2018-04-01

Model based analysis methods are relatively new approaches for processing the output data of radiation detectors in nuclear medicine imaging and spectroscopy. A class of such methods requires fast algorithms for fitting pulse models to experimental data. In order to apply integral-equation based methods for processing the preamplifier output pulses, this article proposes a fast and simple method for estimating the parameters of the well-known bi-exponential pulse model by solving an integral equation. The proposed method needs samples from only three points of the recorded pulse as well as its first and second order integrals. After optimizing the sampling points, the estimation results were calculated and compared with two traditional integration-based methods. Different noise levels (signal-to-noise ratios from 10 to 3000) were simulated for testing the functionality of the proposed method, then it was applied to a set of experimental pulses. Finally, the effect of quantization noise was assessed by studying different sampling rates. Promising results by the proposed method endorse it for future real-time applications.

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...

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.

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

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

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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.

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.

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

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.

Explicit Solutions for the (2 + 1-Dimensional Jaulent–Miodek Equation Using the Integrating Factors Method in an Unbounded Domain

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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.

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.

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)

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.

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.)

Expansion methods for solving integral equations with multiple time lags using Bernstein polynomial of the second kind

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Mahmoud Paripour

2014-08-01

Full Text Available In this paper, the Bernstein polynomials are used to approximatethe solutions of linear integral equations with multiple time lags (IEMTL through expansion methods (collocation method, partition method, Galerkin method. The method is discussed in detail and illustrated by solving some numerical examples. Comparison between the exact and approximated results obtained from these methods is carried out

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.

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

Nodal spectrum method for solving neutron diffusion equation

International Nuclear Information System (INIS)

Sanchez, D.; Garcia, C. R.; Barros, R. C. de; Milian, D.E.

1999-01-01

Presented here is a new numerical nodal method for solving static multidimensional neutron diffusion equation in rectangular geometry. Our method is based on a spectral analysis of the nodal diffusion equations. These equations are obtained by integrating the diffusion equation in X, Y directions and then considering flat approximations for the current. These flat approximations are the only approximations that are considered in this method, as a result the numerical solutions are completely free from truncation errors. We show numerical results to illustrate the methods accuracy for coarse mesh calculations

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...

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.

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

Lagrange-Noether method for solving second-order differential equations

Institute of Scientific and Technical Information of China (English)

Wu Hui-Bin; Wu Run-Heng

2009-01-01

The purpose of this paper is to provide a new method called the Lagrange-Noether method for solving second-order differential equations. The method is,firstly,to write the second-order differential equations completely or partially in the form of Lagrange equations,and secondly,to obtain the integrals of the equations by using the Noether theory of the Lagrange system. An example is given to illustrate the application of the result.

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.

Application of the heat-balance and refined integral methods to the Korteweg-de Vries equation

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Myers Timothy G.

2009-01-01

Full Text Available In this paper we consider approximate travelling wave solutions to the Korteweg-de Vries equation. The heat-balance integral method is first applied to the problem, using two different quartic approximating functions, and then the refined integral method is investigated. We examine two types of solution, chosen by matching the wave speed to that of the exact solution and by imposing the same area. The first set of solutions is generally better with an error that is fixed in time. The second set of solutions has an error that grows with time. This is shown to be due to slight discrepancies in the wave speed.

Babenko’s Approach to Abel’s Integral Equations

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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.

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

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

Modified homotopy perturbation method for solving hypersingular integral equations of the first kind.

Science.gov (United States)

Eshkuvatov, Z K; Zulkarnain, F S; Nik Long, N M A; Muminov, Z

2016-01-01

Modified homotopy perturbation method (HPM) was used to solve the hypersingular integral equations (HSIEs) of the first kind on the interval [-1,1] with the assumption that the kernel of the hypersingular integral is constant on the diagonal of the domain. Existence of inverse of hypersingular integral operator leads to the convergence of HPM in certain cases. Modified HPM and its norm convergence are obtained in Hilbert space. Comparisons between modified HPM, standard HPM, Bernstein polynomials approach Mandal and Bhattacharya (Appl Math Comput 190:1707-1716, 2007), Chebyshev expansion method Mahiub et al. (Int J Pure Appl Math 69(3):265-274, 2011) and reproducing kernel Chen and Zhou (Appl Math Lett 24:636-641, 2011) are made by solving five examples. Theoretical and practical examples revealed that the modified HPM dominates the standard HPM and others. Finally, it is found that the modified HPM is exact, if the solution of the problem is a product of weights and polynomial functions. For rational solution the absolute error decreases very fast by increasing the number of collocation points.

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.)

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…

A variable timestep generalized Runge-Kutta method for the numerical integration of the space-time diffusion equations

International Nuclear Information System (INIS)

Aviles, B.N.; Sutton, T.M.; Kelly, D.J. III.

1991-09-01

A generalized Runge-Kutta method has been employed in the numerical integration of the stiff space-time diffusion equations. The method is fourth-order accurate, using an embedded third-order solution to arrive at an estimate of the truncation error for automatic timestep control. The efficiency of the Runge-Kutta method is enhanced by a block-factorization technique that exploits the sparse structure of the matrix system resulting from the space and energy discretized form of the time-dependent neutron diffusion equations. Preliminary numerical evaluation using a one-dimensional finite difference code shows the sparse matrix implementation of the generalized Runge-Kutta method to be highly accurate and efficient when compared to an optimized iterative theta method. 12 refs., 5 figs., 4 tabs

A boundary integral equation method using auxiliary interior surface approach for acoustic radiation and scattering in two dimensions.

Science.gov (United States)

Yang, S A

2002-10-01

This paper presents an effective solution method for predicting acoustic radiation and scattering fields in two dimensions. The difficulty of the fictitious characteristic frequency is overcome by incorporating an auxiliary interior surface that satisfies certain boundary condition into the body surface. This process gives rise to a set of uniquely solvable boundary integral equations. Distributing monopoles with unknown strengths over the body and interior surfaces yields the simple source formulation. The modified boundary integral equations are further transformed to ordinary ones that contain nonsingular kernels only. This implementation allows direct application of standard quadrature formulas over the entire integration domain; that is, the collocation points are exactly the positions at which the integration points are located. Selecting the interior surface is an easy task. Moreover, only a few corresponding interior nodal points are sufficient for the computation. Numerical calculations consist of the acoustic radiation and scattering by acoustically hard elliptic and rectangular cylinders. Comparisons with analytical solutions are made. Numerical results demonstrate the efficiency and accuracy of the current solution method.

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)

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

Implementing Families of Implicit Chebyshev Methods with Exact Coefficients for the Numerical Integration of First- and Second-Order Differential Equations

National Research Council Canada - National Science Library

Mitchell, Jason

2002-01-01

A method is presented for the generation of exact numerical coefficients found in two families of implicit Chebyshev methods for the numerical integration of first- and second-order ordinary differential equations...

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

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)

A Lie-admissible method of integration of Fokker-Planck equations with non-linear coefficients (exact and numerical solutions)

International Nuclear Information System (INIS)

Fronteau, J.; Combis, P.

1984-08-01

A Lagrangian method is introduced for the integration of non-linear Fokker-Planck equations. Examples of exact solutions obtained in this way are given, and also the explicit scheme used for the computation of numerical solutions. The method is, in addition, shown to be of a Lie-admissible type

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

Modeling of Graphene Planar Grating in the THz Range by the Method of Singular Integral Equations

Science.gov (United States)

Kaliberda, Mstislav E.; Lytvynenko, Leonid M.; Pogarsky, Sergey A.

2018-04-01

Diffraction of the H-polarized electromagnetic wave by the planar graphene grating in the THz range is considered. The scattering and absorption characteristics are studied. The scattered field is represented in the spectral domain via unknown spectral function. The mathematical model is based on the graphene surface impedance and the method of singular integral equations. The numerical solution is obtained by the Nystrom-type method of discrete singularities.

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.

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

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.''

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)

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

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.

Implicit Boundary Integral Methods for the Helmholtz Equation in Exterior Domains

Science.gov (United States)

2016-06-01

solve the Helmholtz equation as ∂Ω goes through significant change in its shape and topology — applications for which implicit representation of the...boundary-value problems for the wave equation and maxwell’s equations. Russian Math . Surv., 1965. [16] S. Reutskiy. The method of fundamental

A nonlinear boundary integral equations method for the solving of quasistatic elastic contact problem with Coulomb friction

Directory of Open Access Journals (Sweden)

Yurii M. Streliaiev

2016-06-01

Full Text Available Three-dimensional quasistatic contact problem of two linearly elastic bodies' interaction with Coulomb friction taken into account is considered. The boundary conditions of the problem have been simplified by the modification of the Coulomb's law of friction. This modification is based on the introducing of a delay in normal contact tractions that bound tangent contact tractions in the Coulomb's law of friction expressions. At this statement the problem is reduced to a sequence of similar systems of nonlinear integral equations describing bodies' interaction at each step of loading. A method for an approximate solution of the integral equations system corresponded to each step of loading is applied. This method consists of system regularization, discretization of regularized system and iterative process application for solving the discretized system. A numerical solution of a contact problem of an elastic sphere with an elastic half-space interaction under increasing and subsequently decreasing normal compressive force has been obtained.

A multi-domain spectral method for time-fractional differential equations

Science.gov (United States)

Chen, Feng; Xu, Qinwu; Hesthaven, Jan S.

2015-07-01

This paper proposes an approach for high-order time integration within a multi-domain setting for time-fractional differential equations. Since the kernel is singular or nearly singular, two main difficulties arise after the domain decomposition: how to properly account for the history/memory part and how to perform the integration accurately. To address these issues, we propose a novel hybrid approach for the numerical integration based on the combination of three-term-recurrence relations of Jacobi polynomials and high-order Gauss quadrature. The different approximations used in the hybrid approach are justified theoretically and through numerical examples. Based on this, we propose a new multi-domain spectral method for high-order accurate time integrations and study its stability properties by identifying the method as a generalized linear method. Numerical experiments confirm hp-convergence for both time-fractional differential equations and time-fractional partial differential equations.

Application of an analytical method for solution of thermal hydraulic conservation equations

Energy Technology Data Exchange (ETDEWEB)

Fakory, M.R. [Simulation, Systems & Services Technologies Company (S3 Technologies), Columbia, MD (United States)

1995-09-01

An analytical method has been developed and applied for solution of two-phase flow conservation equations. The test results for application of the model for simulation of BWR transients are presented and compared with the results obtained from application of the explicit method for integration of conservation equations. The test results show that with application of the analytical method for integration of conservation equations, the Courant limitation associated with explicit Euler method of integration was eliminated. The results obtained from application of the analytical method (with large time steps) agreed well with the results obtained from application of explicit method of integration (with time steps smaller than the size imposed by Courant limitation). The results demonstrate that application of the analytical approach significantly improves the numerical stability and computational efficiency.

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.)

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.

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.

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

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

A systematic method for constructing time discretizations of integrable lattice systems: local equations of motion

International Nuclear Information System (INIS)

Tsuchida, Takayuki

2010-01-01

We propose a new method for discretizing the time variable in integrable lattice systems while maintaining the locality of the equations of motion. The method is based on the zero-curvature (Lax pair) representation and the lowest-order 'conservation laws'. In contrast to the pioneering work of Ablowitz and Ladik, our method allows the auxiliary dependent variables appearing in the stage of time discretization to be expressed locally in terms of the original dependent variables. The time-discretized lattice systems have the same set of conserved quantities and the same structures of the solutions as the continuous-time lattice systems; only the time evolution of the parameters in the solutions that correspond to the angle variables is discretized. The effectiveness of our method is illustrated using examples such as the Toda lattice, the Volterra lattice, the modified Volterra lattice, the Ablowitz-Ladik lattice (an integrable semi-discrete nonlinear Schroedinger system) and the lattice Heisenberg ferromagnet model. For the modified Volterra lattice, we also present its ultradiscrete analogue.

A high-order method for the integration of the Galerkin semi-discretized nuclear reactor kinetics equations

International Nuclear Information System (INIS)

Vargas, L.

1988-01-01

The numerical approximate solution of the space-time nuclear reactor kinetics equation is investigated using a finite-element discretization of the space variable and a high order integration scheme for the resulting semi-discretized parabolic equation. The Galerkin method with spatial piecewise polynomial Lagrange basis functions are used to obtained a continuous time semi-discretized form of the space-time reactor kinetics equation. A temporal discretization is then carried out with a numerical scheme based on the Iterated Defect Correction (IDC) method using piecewise quadratic polynomials or exponential functions. The kinetics equations are thus solved with in a general finite element framework with respect to space as well as time variables in which the order of convergence of the spatial and temporal discretizations is consistently high. A computer code GALFEM/IDC is developed, to implement the numerical schemes described above. This issued to solve a one space dimensional benchmark problem. The results of the numerical experiments confirm the theoretical arguments and show that the convergence is very fast and the overall procedure is quite efficient. This is due to the good asymptotic properties of the numerical scheme which is of third order in the time interval

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

The two-wave X-ray field calculated by means of integral-equation methods

International Nuclear Information System (INIS)

Bremer, J.

1984-01-01

The problem of calculating the two-wave X-ray field on the basis of the Takagi-Taupin equations is discussed for the general case of curved lattice planes. A two-dimensional integral equation which incorporates the nature of the incoming radiation, the form of the crystal/vacuum boundary, and the curvature of the structure, is deduced. Analytical solutions for the symmetrical Laue case with incoming plane waves are obtained directly for perfect crystals by means of iteration. The same method permits a simple derivation of the narrow-wave Laue and Bragg cases. Modulated wave fronts are discussed, and it is shown that a cut-off in the width of an incoming plane wave leads to lateral oscillations which are superimposed on the Pendelloesung fringes. Bragg and Laue shadow fields are obtained. The influence of a non-zero kernel is discussed and a numerical procedure for calculating wave amplitudes in curved crystals is presented. (Auth.)

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.

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.

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

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...

Waveform relaxation methods for implicit differential equations

NARCIS (Netherlands)

P.J. van der Houwen; W.A. van der Veen

1996-01-01

textabstractWe apply a Runge-Kutta-based waveform relaxation method to initial-value problems for implicit differential equations. In the implementation of such methods, a sequence of nonlinear systems has to be solved iteratively in each step of the integration process. The size of these systems

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.

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.

Kinetic equation solution by inverse kinetic method

International Nuclear Information System (INIS)

Salas, G.

1983-01-01

We propose a computer program (CAMU) which permits to solve the inverse kinetic equation. The CAMU code is written in HPL language for a HP 982 A microcomputer with a peripheral interface HP 9876 A ''thermal graphic printer''. The CAMU code solves the inverse kinetic equation by taking as data entry the output of the ionization chambers and integrating the equation with the help of the Simpson method. With this program we calculate the evolution of the reactivity in time for a given disturbance

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)

Spline Collocation Method for Nonlinear Multi-Term Fractional Differential Equation

OpenAIRE

Choe, Hui-Chol; Kang, Yong-Suk

2013-01-01

We study an approximation method to solve nonlinear multi-term fractional differential equations with initial conditions or boundary conditions. First, we transform the nonlinear multi-term fractional differential equations with initial conditions and boundary conditions to nonlinear fractional integral equations and consider the relations between them. We present a Spline Collocation Method and prove the existence, uniqueness and convergence of approximate solution as well as error estimatio...

Nonlinear moments method for the isotropic Boltzmann equation and the invariance of collision integral

International Nuclear Information System (INIS)

Ehnder, A.Ya.; Ehnder, I.A.

1999-01-01

A new approach to develop nonlinear moment method to solve the Boltzmann equation is presented. This approach is based on the invariance of collision integral as to the selection of the base functions. The Sonin polynomials with the Maxwell weighting function are selected to serve as the base functions. It is shown that for the arbitrary cross sections of the interaction the matrix elements corresponding to the moments from the nonlinear integral of collisions are bound by simple recurrent bonds enabling to express all nonlinear matrix elements in terms of the linear ones. As a result, high-efficiency numerical pattern to calculate nonlinear matrix elements is obtained. The presented approach offers possibilities both to calculate relaxation processes within high speed range and to some more complex kinetic problems [ru

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

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.

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)

Exponential and Bessel fitting methods for the numerical solution of the Schroedinger equation

International Nuclear Information System (INIS)

Raptis, A.D.; Cash, J.R.

1987-01-01

A new method is developed for the numerical integration of the one dimensional radial Schroedinger equation. This method involves using different integration formulae in different parts of the range of integration rather than using the same integration formula throughout. Two new integration formulae are derived, one which integrates Bessel and Neumann functions exactly and another which exactly integrates certain exponential functions. It is shown that, for large r, these new formulae are much more accurate than standard integration methods for the Schroedinger equation. The benefit of using this new approach is demonstrated by considering some numerical examples based on the Lennard-Jones potential. (orig.)

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.

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)

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)

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.

Workshop on Numerical Methods for Ordinary Differential Equations

CERN Document Server

Gear, Charles; Russo, Elvira

1989-01-01

Developments in numerical initial value ode methods were the focal topic of the meeting at L'Aquila which explord the connections between the classical background and new research areas such as differental-algebraic equations, delay integral and integro-differential equations, stability properties, continuous extensions (interpolants for Runge-Kutta methods and their applications, effective stepsize control, parallel algorithms for small- and large-scale parallel architectures). The resulting proceedings address many of these topics in both research and survey papers.

Modelling of fluid flow in fractured porous media by the singular integral equations method

International Nuclear Information System (INIS)

Vu, M.N.

2012-01-01

This thesis aims to develop a method for numerical modelling of fluid flow through fractured porous media and for determination of their effective permeability by taking advantage of recent results based on formulation of the problem by Singular Integral Equations. In parallel, it was also an occasion to continue on the theoretical development and to obtain new results in this area. The governing equations for flow in such materials are reviewed first and mass conservation at the fracture intersections is expressed explicitly. Using the theory of potential, the general potential solutions are proposed in the form of a singular integral equation that describes the steady-state flow in and around several fractures embedded in an infinite porous matrix under a far-field pressure condition. These solutions represent the pressure field in the whole body as functions of the infiltration in the fractures, which fully take into account the fracture interaction and intersections. Closed-form solutions for the fundamental problem of fluid flow around a single fracture are derived, which are considered as the benchmark problems to validate the numerical solutions. In particular, the solution obtained for the case of an elliptical disc-shaped crack obeying to the Poiseuille law has been compared to that obtained for ellipsoidal inclusions with Darcy law.The numerical programs have been developed based on the singular integral equations method to resolve the general potential equations. These allow modeling the fluid flow through a porous medium containing a great number of fractures. Besides, this formulation of the problem also allows obtaining a semi-analytical infiltration solution over a single fracture depending on the matrice permeability, the fracture conductivity and the fracture geometry. This result is the important key to up-scaling the effective permeability of a fractured porous medium by using different homogenisation schemes. The results obtained by the self

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

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...

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

The hyperspherical-harmonics expansion method and the integral-equation approach to solving the few-body problem in momentum space

International Nuclear Information System (INIS)

Liu, F.-Q.; Lim, T.K.

1988-01-01

The Faddeev and Faddeev-Yakubovsky equations for three- and four-body systems are solved by applying the hyperspherical-harmonics expansion to them in momentum space. This coupling of two popular approaches to the few-body problem together with the use of the so-called Raynal-Revai transformation, which relates hyperspherical functions, allows the few-body equations to be written as one-dimensional coupled integral equations. Numerical solutions for these are achieved through standard matrix methods; these are made straightforward, because a second transformation renders potential multipoles easily calculable. For sample potentials and a restricted size of matrix in each case, the binding energies extracted match those previously obtained in solving the Schroedinger equation through the hyperspherical-harmonics expansion in coordinate space. 9 refs

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)

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.

Comparative analysis of solution methods of the punctual kinetic equations

International Nuclear Information System (INIS)

Hernandez S, A.

2003-01-01

The following one written it presents a comparative analysis among different analytical solutions for the punctual kinetics equation, which present two variables of interest: a) the temporary behavior of the neutronic population, and b) The temporary behavior of the different groups of precursors of delayed neutrons. The first solution is based on a method that solves the transfer function of the differential equation for the neutronic population, in which intends to obtain the different poles that give the stability of this transfer function. In this section it is demonstrated that the temporary variation of the reactivity of the system can be managed as it is required, since the integration time for this method doesn't affect the result. However, the second solution is based on an iterative method like that of Runge-Kutta or the Euler method where the algorithm was only used to solve first order differential equations giving this way solution to each differential equation that conforms the equations of punctual kinetics. In this section it is demonstrated that only it can obtain a correct temporary behavior of the neutronic population when it is integrated on an interval of very short time, forcing to the temporary variation of the reactivity to change very quick way without one has some control about the time. In both methods the same change is used so much in the reactivity of the system like in the integration times, giving validity to the results graph the one the temporary behavior of the neutronic population vs. time. (Author)

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

Theoretical analysis of integral neutron transport equation using collision probability method with quadratic flux approach

International Nuclear Information System (INIS)

Shafii, Mohammad Ali; Meidianti, Rahma; Wildian,; Fitriyani, Dian; Tongkukut, Seni H. J.; Arkundato, Artoto

2014-01-01

Theoretical analysis of integral neutron transport equation using collision probability (CP) method with quadratic flux approach has been carried out. In general, the solution of the neutron transport using the CP method is performed with the flat flux approach. In this research, the CP method is implemented in the cylindrical nuclear fuel cell with the spatial of mesh being conducted into non flat flux approach. It means that the neutron flux at any point in the nuclear fuel cell are considered different each other followed the distribution pattern of quadratic flux. The result is presented here in the form of quadratic flux that is better understanding of the real condition in the cell calculation and as a starting point to be applied in computational calculation

Theoretical analysis of integral neutron transport equation using collision probability method with quadratic flux approach

Energy Technology Data Exchange (ETDEWEB)

Shafii, Mohammad Ali, E-mail: mashafii@fmipa.unand.ac.id; Meidianti, Rahma, E-mail: mashafii@fmipa.unand.ac.id; Wildian,, E-mail: mashafii@fmipa.unand.ac.id; Fitriyani, Dian, E-mail: mashafii@fmipa.unand.ac.id [Department of Physics, Andalas University Padang West Sumatera Indonesia (Indonesia); Tongkukut, Seni H. J. [Department of Physics, Sam Ratulangi University Manado North Sulawesi Indonesia (Indonesia); Arkundato, Artoto [Department of Physics, Jember University Jember East Java Indonesia (Indonesia)

2014-09-30

Theoretical analysis of integral neutron transport equation using collision probability (CP) method with quadratic flux approach has been carried out. In general, the solution of the neutron transport using the CP method is performed with the flat flux approach. In this research, the CP method is implemented in the cylindrical nuclear fuel cell with the spatial of mesh being conducted into non flat flux approach. It means that the neutron flux at any point in the nuclear fuel cell are considered different each other followed the distribution pattern of quadratic flux. The result is presented here in the form of quadratic flux that is better understanding of the real condition in the cell calculation and as a starting point to be applied in computational calculation.

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.

Runge–Kutta type methods with special properties for the numerical integration of ordinary differential equations

International Nuclear Information System (INIS)

Kalogiratou, Z.; Monovasilis, Th.; Psihoyios, G.; Simos, T.E.

2014-01-01

In this work we review single step methods of the Runge–Kutta type with special properties. Among them are methods specially tuned to integrate problems that exhibit a pronounced oscillatory character and such problems arise often in celestial mechanics and quantum mechanics. Symplectic methods, exponentially and trigonometrically fitted methods, minimum phase-lag and phase-fitted methods are presented. These are Runge–Kutta, Runge–Kutta–Nyström and Partitioned Runge–Kutta methods. The theory of constructing such methods is given as well as several specific methods. In order to present the performance of the methods we have tested 58 methods from all categories. We consider the two dimensional harmonic oscillator, the two body problem, the pendulum problem and the orbital problem studied by Stiefel and Bettis. Also we have tested the methods on the computation of the eigenvalues of the one dimensional time independent Schrödinger equation with the harmonic oscillator, the doubly anharmonic oscillator and the exponential potentials

Method of moments solution of volume integral equations using higher-order hierarchical Legendre basis functions

DEFF Research Database (Denmark)

Kim, Oleksiy S.; Jørgensen, Erik; Meincke, Peter

2004-01-01

An efficient higher-order method of moments (MoM) solution of volume integral equations is presented. The higher-order MoM solution is based on higher-order hierarchical Legendre basis functions and higher-order geometry modeling. An unstructured mesh composed of 8-node trilinear and/or curved 27...... of magnitude in comparison to existing higher-order hierarchical basis functions. Consequently, an iterative solver can be applied even for high expansion orders. Numerical results demonstrate excellent agreement with the analytical Mie series solution for a dielectric sphere as well as with results obtained...

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

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

Iterative method of the parameter variation for solution of nonlinear functional equations

International Nuclear Information System (INIS)

Davidenko, D.F.

1975-01-01

The iteration method of parameter variation is used for solving nonlinear functional equations in Banach spaces. The authors consider some methods for numerical integration of ordinary first-order differential equations and construct the relevant iteration methods of parameter variation, both one- and multifactor. They also discuss problems of mathematical substantiation of the method, study the conditions and rate of convergence, estimate the error. The paper considers the application of the method to specific functional equations

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.

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

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.

Series: Utilization of Differential Equations and Methods for Solving Them in Medical Physics (4).

Science.gov (United States)

Murase, Kenya

2016-01-01

Partial differential equations are often used in the field of medical physics. In this (final) issue, the methods for solving the partial differential equations were introduced, which include separation of variables, integral transform (Fourier and Fourier-sine transforms), Green's function, and series expansion methods. Some examples were also introduced, in which the integral transform and Green's function methods were applied to solving Pennes' bioheat transfer equation and the Fourier series expansion method was applied to Navier-Stokes equation for analyzing the wall shear stress in blood vessels.Finally, the author hopes that this series will be helpful for people who engage in medical physics.

Methods of mathematical modelling continuous systems and differential equations

CERN Document Server

Witelski, Thomas

2015-01-01

This book presents mathematical modelling and the integrated process of formulating sets of equations to describe real-world problems. It describes methods for obtaining solutions of challenging differential equations stemming from problems in areas such as chemical reactions, population dynamics, mechanical systems, and fluid mechanics. Chapters 1 to 4 cover essential topics in ordinary differential equations, transport equations and the calculus of variations that are important for formulating models. Chapters 5 to 11 then develop more advanced techniques including similarity solutions, matched asymptotic expansions, multiple scale analysis, long-wave models, and fast/slow dynamical systems. Methods of Mathematical Modelling will be useful for advanced undergraduate or beginning graduate students in applied mathematics, engineering and other applied sciences.

Integral Methods in Science and Engineering

CERN Document Server

Constanda, Christian

2011-01-01

An enormous array of problems encountered by scientists and engineers are based on the design of mathematical models using many different types of ordinary differential, partial differential, integral, and integro-differential equations. Accordingly, the solutions of these equations are of great interest to practitioners and to science in general. Presenting a wealth of cutting-edge research by a diverse group of experts in the field, Integral Methods in Science and Engineering: Computational and Analytic Aspects gives a vivid picture of both the development of theoretical integral techniques

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.

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.

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

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

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.

Boundary integral equation methods and numerical solutions thin plates on an elastic foundation

CERN Document Server

Constanda, Christian; Hamill, William

2016-01-01

This book presents and explains a general, efficient, and elegant method for solving the Dirichlet, Neumann, and Robin boundary value problems for the extensional deformation of a thin plate on an elastic foundation. The solutions of these problems are obtained both analytically—by means of direct and indirect boundary integral equation methods (BIEMs)—and numerically, through the application of a boundary element technique. The text discusses the methodology for constructing a BIEM, deriving all the attending mathematical properties with full rigor. The model investigated in the book can serve as a template for the study of any linear elliptic two-dimensional problem with constant coefficients. The representation of the solution in terms of single-layer and double-layer potentials is pivotal in the development of a BIEM, which, in turn, forms the basis for the second part of the book, where approximate solutions are computed with a high degree of accuracy. The book is intended for graduate students and r...

Efficient method for the solution of the energy dependent integral Boltzmann transport equation in the resolved resonance energy region

International Nuclear Information System (INIS)

Schwenk, G.A. Jr.

1980-01-01

The calculation of neutron-nuclei reaction rates in the lower resolved resonance region (167 eV - 1.855 eV) is considered in this dissertation. Particular emphasis is placed on the calculation of these reaction rates for tight lattices where their accuracy is most important. The results of the continuous energy Monte Carlo code, VIM, are chosen as reference values for this study. The primary objective of this work is to develop a method for calculating resonance reaction rates which agree well with the reference solution, yet is efficient enough to be used by nuclear reactor fuel cycle designers on a production basis. A very efficient multigroup solution of the two spatial region energy dependent integral transport equation is developed. This solution, denoted the Broad Group Integral Method (BGIM), uses escape probabilities to obtain the spatial coupling between regions and uses an analytical flux shape within a multigroup to obtain weighted cross sections which account for the rapidly varying resonance cross sections. The multigroup lethargy widths chosen for the numerical integration of the two region energy-dependent neutron continuity equations can be chosen much wider (a factor of 30 larger) than in the direct numerical integration methods since the analytical flux shape is used to account for fine structure effects. The BGIM solution is made highly efficient through the use of these broad groups. It is estimated that for a 10 step unit cell fuel cycle depletion calculation, the computer running time for a production code such as EPRI-LEOPARD would be increased by only 6% through the use of the more accurate and intricate BGIM method in the lower resonance energy region

Inverse Free Iterative Methods for Nonlinear Ill-Posed Operator Equations

Directory of Open Access Journals (Sweden)

Ioannis K. Argyros

2014-01-01

ill-posed operator equation F(x=y. The proposed method is a modified form of Tikhonov gradient (TIGRA method considered by Ramlau (2003. The regularization parameter is chosen according to the balancing principle considered by Pereverzev and Schock (2005. The error estimate is derived under a general source condition and is of optimal order. Some numerical examples involving integral equations are also given in this paper.

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.

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.

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

Evolution operator equation: Integration with algebraic and finite difference methods. Applications to physical problems in classical and quantum mechanics and quantum field theory

Energy Technology Data Exchange (ETDEWEB)

Dattoli, Giuseppe; Torre, Amalia [ENEA, Centro Ricerche Frascati, Rome (Italy). Dipt. Innovazione; Ottaviani, Pier Luigi [ENEA, Centro Ricerche Bologna (Italy); Vasquez, Luis [Madris, Univ. Complutense (Spain). Dept. de Matemateca Aplicado

1997-10-01

The finite-difference based integration method for evolution-line equations is discussed in detail and framed within the general context of the evolution operator picture. Exact analytical methods are described to solve evolution-like equations in a quite general physical context. The numerical technique based on the factorization formulae of exponential operator is then illustrated and applied to the evolution-operator in both classical and quantum framework. Finally, the general view to the finite differencing schemes is provided, displaying the wide range of applications from the classical Newton equation of motion to the quantum field theory.

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.

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

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.)

Matrix Methods for Solving Hartree-Fock Equations in Atomic Structure Calculations and Line Broadening

Directory of Open Access Journals (Sweden)

Thomas Gomez

2018-04-01

Full Text Available Atomic structure of N-electron atoms is often determined by solving the Hartree-Fock equations, which are a set of integro-differential equations. The integral part of the Hartree-Fock equations treats electron exchange, but the Hartree-Fock equations are not often treated as an integro-differential equation. The exchange term is often approximated as an inhomogeneous or an effective potential so that the Hartree-Fock equations become a set of ordinary differential equations (which can be solved using the usual shooting methods. Because the Hartree-Fock equations are an iterative-refinement method, the inhomogeneous term relies on the previous guess of the wavefunction. In addition, there are numerical complications associated with solving inhomogeneous differential equations. This work uses matrix methods to solve the Hartree-Fock equations as an integro-differential equation. It is well known that a derivative operator can be expressed as a matrix made of finite-difference coefficients; energy eigenvalues and eigenvectors can be obtained by using linear-algebra packages. The integral (exchange part of the Hartree-Fock equation can be approximated as a sum and written as a matrix. The Hartree-Fock equations can be solved as a matrix that is the sum of the differential and integral matrices. We compare calculations using this method against experiment and standard atomic structure calculations. This matrix method can also be used to solve for free-electron wavefunctions, thus improving how the atoms and free electrons interact. This technique is important for spectral line broadening in two ways: it improves the atomic structure calculations, and it improves the motion of the plasma electrons that collide with the atom.

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

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.

A hybrid finite-difference and integral-equation method for modeling and inversion of marine controlled-source electromagnetic data

DEFF Research Database (Denmark)

Yoon, Daeung; Zhdanov, Michael; Mattsson, Johan

2016-01-01

One of the major problems in the modeling and inversion of marine controlled-source electromagnetic (CSEM) data is related to the need for accurate representation of very complex geoelectrical models typical for marine environment. At the same time, the corresponding forward-modeling algorithms...... should be powerful and fast enough to be suitable for repeated use in hundreds of iterations of the inversion and for multiple transmitter/receiver positions. To this end, we have developed a novel 3D modeling and inversion approach, which combines the advantages of the finite-difference (FD......) and integral-equation (IE) methods. In the framework of this approach, we have solved Maxwell’s equations for anomalous electric fields using the FD approximation on a staggered grid. Once the unknown electric fields in the computation domain of the FD method are computed, the electric and magnetic fields...

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)

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.

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.

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.)

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)

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

Directory of Open Access Journals (Sweden)

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 .

Partial differential equations with numerical methods

CERN Document Server

Larsson, Stig

2003-01-01

The book is suitable for advanced undergraduate and beginning graduate students of applied mathematics and engineering. The main theme is the integration of the theory of linear PDEs and the numerical solution of such equations. For each type of PDE, elliptic, parabolic, and hyperbolic, the text contains one chapter on the mathematical theory of the differential equation, followed by one chapter on finite difference methods and one on finite element methods. As preparation, the two-point boundary value problem and the initial-value problem for ODEs are discussed in separate chapters. There is also one chapter on the elliptic eigenvalue problem and eigenfunction expansion. The presentation does not presume a deep knowledge of mathematical and functional analysis. Some background on linear functional analysis and Sobolev spaces, and also on numerical linear algebra, is reviewed in two appendices.

A Laplace transform certified reduced basis method; application to the heat equation and wave equation

OpenAIRE

Knezevic, David; Patera, Anthony T.; Huynh, Dinh Bao Phuong

2010-01-01

We present a certified reduced basis (RB) method for the heat equation and wave equation. The critical ingredients are certified RB approximation of the Laplace transform; the inverse Laplace transform to develop the time-domain RB output approximation and rigorous error bound; a (Butterworth) filter in time to effect the necessary “modal” truncation; RB eigenfunction decomposition and contour integration for Offline–Online decomposition. We present numerical results to demonstrate the accura...

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)

A Fortran program for the numerical integration of the one-dimensional Schroedinger equation using exponential and Bessel fitting methods

International Nuclear Information System (INIS)

Cash, J.R.; Raptis, A.D.; Simos, T.E.

1990-01-01

An efficient algorithm is described for the accurate numerical integration of the one-dimensional Schroedinger equation. This algorithm uses a high-order, variable step Runge-Kutta like method in the region where the potential term dominates, and an exponential or Bessel fitted method in the asymptotic region. This approach can be used to compute scattering phase shifts in an efficient and reliable manner. A Fortran program which implements this algorithm is provided and some test results are given. (orig.)

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...

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.

Low-mode truncation methods in the sine-Gordon equation

International Nuclear Information System (INIS)

Xiong Chuyu.

1991-01-01

In this dissertation, the author studies the chaotic and coherent motions (i.e., low-dimensional chaotic attractor) in some near integrable partial differential equations, particularly the sine-Gordon equation and the nonlinear Schroedinger equation. In order to study the motions, he uses low mode truncation methods to reduce these partial differential equations to some truncated models (low-dimensional ordinary differential equations). By applying many methods available to low-dimensional ordinary differential equations, he can understand the low-dimensional chaotic attractor of PDE's much better. However, there are two important questions one needs to answer: (1) How many modes is good enough for the low mode truncated models to capture the dynamics uniformly? (2) Is the chaotic attractor in a low mode truncated model close to the chaotic attractor in the original PDE? And how close is? He has developed two groups of powerful methods to help to answer these two questions. They are the computation methods of continuation and local bifurcation, and local Lyapunov exponents and Lyapunov exponents. Using these methods, he concludes that the 2N-nls ODE is a good model for the sine-Gordon equation and the nonlinear Schroedinger equation provided one chooses a 'good' basis and uses 'enough' modes (where 'enough' depends on the parameters of the system but is small for the parameter studied here). Therefore, one can use 2N-nls ODE to study the chaos of PDE's in more depth

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...

Analysis of Leaky Modes in Photonic Crystal Fibers Using the Surface Integral Equation Method

Directory of Open Access Journals (Sweden)

Jung-Sheng Chiang

2018-04-01

Full Text Available A fully vectorial algorithm based on the surface integral equation method for the modelling of leaky modes in photonic crystal fibers (PCFs by solely solving the complex propagation constants of characteristic equations is presented. It can be used for calculations of the complex effective index and confinement losses of photonic crystal fibers. As complex root examination is the key technique in the solution, the new algorithm which possesses this technique can be used to solve the leaky modes of photonic crystal fibers. The leaky modes of solid-core PCFs with a hexagonal lattice of circular air-holes are reported and discussed. The simulation results indicate how the confinement loss by the imaginary part of the effective index changes with air-hole size, the number of rings of air-holes, and wavelength. Confinement loss reductions can be realized by increasing the air-hole size and the number of air-holes. The results show that the confinement loss rises with wavelength, implying that the light leaks more easily for longer wavelengths; meanwhile, the losses are decreased significantly as the air-hole size d/Λ is increased.

Application of the trial equation method for solving some nonlinear ...

Indian Academy of Sciences (India)

Therefore, our aim is just to find the function F. Liu has obtained a number of exact solutions to many nonlinear differential equations when F(u) is a polynomial or a rational function. ... In this study, we apply the trial equation method to seek exact solutions of the ... twice and setting the integration constant to zero, we have.

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.

Practical splitting methods for the adaptive integration of nonlinear evolution equations. Part I: Construction of optimized schemes and pairs of schemes

KAUST Repository

Auzinger, Winfried; Hofstä tter, Harald; Ketcheson, David I.; Koch, Othmar

2016-01-01

We present a number of new contributions to the topic of constructing efficient higher-order splitting methods for the numerical integration of evolution equations. Particular schemes are constructed via setup and solution of polynomial systems for the splitting coefficients. To this end we use and modify a recent approach for generating these systems for a large class of splittings. In particular, various types of pairs of schemes intended for use in adaptive integrators are constructed.

Practical splitting methods for the adaptive integration of nonlinear evolution equations. Part I: Construction of optimized schemes and pairs of schemes

KAUST Repository

Auzinger, Winfried

2016-07-28

We present a number of new contributions to the topic of constructing efficient higher-order splitting methods for the numerical integration of evolution equations. Particular schemes are constructed via setup and solution of polynomial systems for the splitting coefficients. To this end we use and modify a recent approach for generating these systems for a large class of splittings. In particular, various types of pairs of schemes intended for use in adaptive integrators are constructed.

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

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

Methods for Equating Mental Tests.

Science.gov (United States)

1984-11-01

1983) compared conventional and IRT methods for equating the Test of English as a Foreign Language ( TOEFL ) after chaining. Three conventional and...three IRT equating methods were examined in this study; two sections of TOEFL were each (separately) equated. The IRT methods included the following: (a...group. A separate base form was established for each of the six equating methods. Instead of equating the base-form TOEFL to itself, the last (eighth

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...

Discrete nodal integral transport-theory method for multidimensional reactor physics and shielding calculations

International Nuclear Information System (INIS)

Lawrence, R.D.; Dorning, J.J.

1980-01-01

A coarse-mesh discrete nodal integral transport theory method has been developed for the efficient numerical solution of multidimensional transport problems of interest in reactor physics and shielding applications. The method, which is the discrete transport theory analogue and logical extension of the nodal Green's function method previously developed for multidimensional neutron diffusion problems, utilizes the same transverse integration procedure to reduce the multidimensional equations to coupled one-dimensional equations. This is followed by the conversion of the differential equations to local, one-dimensional, in-node integral equations by integrating back along neutron flight paths. One-dimensional and two-dimensional transport theory test problems have been systematically studied to verify the superior computational efficiency of the new method

Gaussian-windowed frame based method of moments formulation of surface-integral-equation for extended apertures

Energy Technology Data Exchange (ETDEWEB)

Shlivinski, A., E-mail: amirshli@ee.bgu.ac.il [Department of Electrical and Computer Engineering, Ben-Gurion University of the Negev, Beer-Sheva 84105 (Israel); Lomakin, V., E-mail: vlomakin@eng.ucsd.edu [Department of Electrical and Computer Engineering, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0407 (United States)

2016-03-01

Scattering or coupling of electromagnetic beam-field at a surface discontinuity separating two homogeneous or inhomogeneous media with different propagation characteristics is formulated using surface integral equation, which are solved by the Method of Moments with the aid of the Gabor-based Gaussian window frame set of basis and testing functions. The application of the Gaussian window frame provides (i) a mathematically exact and robust tool for spatial-spectral phase-space formulation and analysis of the problem; (ii) a system of linear equations in a transmission-line like form relating mode-like wave objects of one medium with mode-like wave objects of the second medium; (iii) furthermore, an appropriate setting of the frame parameters yields mode-like wave objects that blend plane wave properties (as if solving in the spectral domain) with Green's function properties (as if solving in the spatial domain); and (iv) a representation of the scattered field with Gaussian-beam propagators that may be used in many large (in terms of wavelengths) systems.

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.

Analysis of spectral methods for the homogeneous Boltzmann equation

KAUST Repository

Filbet, Francis; Mouhot, Clé ment

2011-01-01

The development of accurate and fast algorithms for the Boltzmann collision integral and their analysis represent a challenging problem in scientific computing and numerical analysis. Recently, several works were devoted to the derivation of spectrally accurate schemes for the Boltzmann equation, but very few of them were concerned with the stability analysis of the method. In particular there was no result of stability except when the method was modified in order to enforce the positivity preservation, which destroys the spectral accuracy. In this paper we propose a new method to study the stability of homogeneous Boltzmann equations perturbed by smoothed balanced operators which do not preserve positivity of the distribution. This method takes advantage of the "spreading" property of the collision, together with estimates on regularity and entropy production. As an application we prove stability and convergence of spectral methods for the Boltzmann equation, when the discretization parameter is large enough (with explicit bound). © 2010 American Mathematical Society.

Analysis of spectral methods for the homogeneous Boltzmann equation

KAUST Repository

Filbet, Francis

2011-04-01

The development of accurate and fast algorithms for the Boltzmann collision integral and their analysis represent a challenging problem in scientific computing and numerical analysis. Recently, several works were devoted to the derivation of spectrally accurate schemes for the Boltzmann equation, but very few of them were concerned with the stability analysis of the method. In particular there was no result of stability except when the method was modified in order to enforce the positivity preservation, which destroys the spectral accuracy. In this paper we propose a new method to study the stability of homogeneous Boltzmann equations perturbed by smoothed balanced operators which do not preserve positivity of the distribution. This method takes advantage of the "spreading" property of the collision, together with estimates on regularity and entropy production. As an application we prove stability and convergence of spectral methods for the Boltzmann equation, when the discretization parameter is large enough (with explicit bound). © 2010 American Mathematical Society.

Solution and study of nodal neutron transport equation applying the LTSN-DiagExp method

International Nuclear Information System (INIS)

Hauser, Eliete Biasotto; Pazos, Ruben Panta; Vilhena, Marco Tullio de; Barros, Ricardo Carvalho de

2003-01-01

In this paper we report advances about the three-dimensional nodal discrete-ordinates approximations of neutron transport equation for Cartesian geometry. We use the combined collocation method of the angular variables and nodal approach for the spatial variables. By nodal approach we mean the iterated transverse integration of the S N equations. This procedure leads to the set of one-dimensional averages angular fluxes in each spatial variable. The resulting system of equations is solved with the LTS N method, first applying the Laplace transform to the set of the nodal S N equations and then obtained the solution by symbolic computation. We include the LTS N method by diagonalization to solve the nodal neutron transport equation and then we outline the convergence of these nodal-LTS N approximations with the help of a norm associated to the quadrature formula used to approximate the integral term of the neutron transport equation. (author)

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

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)

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)

Green`s function of Maxwell`s equations and corresponding implications for iterative methods

Energy Technology Data Exchange (ETDEWEB)

Singer, B.S. [Macquarie Univ., Sydney (Australia); Fainberg, E.B. [Inst. of Physics of the Earth, Moscow (Russian Federation)

1996-12-31

Energy conservation law imposes constraints on the norm and direction of the Hilbert space vector representing a solution of Maxwell`s equations. In this paper, we derive these constrains and discuss the corresponding implications for the Green`s function of Maxwell`s equations in a dissipative medium. It is shown that Maxwell`s equations can be reduced to an integral equation with a contracting kernel. The equation can be solved using simple iterations. Software based on this algorithm have successfully been applied to a wide range of problems dealing with high contrast models. The matrix corresponding to the integral equation has a well defined spectrum. The equation can be symmetrized and solved using different approaches, for instance one of the conjugate gradient methods.

HAM-Based Adaptive Multiscale Meshless Method for Burgers Equation

Directory of Open Access Journals (Sweden)

Shu-Li Mei

2013-01-01

Full Text Available Based on the multilevel interpolation theory, we constructed a meshless adaptive multiscale interpolation operator (MAMIO with the radial basis function. Using this operator, any nonlinear partial differential equations such as Burgers equation can be discretized adaptively in physical spaces as a nonlinear matrix ordinary differential equation. In order to obtain the analytical solution of the system of ODEs, the homotopy analysis method (HAM proposed by Shijun Liao was developed to solve the system of ODEs by combining the precise integration method (PIM which can be employed to get the analytical solution of linear system of ODEs. The numerical experiences show that HAM is not sensitive to the time step, and so the arithmetic error is mainly derived from the discrete in physical space.

Application of heat-balance integral method to conjugate thermal explosion

Directory of Open Access Journals (Sweden)

Novozhilov Vasily

2009-01-01

Full Text Available Conjugate thermal explosion is an extension of the classical theory, proposed and studied recently by the author. The paper reports application of heat-balance integral method for developing phase portraits for systems undergoing conjugate thermal explosion. The heat-balance integral method is used as an averaging method reducing partical differential equation problem to the set of first-order ordinary differential equations. The latter reduced problem allows natural interpretation in appropriately chosen phase space. It is shown that, with the help of heat-balance integral technique, conjugate thermal explosion problem can be described with a good accuracy by the set of non-linear first-order differential equations involving complex error function. Phase trajectories are presented for typical regimes emerging in conjugate thermal explosion. Use of heat-balance integral as a spatial averaging method allows efficient description of system evolution to be developed.

Runge-Kutta Methods for Linear Ordinary Differential Equations

Science.gov (United States)

Zingg, David W.; Chisholm, Todd T.

1997-01-01

Three new Runge-Kutta methods are presented for numerical integration of systems of linear inhomogeneous ordinary differential equations (ODES) with constant coefficients. Such ODEs arise in the numerical solution of the partial differential equations governing linear wave phenomena. The restriction to linear ODEs with constant coefficients reduces the number of conditions which the coefficients of the Runge-Kutta method must satisfy. This freedom is used to develop methods which are more efficient than conventional Runge-Kutta methods. A fourth-order method is presented which uses only two memory locations per dependent variable, while the classical fourth-order Runge-Kutta method uses three. This method is an excellent choice for simulations of linear wave phenomena if memory is a primary concern. In addition, fifth- and sixth-order methods are presented which require five and six stages, respectively, one fewer than their conventional counterparts, and are therefore more efficient. These methods are an excellent option for use with high-order spatial discretizations.

On the evolution equations, solvable through the inverse scattering method

International Nuclear Information System (INIS)

Gerdjikov, V.S.; Khristov, E.Kh.

1979-01-01

The nonlinear evolution equations (NLEE), related to the one-parameter family of Dirac operators are considered in a uniform manner. The class of NLEE solvable through the inverse scatterina method and their conservation laws are described. The description of the hierarchy of Hamiltonian structures and the proof of complete integrability of the NLEE is presented. The class of Baecklund transformations for these NLEE is derived. The general formulae are illustrated by two important examples: the nonlinear Schroedinger equation and the sine-Gordon equation

Lagrange–Galerkin methods for the incompressible Navier-Stokes equations: a review

Directory of Open Access Journals (Sweden)

Bermejo Rodolfo

2016-09-01

Full Text Available We review in this paper the development of Lagrange-Galerkin (LG methods to integrate the incompressible Navier-Stokes equations (NSEs for engineering applications. These methods were introduced in the computational fluid dynamics community in the early eighties of the past century, and at that time they were considered good methods for both their theoretical stability properties and the way of dealing with the nonlinear terms of the equations; however, the numerical experience gained with the application of LG methods to different problems has identified drawbacks of them, such as the calculation of specific integrals that arise in their formulation and the calculation of the ow trajectories, which somehow have hampered the applicability of LG methods. In this paper, we focus on these issues and summarize the convergence results of LG methods; furthermore, we shall briefly introduce a new stabilized LG method suitable for high Reynolds numbers.

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.

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.

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

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

Matrix-type multiple reciprocity boundary element method for solving three-dimensional two-group neutron diffusion equations

International Nuclear Information System (INIS)

Itagaki, Masafumi; Sahashi, Naoki.

1997-01-01

The multiple reciprocity boundary element method has been applied to three-dimensional two-group neutron diffusion problems. A matrix-type boundary integral equation has been derived to solve the first and the second group neutron diffusion equations simultaneously. The matrix-type fundamental solutions used here satisfy the equation which has a point source term and is adjoint to the neutron diffusion equations. A multiple reciprocity method has been employed to transform the matrix-type domain integral related to the fission source into an equivalent boundary one. The higher order fundamental solutions required for this formulation are composed of a series of two types of analytic functions. The eigenvalue itself is also calculated using only boundary integrals. Three-dimensional test calculations indicate that the present method provides stable and accurate solutions for criticality problems. (author)

Comparison of Kernel Equating and Item Response Theory Equating Methods

Science.gov (United States)

Meng, Yu

2012-01-01

The kernel method of test equating is a unified approach to test equating with some advantages over traditional equating methods. Therefore, it is important to evaluate in a comprehensive way the usefulness and appropriateness of the Kernel equating (KE) method, as well as its advantages and disadvantages compared with several popular item…

A constrained Hartree-Fock-Bogoliubov equation derived from the double variational method

International Nuclear Information System (INIS)

Onishi, Naoki; Horibata, Takatoshi.

1980-01-01

The double variational method is applied to the intrinsic state of the generalized BCS wave function. A constrained Hartree-Fock-Bogoliubov equation is derived explicitly in the form of an eigenvalue equation. A method of obtaining approximate overlap and energy overlap integrals is proposed. This will help development of numerical calculations of the angular momentum projection method, especially for general intrinsic wave functions without any symmetry restrictions. (author)

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)

Explicit Finite Difference Methods for the Delay Pseudoparabolic Equations

Directory of Open Access Journals (Sweden)

I. Amirali

2014-01-01

Full Text Available Finite difference technique is applied to numerical solution of the initial-boundary value problem for the semilinear delay Sobolev or pseudoparabolic equation. By the method of integral identities two-level difference scheme is constructed. For the time integration the implicit rule is being used. Based on the method of energy estimates the fully discrete scheme is shown to be absolutely stable and convergent of order two in space and of order one in time. The error estimates are obtained in the discrete norm. Some numerical results confirming the expected behavior of the method are shown.

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

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

Simple equation method for nonlinear partial differential equations and its applications

Directory of Open Access Journals (Sweden)

Taher A. Nofal

2016-04-01

Full Text Available In this article, we focus on the exact solution of the some nonlinear partial differential equations (NLPDEs such as, Kodomtsev–Petviashvili (KP equation, the (2 + 1-dimensional breaking soliton equation and the modified generalized Vakhnenko equation by using the simple equation method. In the simple equation method the trial condition is the Bernoulli equation or the Riccati equation. It has been shown that the method provides a powerful mathematical tool for solving nonlinear wave equations in mathematical physics and engineering problems.

Nystro¨m method applied to integral formulation of the neutron transport equation in X-Y geometry

Energy Technology Data Exchange (ETDEWEB)

Azevedo, Fabio S.; Sauter, Esequia; Konzen, Pedro H.A.; Barichello, Liliane B., E-mail: fabio.azevedo@ufrgs.br, E-mail: esequia.sauter@ufrgs.br, E-mail: pedro.konzen@ufrgs.br, E-mail: lbaric@mat.ufrgs.br [Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, RS (Brazil). Departamento de Matem´atica Pura e Aplicada

2017-07-01

Neutron transport problems in X-Y geometry have been solved with several techniques in last decades but it is still a challenge to produce a good balance between computational efficiency and accuracy. In this work, we address this problem by efficiently applying the Nystr¨om method to the integral formulation of the transport equation. Analytical techniques, modern numerical packages and optimized implementation were applied to reduce the computational time. This method presented results free of ray effects leading to high accurate numerical results for two-dimensional scalar flux. Our implementation simulates homogeneous problems with vacuum and reflective boundary conditions. Results were validated with up to seven significant digits and compared with those available in the literature. (author)

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.)

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)

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

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.

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.

Modelling the Influence of Ground Surface Relief on Electric Sounding Curves Using the Integral Equations Method

Directory of Open Access Journals (Sweden)

Balgaisha Mukanova

2017-01-01

Full Text Available The problem of electrical sounding of a medium with ground surface relief is modelled using the integral equations method. This numerical method is based on the triangulation of the computational domain, which is adapted to the shape of the relief and the measuring line. The numerical algorithm is tested by comparing the results with the known solution for horizontally layered media with two layers. Calculations are also performed to verify the fulfilment of the “reciprocity principle” for the 4-electrode installations in our numerical model. Simulations are then performed for a two-layered medium with a surface relief. The quantitative influences of the relief, the resistivity ratios of the contacting media, and the depth of the second layer on the apparent resistivity curves are established.

Telescopic projective methods for parabolic differential equations

CERN Document Server

Gear, C W

2003-01-01

Projective methods were introduced in an earlier paper [C.W. Gear, I.G. Kevrekidis, Projective Methods for Stiff Differential Equations: problems with gaps in their eigenvalue spectrum, NEC Research Institute Report 2001-029, available from http://www.neci.nj.nec.com/homepages/cwg/projective.pdf Abbreviated version to appear in SISC] as having potential for the efficient integration of problems with a large gap between two clusters in their eigenvalue spectrum, one cluster containing eigenvalues corresponding to components that have already been damped in the numerical solution and one corresponding to components that are still active. In this paper we introduce iterated projective methods that allow for explicit integration of stiff problems that have a large spread of eigenvalues with no gaps in their spectrum as arise in the semi-discretization of PDEs with parabolic components.

Telescopic projective methods for parabolic differential equations

International Nuclear Information System (INIS)

Gear, C.W.; Kevrekidis, Ioannis G.

2003-01-01

Projective methods were introduced in an earlier paper [C.W. Gear, I.G. Kevrekidis, Projective Methods for Stiff Differential Equations: problems with gaps in their eigenvalue spectrum, NEC Research Institute Report 2001-029, available from http://www.neci.nj.nec.com/homepages/cwg/projective.pdf Abbreviated version to appear in SISC] as having potential for the efficient integration of problems with a large gap between two clusters in their eigenvalue spectrum, one cluster containing eigenvalues corresponding to components that have already been damped in the numerical solution and one corresponding to components that are still active. In this paper we introduce iterated projective methods that allow for explicit integration of stiff problems that have a large spread of eigenvalues with no gaps in their spectrum as arise in the semi-discretization of PDEs with parabolic components

On the advantage of the finite fourier transformation method for the solution of a multigroup transport equation by the spherical harmonics method

International Nuclear Information System (INIS)

Kobayashi, K.

1995-01-01

A simple formulation to derive second order differential equations of the spherical harmonics method is described, and it is shown that there is difficulty in deriving finite difference equations for these differential equations at material interfaces, because the second order differential terms of higher order moments must be expressed by the values in a mesh box. On the other hand, it is shown that there is no such difficulty, if the author uses the finite Fourier transformation method, since the differential equations are transformed into integral equations and the differentiation corresponds simply to a multiplication by the transformation parameter. Solution can be obtained in the form of Fourier series without making use of the inversion integral

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.

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.

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.

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.

Weak Second Order Explicit Stabilized Methods for Stiff Stochastic Differential Equations

KAUST Repository

Abdulle, Assyr; Vilmart, Gilles; Zygalakis, Konstantinos C.

2013-01-01

We introduce a new family of explicit integrators for stiff Itô stochastic differential equations (SDEs) of weak order two. These numerical methods belong to the class of one-step stabilized methods with extended stability domains and do not suffer

A numerical method for resonance integral calculations

International Nuclear Information System (INIS)

Tanbay, Tayfun; Ozgener, Bilge

2013-01-01

A numerical method has been proposed for resonance integral calculations and a cubic fit based on least squares approximation to compute the optimum Bell factor is given. The numerical method is based on the discretization of the neutron slowing down equation. The scattering integral is approximated by taking into account the location of the upper limit in energy domain. The accuracy of the method has been tested by performing computations of resonance integrals for uranium dioxide isolated rods and comparing the results with empirical values. (orig.)

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.

An efficient fringe integral equation method for optimizing the antenna location on complex bodies

DEFF Research Database (Denmark)

Jørgensen, Erik; Meincke, Peter; Breinbjerg, Olav

2001-01-01

The radiation pattern of an antenna mounted nearby, or directly on, a complex three-dimensional (3D) structure can be significantly influenced by this structure. Integral equations combined with the method of moments (MoM) provide an accurate means for calculating the scattering from the structures...... in such applications. The structure is then modelled by triangular or rectangular surface patches with corresponding surface current expansion functions. A MoM matrix which is independent of the antenna location can be obtained by modelling the antenna as an impressed electric or magnetic source, e.g., a slot antenna...... can be modelled by a magnetic Hertzian dipole. For flush-mounted antennas, or antennas mounted in close vicinity of the scattering structure, the nearby impressed source induces a highly peaked surface current on the scattering structure. For the low-order basis functions usually applied...

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

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.)

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

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).

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.

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

Conservative multi-implicit integral deferred correction methods with adaptive mesh refinement

International Nuclear Information System (INIS)

Layton, A.T.

2004-01-01

In most models of reacting gas dynamics, the characteristic time scales of chemical reactions are much shorter than the hydrodynamic and diffusive time scales, rendering the reaction part of the model equations stiff. Moreover, nonlinear forcings may introduce into the solutions sharp gradients or shocks, the robust behavior and correct propagation of which require the use of specialized spatial discretization procedures. This study presents high-order conservative methods for the temporal integration of model equations of reacting flows. By means of a method of lines discretization on the flux difference form of the equations, these methods compute approximations to the cell-averaged or finite-volume solution. The temporal discretization is based on a multi-implicit generalization of integral deferred correction methods. The advection term is integrated explicitly, and the diffusion and reaction terms are treated implicitly but independently, with the splitting errors present in traditional operator splitting methods reduced via the integral deferred correction procedure. To reduce computational cost, time steps used to integrate processes with widely-differing time scales may differ in size. (author)

Selective Integration in the Material-Point Method

DEFF Research Database (Denmark)

Andersen, Lars; Andersen, Søren; Damkilde, Lars

2009-01-01

The paper deals with stress integration in the material-point method. In order to avoid parasitic shear in bending, a formulation is proposed, based on selective integration in the background grid that is used to solve the governing equations. The suggested integration scheme is compared...... to a traditional material-point-method computation in which the stresses are evaluated at the material points. The deformation of a cantilever beam is analysed, assuming elastic or elastoplastic material behaviour....

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.

Auxiliary equation method for solving nonlinear partial differential equations

International Nuclear Information System (INIS)

Sirendaoreji,; Jiong, Sun

2003-01-01

By using the solutions of an auxiliary ordinary differential equation, a direct algebraic method is described to construct several kinds of exact travelling wave solutions for some nonlinear partial differential equations. By this method some physically important nonlinear equations are investigated and new exact travelling wave solutions are explicitly obtained with the aid of symbolic computation

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

Science.gov (United States)

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.

Free-complement local-Schrödinger-equation method for solving the Schrödinger equation of atoms and molecules: Basic theories and features

Science.gov (United States)

Nakatsuji, Hiroshi; Nakashima, Hiroyuki

2015-02-01

The free-complement (FC) method is a general method for solving the Schrödinger equation (SE): The produced wave function has the potentially exact structure as the solution of the Schrödinger equation. The variables included are determined either by using the variational principle (FC-VP) or by imposing the local Schrödinger equations (FC-LSE) at the chosen set of the sampling points. The latter method, referred to as the local Schrödinger equation (LSE) method, is integral-free and therefore applicable to any atom and molecule. The purpose of this paper is to formulate the basic theories of the LSE method and explain their basic features. First, we formulate three variants of the LSE method, the AB, HS, and HTQ methods, and explain their properties. Then, the natures of the LSE methods are clarified in some detail using the simple examples of the hydrogen atom and the Hooke's atom. Finally, the ideas obtained in this study are applied to solving the SE of the helium atom highly accurately with the FC-LSE method. The results are very encouraging: we could get the world's most accurate energy of the helium atom within the sampling-type methodologies, which is comparable to those obtained with the FC-VP method. Thus, the FC-LSE method is an easy and yet a powerful integral-free method for solving the Schrödinger equation of general atoms and molecules.

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 .

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.

Numerical method for the nonlinear Fokker-Planck equation

International Nuclear Information System (INIS)

Zhang, D.S.; Wei, G.W.; Kouri, D.J.; Hoffman, D.K.

1997-01-01

A practical method based on distributed approximating functionals (DAFs) is proposed for numerically solving a general class of nonlinear time-dependent Fokker-Planck equations. The method relies on a numerical scheme that couples the usual path-integral concept to the DAF idea. The high accuracy and reliability of the method are illustrated by applying it to an exactly solvable nonlinear Fokker-Planck equation, and the method is compared with the accurate K-point Stirling interpolation formula finite-difference method. The approach is also used successfully to solve a nonlinear self-consistent dynamic mean-field problem for which both the cumulant expansion and scaling theory have been found by Drozdov and Morillo [Phys. Rev. E 54, 931 (1996)] to be inadequate to describe the occurrence of a long-lived transient bimodality. The standard interpretation of the transient bimodality in terms of the flat region in the kinetic potential fails for the present case. An alternative analysis based on the effective potential of the Schroedinger-like Fokker-Planck equation is suggested. Our analysis of the transient bimodality is strongly supported by two examples that are numerically much more challenging than other examples that have been previously reported for this problem. copyright 1997 The American Physical Society

Iterative Splitting Methods for Differential Equations

CERN Document Server

Geiser, Juergen

2011-01-01

Iterative Splitting Methods for Differential Equations explains how to solve evolution equations via novel iterative-based splitting methods that efficiently use computational and memory resources. It focuses on systems of parabolic and hyperbolic equations, including convection-diffusion-reaction equations, heat equations, and wave equations. In the theoretical part of the book, the author discusses the main theorems and results of the stability and consistency analysis for ordinary differential equations. He then presents extensions of the iterative splitting methods to partial differential

Weak Second Order Explicit Stabilized Methods for Stiff Stochastic Differential Equations

KAUST Repository

Abdulle, Assyr

2013-01-01

We introduce a new family of explicit integrators for stiff Itô stochastic differential equations (SDEs) of weak order two. These numerical methods belong to the class of one-step stabilized methods with extended stability domains and do not suffer from the step size reduction faced by standard explicit methods. The family is based on the standard second order orthogonal Runge-Kutta-Chebyshev (ROCK2) methods for deterministic problems. The convergence, meansquare, and asymptotic stability properties of the methods are analyzed. Numerical experiments, including applications to nonlinear SDEs and parabolic stochastic partial differential equations are presented and confirm the theoretical results. © 2013 Society for Industrial and Applied Mathematics.

High-order integral equation methods for problems of scattering by bumps and cavities on half-planes.

Science.gov (United States)

Pérez-Arancibia, Carlos; Bruno, Oscar P

2014-08-01

This paper presents high-order integral equation methods for the evaluation of electromagnetic wave scattering by dielectric bumps and dielectric cavities on perfectly conducting or dielectric half-planes. In detail, the algorithms introduced in this paper apply to eight classical scattering problems, namely, scattering by a dielectric bump on a perfectly conducting or a dielectric half-plane, and scattering by a filled, overfilled, or void dielectric cavity on a perfectly conducting or a dielectric half-plane. In all cases field representations based on single-layer potentials for appropriately chosen Green functions are used. The numerical far fields and near fields exhibit excellent convergence as discretizations are refined-even at and around points where singular fields and infinite currents exist.

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.

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.

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

On the comparison of numerical methods for the integration of kinetic equations in atmospheric chemistry and transport models

Science.gov (United States)

Saylor, Rick D.; Ford, Gregory D.

The integration of systems of ordinary differential equations (ODEs) that arise in atmospheric photochemistry is of significant concern to tropospheric and stratospheric chemistry modelers. As a consequence of the stiff nature of these ODE systems, their solution requires a large fraction of the total computational effort in three-dimensional chemical model simulations. Several integration techniques have been proposed and utilized over the years in an attempt to provide computationally efficient, yet accurate, solutions to chemical kinetics ODES. In this work, we present a comparison of some of these techniques and argue that valid comparisons of ODE solvers must take into account the trade-off between solution accuracy and computational efficiency. Misleading comparison results can be obtained by neglecting the fact that any ODE solution method can be made faster or slower by manipulation of the appropriate error tolerances or time steps. Comparisons among ODE solution techniques should therefore attempt to identify which technique can provide the most accurate solution with the least computational effort over the entire range of behavior of each technique. We present here a procedure by which ODE solver comparisons can achieve this goal. Using this methodology, we compare a variety of integration techniques, including methods proposed by Hesstvedt et al. (1978, Int. J. Chem. Kinet.10, 971-994), Gong and Cho (1993, Atmospheric Environment27A, 2147-2160), Young and Boris (1977, J. phys. Chem.81, 2424-2427) and Hindmarsh (1983, In Scientific Computing (edited by Stepleman R. S. et al.), pp. 55-64. North-Holland, Amsterdam). We find that Gear-type solvers such as the Livermore Solver for ordinary differential equations (LSODE) and the sparse-matrix version of LSODE (LSODES) provide the most accurate solution of our test problems with the least computational effort.

Isotope decay equations solved by means of a recursive method

International Nuclear Information System (INIS)

Grant, Carlos

2009-01-01

The isotope decay equations have been solved using forward finite differences taking small time steps, among other methods. This is the case of the cell code WIMS, where it is assumed that concentrations of all fissionable isotopes remain constant during the integration interval among other simplifications. Even when the problem could be solved running through a logical tree, all algorithms used for resolution of these equations used an iterative programming formulation. That happened because nearly all computer languages used up to a recent past by the scientific programmers did not support recursion, such as the case of the old versions of FORTRAN or BASIC. Nowadays also an integral form of the depletion equations is used in Monte Carlo simulation. In this paper we propose another programming solution using a recursive algorithm, running through all descendants of each isotope and adding their contributions to all isotopes in each generation. The only assumption made for this solution is that fluxes remain constant during the whole time step. Recursive process is interrupted when a stable isotope was attained or the calculated contributions are smaller than a given precision. These algorithms can be solved by means an exact analytic method that can have some problems when circular loops appear for isotopes with alpha decay, and a more general polynomial method. Both methods are shown. (author)

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.

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)

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

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)

Exponential-fitted methods for integrating stiff systems of ordinary differential equations: Applications to homogeneous gas-phase chemical kinetics

Science.gov (United States)

Pratt, D. T.

1984-01-01

Conventional algorithms for the numerical integration of ordinary differential equations (ODEs) are based on the use of polynomial functions as interpolants. However, the exact solutions of stiff ODEs behave like decaying exponential functions, which are poorly approximated by polynomials. An obvious choice of interpolant are the exponential functions themselves, or their low-order diagonal Pade (rational function) approximants. A number of explicit, A-stable, integration algorithms were derived from the use of a three-parameter exponential function as interpolant, and their relationship to low-order, polynomial-based and rational-function-based implicit and explicit methods were shown by examining their low-order diagonal Pade approximants. A robust implicit formula was derived by exponential fitting the trapezoidal rule. Application of these algorithms to integration of the ODEs governing homogenous, gas-phase chemical kinetics was demonstrated in a developmental code CREK1D, which compares favorably with the Gear-Hindmarsh code LSODE in spite of the use of a primitive stepsize control strategy.

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)

Integration of Lax and Zakharov-Schabat equations by means of algebraic geometry's methods

International Nuclear Information System (INIS)

Gozman, N.Ja.; Latyshev, A.V.; Savostjanov, M.V.; Lebedev, D.R.

1982-01-01

The solutions of nonlinear partial differential equations of Lax and Zakharov-Schabat types are obtained with the help of algebro-geometric method. The Krichever-Drinfeld bimodule for rational curve with cusp point is constructed. It is noted that rational solutions of Zakharov-Schabat equations can be found by means of constructed bimodule in the case of rank 1 only. The evolution of the poles of these solutions is investigated

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

Solution and study of nodal neutron transport equation applying the LTS{sub N}-DiagExp method

Energy Technology Data Exchange (ETDEWEB)

Hauser, Eliete Biasotto; Pazos, Ruben Panta [Pontificia Univ. Catolica do Rio Grande do Sul, Porto Alegre, RS (Brazil). Faculdade de Matematica]. E-mail: eliete@pucrs.br; rpp@mat.pucrs.br; Vilhena, Marco Tullio de [Pontificia Univ. Catolica do Rio Grande do Sul, Porto Alegre, RS (Brazil). Instituto de Matematica]. E-mail: vilhena@mat.ufrgs.br; Barros, Ricardo Carvalho de [Universidade do Estado, Nova Friburgo, RJ (Brazil). Instituto Politecnico]. E-mail: ricardo@iprj.uerj.br

2003-07-01

In this paper we report advances about the three-dimensional nodal discrete-ordinates approximations of neutron transport equation for Cartesian geometry. We use the combined collocation method of the angular variables and nodal approach for the spatial variables. By nodal approach we mean the iterated transverse integration of the S{sub N} equations. This procedure leads to the set of one-dimensional averages angular fluxes in each spatial variable. The resulting system of equations is solved with the LTS{sub N} method, first applying the Laplace transform to the set of the nodal S{sub N} equations and then obtained the solution by symbolic computation. We include the LTS{sub N} method by diagonalization to solve the nodal neutron transport equation and then we outline the convergence of these nodal-LTS{sub N} approximations with the help of a norm associated to the quadrature formula used to approximate the integral term of the neutron transport equation. (author)

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)

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)

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.

Analysing radio-frequency coil arrays in high-field magnetic resonance imaging by the combined field integral equation method

Energy Technology Data Exchange (ETDEWEB)

Wang Shumin; Duyn, Jeff H [Laboratory of Functional and Molecular Imaging, National Institute of Neurological Disorders and Stroke, National Institutes of Health, 10 Center Drive, 10/B1D728, Bethesda, MD 20892 (United States)

2006-06-21

We present the combined field integral equation (CFIE) method for analysing radio-frequency coil arrays in high-field magnetic resonance imaging (MRI). Three-dimensional models of coils and the human body were used to take into account the electromagnetic coupling. In the method of moments formulation, we applied triangular patches and the Rao-Wilton-Glisson basis functions to model arbitrarily shaped geometries. We first examined a rectangular loop coil to verify the CFIE method and also demonstrate its efficiency and accuracy. We then studied several eight-channel receive-only head coil arrays for 7.0 T SENSE functional MRI. Numerical results show that the signal dropout and the average SNR are two major concerns in SENSE coil array design. A good design should be a balance of these two factors.

High Weak Order Methods for Stochastic Differential Equations Based on Modified Equations

KAUST Repository

Abdulle, Assyr; Cohen, David; Vilmart, Gilles; Zygalakis, Konstantinos C.

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

Computational methods and modeling. 3. Adaptive Mesh Refinement for the Nodal Integral Method and Application to the Convection-Diffusion Equation

International Nuclear Information System (INIS)

Torej, Allen J.; Rizwan-Uddin

2001-01-01

The nodal integral method (NIM) has been developed for several problems, including the Navier-Stokes equations, the convection-diffusion equation, and the multigroup neutron diffusion equations. The coarse-mesh efficiency of the NIM is not fully realized in problems characterized by a wide range of spatial scales. However, the combination of adaptive mesh refinement (AMR) capability with the NIM can recover the coarse mesh efficiency by allowing high degrees of resolution in specific localized areas where it is needed and by using a lower resolution everywhere else. Furthermore, certain features of the NIM can be fruitfully exploited in the application of the AMR process. In this paper, we outline a general approach to couple nodal schemes with AMR and then apply it to the convection-diffusion (energy) equation. The development of the NIM with AMR capability (NIMAMR) is based on the well-known Berger-Oliger method for structured AMR. In general, the main components of all AMR schemes are 1. the solver; 2. the level-grid hierarchy; 3. the selection algorithm; 4. the communication procedures; 5. the governing algorithm. The first component, the solver, consists of the numerical scheme for the governing partial differential equations and the algorithm used to solve the resulting system of discrete algebraic equations. In the case of the NIM-AMR, the solver is the iterative approach to the solution of the set of discrete equations obtained by applying the NIM. Furthermore, in the NIM-AMR, the level-grid hierarchy (the second component) is based on the Hierarchical Adaptive Mesh Refinement (HAMR) system,6 and hence, the details of the hierarchy are omitted here. In the selection algorithm, regions of the domain that require mesh refinement are identified. The criterion to select regions for mesh refinement can be based on the magnitude of the gradient or on the Richardson truncation error estimate. Although an excellent choice for the selection criterion, the Richardson

Null Space Integration Method for Constrained Multibody Systems with No Constraint Violation

International Nuclear Information System (INIS)

Terze, Zdravko; Lefeber, Dirk; Muftic, Osman

2001-01-01

A method for integrating equations of motion of constrained multibody systems with no constraint violation is presented. A mathematical model, shaped as a differential-algebraic system of index 1, is transformed into a system of ordinary differential equations using the null-space projection method. Equations of motion are set in a non-minimal form. During integration, violations of constraints are corrected by solving constraint equations at the position and velocity level, utilizing the metric of the system's configuration space, and projective criterion to the coordinate partitioning method. The method is applied to dynamic simulation of 3D constrained biomechanical system. The simulation results are evaluated by comparing them to the values of characteristic parameters obtained by kinematics analysis of analyzed motion based unmeasured kinematics data

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.)

Numerical Methods for Partial Differential Equations

CERN Document Server

Guo, Ben-yu

1987-01-01

These Proceedings of the first Chinese Conference on Numerical Methods for Partial Differential Equations covers topics such as difference methods, finite element methods, spectral methods, splitting methods, parallel algorithm etc., their theoretical foundation and applications to engineering. Numerical methods both for boundary value problems of elliptic equations and for initial-boundary value problems of evolution equations, such as hyperbolic systems and parabolic equations, are involved. The 16 papers of this volume present recent or new unpublished results and provide a good overview of current research being done in this field in China.

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.

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.

analysis of large electromagnetic pulse simulators using the electric field integral equation method in time domain

International Nuclear Information System (INIS)

Jamali, J.; Aghajafari, R.; Moini, R.; Sadeghi, H.

2002-01-01

A time-domain approach is presented to calculate electromagnetic fields inside a large Electromagnetic Pulse (EMP) simulator. This type of EMP simulator is used for studying the effect of electromagnetic pulses on electrical apparatus in various structures such as vehicles, a reoplanes, etc. The simulator consists of three planar transmission lines. To solve the problem, we first model the metallic structure of the simulator as a grid of conducting wires. The numerical solution of the governing electric field integral equation is then obtained using the method of moments in time domain. To demonstrate the accuracy of the model, we consider a typical EMP simulator. The comparison of our results with those obtained experimentally in the literature validates the model introduced in this paper

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

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.

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.

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

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.

A Comparison of Kernel Equating and Traditional Equipercentile Equating Methods and the Parametric Bootstrap Methods for Estimating Standard Errors in Equipercentile Equating

Science.gov (United States)

Choi, Sae Il

2009-01-01

This study used simulation (a) to compare the kernel equating method to traditional equipercentile equating methods under the equivalent-groups (EG) design and the nonequivalent-groups with anchor test (NEAT) design and (b) to apply the parametric bootstrap method for estimating standard errors of equating. A two-parameter logistic item response…

The H-N method for solving linear transport equation: theory and application

International Nuclear Information System (INIS)

Kaskas, A.; Gulecyuz, M.C.; Tezcan, C.

2002-01-01

The system of singular integral equation which is obtained from the integro-differential form of the linear transport equation as a result of Placzec lemma is solved. Application are given using the exit distributions and the infinite medium Green's function. The same theoretical results are also obtained with the use of the singular eigenfunction of the method of elementary solutions

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

Evolution equation of Lie-type for finite deformations, time-discrete integration, and incremental methods

Czech Academy of Sciences Publication Activity Database

Fiala, Zdeněk

2015-01-01

Roč. 226, č. 1 (2015), s. 17-35 ISSN 0001-5970 R&D Projects: GA ČR(CZ) GA103/09/2101 Institutional support: RVO:68378297 Keywords : solid mechanics * finite deformations * evolution equation of Lie-type * time-discrete integration Subject RIV: BA - General Mathematics OBOR OECD: Statistics and probability Impact factor: 1.694, year: 2015 http://link.springer.com/article/10.1007%2Fs00707-014-1162-9#page-1

Fast pressure-correction method for incompressible Navier-Stokes equations in curvilinear coordinates

Science.gov (United States)

Aithal, Abhiram; Ferrante, Antonino

2017-11-01

In order to perform direct numerical simulations (DNS) of turbulent flows over curved surfaces and axisymmetric bodies, we have developed the numerical methodology to solve the incompressible Navier-Stokes (NS) equations in curvilinear coordinates for orthogonal meshes. The orthogonal meshes are generated by solving a coupled system of non-linear Poisson equations. The NS equations in orthogonal curvilinear coordinates are discretized in space on a staggered mesh using second-order central-difference scheme and are solved with an FFT-based pressure-correction method. The momentum equation is integrated in time using the second-order Adams-Bashforth scheme. The velocity field is advanced in time by applying the pressure correction to the approximate velocity such that it satisfies the divergence free condition. The novelty of the method stands in solving the variable coefficient Poisson equation for pressure using an FFT-based Poisson solver rather than the slower multigrid methods. We present the verification and validation results of the new numerical method and the DNS results of transitional flow over a curved axisymmetric body.

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)

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)

High Energy Laser Beam Propagation in the Atmosphere: The Integral Invariants of the Nonlinear Parabolic Equation and the Method of Moments

Science.gov (United States)

Manning, Robert M.

2012-01-01

The method of moments is used to define and derive expressions for laser beam deflection and beam radius broadening for high-energy propagation through the Earth s atmosphere. These expressions are augmented with the integral invariants of the corresponding nonlinear parabolic equation that describes the electric field of high-energy laser beam to propagation to yield universal equations for the aforementioned quantities; the beam deflection is a linear function of the propagation distance whereas the beam broadening is a quadratic function of distance. The coefficients of these expressions are then derived from a thin screen approximation solution of the nonlinear parabolic equation to give corresponding analytical expressions for a target located outside the Earth s atmospheric layer. These equations, which are graphically presented for a host of propagation scenarios, as well as the thin screen model, are easily amenable to the phase expansions of the wave front for the specification and design of adaptive optics algorithms to correct for the inherent phase aberrations. This work finds application in, for example, the analysis of beamed energy propulsion for space-based vehicles.

On Newton-Kantorovich Method for Solving the Nonlinear Operator Equation

Directory of Open Access Journals (Sweden)

Hameed Husam Hameed

2015-01-01

Full Text Available We develop the Newton-Kantorovich method to solve the system of 2×2 nonlinear Volterra integral equations where the unknown function is in logarithmic form. A new majorant function is introduced which leads to the increment of the convergence interval. The existence and uniqueness of approximate solution are proved and a numerical example is provided to show the validation of the method.

Solution of Nonlinear Partial Differential Equations by New Laplace Variational Iteration Method

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Eman M. A. Hilal

2014-01-01

Full Text Available The aim of this study is to give a good strategy for solving some linear and nonlinear partial differential equations in engineering and physics fields, by combining Laplace transform and the modified variational iteration method. This method is based on the variational iteration method, Laplace transforms, and convolution integral, introducing an alternative Laplace correction functional and expressing the integral as a convolution. Some examples in physical engineering are provided to illustrate the simplicity and reliability of this method. The solutions of these examples are contingent only on the initial conditions.

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…

Solution of two group neutron diffusion equation by using homotopy analysis method

International Nuclear Information System (INIS)

Cavdar, S.

2010-01-01

The Homotopy Analysis Method (HAM), proposed in 1992 by Shi Jun Liao and has been developed since then, is based on differential geometry as well as homotopy which is a fundamental concept in topology. It has proved to be useful for obtaining series solutions of many such problems involving algebraic, linear/non-linear, ordinary/partial differential equations, differential-integral equations, differential-difference equations, and coupled equations of them. Briefly, through HAM, it is possible to construct a continuous mapping of an initial guess approximation to the exact solution of the equation of concern. An auxiliary linear operator is chosen to construct such kind of a continuous mapping and an auxiliary parameter is used to ensure the convergence of series solution. We present the solutions of two-group neutron diffusion equation through HAM in this work. We also compare the results with that obtained by other well-known solution analytical and numeric methods.

A method for exponential propagation of large systems of stiff nonlinear differential equations

Science.gov (United States)

Friesner, Richard A.; Tuckerman, Laurette S.; Dornblaser, Bright C.; Russo, Thomas V.

1989-01-01

A new time integrator for large, stiff systems of linear and nonlinear coupled differential equations is described. For linear systems, the method consists of forming a small (5-15-term) Krylov space using the Jacobian of the system and carrying out exact exponential propagation within this space. Nonlinear corrections are incorporated via a convolution integral formalism; the integral is evaluated via approximate Krylov methods as well. Gains in efficiency ranging from factors of 2 to 30 are demonstrated for several test problems as compared to a forward Euler scheme and to the integration package LSODE.

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)

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.)

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

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.

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.

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

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.

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.

Receptor binding kinetics equations: Derivation using the Laplace transform method.

Science.gov (United States)

Hoare, Sam R J

Measuring unlabeled ligand receptor binding kinetics is valuable in optimizing and understanding drug action. Unfortunately, deriving equations for estimating kinetic parameters is challenging because it involves calculus; integration can be a frustrating barrier to the pharmacologist seeking to measure simple rate parameters. Here, a well-known tool for simplifying the derivation, the Laplace transform, is applied to models of receptor-ligand interaction. The method transforms differential equations to a form in which simple algebra can be applied to solve for the variable of interest, for example the concentration of ligand-bound receptor. The goal is to provide instruction using familiar examples, to enable investigators familiar with handling equilibrium binding equations to derive kinetic equations for receptor-ligand interaction. First, the Laplace transform is used to derive the equations for association and dissociation of labeled ligand binding. Next, its use for unlabeled ligand kinetic equations is exemplified by a full derivation of the kinetics of competitive binding equation. Finally, new unlabeled ligand equations are derived using the Laplace transform. These equations incorporate a pre-incubation step with unlabeled or labeled ligand. Four equations for measuring unlabeled ligand kinetics were compared and the two new equations verified by comparison with numerical solution. Importantly, the equations have not been verified with experimental data because no such experiments are evident in the literature. Equations were formatted for use in the curve-fitting program GraphPad Prism 6.0 and fitted to simulated data. This description of the Laplace transform method will enable pharmacologists to derive kinetic equations for their model or experimental paradigm under study. Application of the transform will expand the set of equations available for the pharmacologist to measure unlabeled ligand binding kinetics, and for other time

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...

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

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.)

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

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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.

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)

Analysis of Buried Dielectric Objects Using Higher-Order MoM for Volume Integral Equations

DEFF Research Database (Denmark)

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

2004-01-01

A higher-order method of moments (MoM) is applied to solve a volume integral equation for dielectric objects in layered media. In comparison to low-order methods, the higher-order MoM, which is based on higher-order hierarchical Legendre vector basis functions and curvilinear hexahedral elements,...

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

Solution of the radiative heat transfer equation with internal energy sources in a slab by the GFD method for anisotropic albedo

Energy Technology Data Exchange (ETDEWEB)

Azevedo, Fabio Souto de, E-mail: fabio.azevedo@ufrgs.b [Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, RS (Brazil). Inst. de Matematica; Sauter, Esequia, E-mail: esequia.sauter@canoas.ifrs.edu.b [Instituto Federal do Rio Grande do Sul (IFRS), Canoas, RS (Brazil); Thompson, Mark; Vilhena, Marco Tulio B., E-mail: mark.thompson@mat.ufrgs.b, E-mail: vilhena@mat.ufrgs.b [Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, RS (Brazil). Programa de Pos-Graduacao em Matematica Aplicada

2011-07-01

In this work we apply the Green Function Decomposition Method the radiative transport equation in a slab. The method consists in converting the radiative transport equation into a integral equation and projecting the integral operators involved into a finite dimensional space. This methodology does not involve an a priori discretization on the angular variable {mu}, requiring only that the kernel is numerically integrated on {mu}. Numerical results are provided for isotropic, linearly anisotropic, and Rayleigh scattering near the unitary albedo. (author)

A New Fractional Projective Riccati Equation Method for Solving Fractional Partial Differential Equations

International Nuclear Information System (INIS)

Feng Qing-Hua

2014-01-01

In this paper, a new fractional projective Riccati equation method is proposed to establish exact solutions for fractional partial differential equations in the sense of modified Riemann—Liouville derivative. This method can be seen as the fractional version of the known projective Riccati equation method. For illustrating the validity of this method, we apply this method to solve the space-time fractional Whitham—Broer—Kaup (WBK) equations and the nonlinear fractional Sharma—Tasso—Olever (STO) equation, and as a result, some new exact solutions for them are obtained. (general)

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%.

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.

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

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

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.

A generalized simplest equation method and its application to the Boussinesq-Burgers equation.

Science.gov (United States)

Sudao, Bilige; Wang, Xiaomin

2015-01-01

In this paper, a generalized simplest equation method is proposed to seek exact solutions of nonlinear evolution equations (NLEEs). In the method, we chose a solution expression with a variable coefficient and a variable coefficient ordinary differential auxiliary equation. This method can yield a Bäcklund transformation between NLEEs and a related constraint equation. By dealing with the constraint equation, we can derive infinite number of exact solutions for NLEEs. These solutions include the traveling wave solutions, non-traveling wave solutions, multi-soliton solutions, rational solutions, and other types of solutions. As applications, we obtained wide classes of exact solutions for the Boussinesq-Burgers equation by using the generalized simplest equation method.

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

Transforming parts of a differential equations system to difference equations as a method for run-time savings in NONMEM.

Science.gov (United States)

Petersson, K J F; Friberg, L E; Karlsson, M O

2010-10-01

Computer models of biological systems grow more complex as computing power increase. Often these models are defined as differential equations and no analytical solutions exist. Numerical integration is used to approximate the solution; this can be computationally intensive, time consuming and be a large proportion of the total computer runtime. The performance of different integration methods depend on the mathematical properties of the differential equations system at hand. In this paper we investigate the possibility of runtime gains by calculating parts of or the whole differential equations system at given time intervals, outside of the differential equations solver. This approach was tested on nine models defined as differential equations with the goal to reduce runtime while maintaining model fit, based on the objective function value. The software used was NONMEM. In four models the computational runtime was successfully reduced (by 59-96%). The differences in parameter estimates, compared to using only the differential equations solver were less than 12% for all fixed effects parameters. For the variance parameters, estimates were within 10% for the majority of the parameters. Population and individual predictions were similar and the differences in OFV were between 1 and -14 units. When computational runtime seriously affects the usefulness of a model we suggest evaluating this approach for repetitive elements of model building and evaluation such as covariate inclusions or bootstraps.

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.

A conservative finite difference method for the numerical solution of plasma fluid equations

International Nuclear Information System (INIS)

Colella, P.; Dorr, M.R.; Wake, D.D.

1999-01-01

This paper describes a numerical method for the solution of a system of plasma fluid equations. The fluid model is similar to those employed in the simulation of high-density, low-pressure plasmas used in semiconductor processing. The governing equations consist of a drift-diffusion model of the electrons, together with an internal energy equation, coupled via Poisson's equation to a system of Euler equations for each ion species augmented with electrostatic force, collisional, and source/sink terms. The time integration of the full system is performed using an operator splitting that conserves space charge and avoids dielectric relaxation timestep restrictions. The integration of the individual ion species and electrons within the time-split advancement is achieved using a second-order Godunov discretization of the hyperbolic terms, modified to account for the significant role of the electric field in the propagation of acoustic waves, combined with a backward Euler discretization of the parabolic terms. Discrete boundary conditions are employed to accommodate the plasma sheath boundary layer on underresolved grids. The algorithm is described for the case of a single Cartesian grid as the first step toward an implementation on a locally refined grid hierarchy in which the method presented here may be applied on each refinement level

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.

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.

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.

Numerical methods for engine-airframe integration

International Nuclear Information System (INIS)

Murthy, S.N.B.; Paynter, G.C.

1986-01-01

Various papers on numerical methods for engine-airframe integration are presented. The individual topics considered include: scientific computing environment for the 1980s, overview of prediction of complex turbulent flows, numerical solutions of the compressible Navier-Stokes equations, elements of computational engine/airframe integrations, computational requirements for efficient engine installation, application of CAE and CFD techniques to complete tactical missile design, CFD applications to engine/airframe integration, and application of a second-generation low-order panel methods to powerplant installation studies. Also addressed are: three-dimensional flow analysis of turboprop inlet and nacelle configurations, application of computational methods to the design of large turbofan engine nacelles, comparison of full potential and Euler solution algorithms for aeropropulsive flow field computations, subsonic/transonic, supersonic nozzle flows and nozzle integration, subsonic/transonic prediction capabilities for nozzle/afterbody configurations, three-dimensional viscous design methodology of supersonic inlet systems for advanced technology aircraft, and a user's technology assessment

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).

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

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.

Test equating methods and practices

CERN Document Server

Kolen, Michael J

1995-01-01

In recent years, many researchers in the psychology and statistical communities have paid increasing attention to test equating as issues of using multiple test forms have arisen and in response to criticisms of traditional testing techniques This book provides a practically oriented introduction to test equating which both discusses the most frequently used equating methodologies and covers many of the practical issues involved The main themes are - the purpose of equating - distinguishing between equating and related methodologies - the importance of test equating to test development and quality control - the differences between equating properties, equating designs, and equating methods - equating error, and the underlying statistical assumptions for equating The authors are acknowledged experts in the field, and the book is based on numerous courses and seminars they have presented As a result, educators, psychometricians, professionals in measurement, statisticians, and students coming to the subject for...