A Boundary Integral Equation Approach for Boundary Problem of Laplace Equation
SUNJian-she; YELiu-qing
2003-01-01
Using the second Green formula, the boundary problem of Laplace equation satisfied by potential function of static electric field is transformed to the problem of the boundary integral equation,and then a boundary integral equation approach is established by partitioning boundary using linear boundary element.
Iterative solution of Hermite boundary integral equations
Gray, Leonard J [ORNL; Nintcheu Fata, Sylvain [ORNL; Ma, Ding [ORNL
2008-01-01
An efficient iterative method for the solution of the linear equations arising from a Hermite boundary integral approximation has been developed. Along with equations for the boundary unknowns, the Hermite system incorporates equations for the first-order surface derivatives (gradient) of the potential, and is therefore substantially larger than the matrix for a corresponding linear approximation. However, by exploiting the structure of the Hermite matrix, a two-level iterative algorithm has been shown to provide a very efficient solution algorithm. In this approach, the boundary function unknowns are treated separately from the gradient, taking advantage of the sparsity and near-positive definiteness of the gradient equations. In test problems, the new algorithm significantly reduced computation time compared to iterative solution applied to the full matrix. This approach should prove to be even more effective for the larger systems encountered in three-dimensional analysis, and increased efficiency should come from pre-conditioning of the non-sparse matrix component.
A NOVEL BOUNDARY INTEGRAL EQUATION METHOD FOR LINEAR ELASTICITY--NATURAL BOUNDARY INTEGRAL EQUATION
Niu Zhongrong; Wang Xiuxi; Zhou Huanlin; Zhang Chenli
2001-01-01
The boundary integral equation (BIE) of displacement derivatives is put at a disadvantage for the difficulty involved in the evaluation of the hypersingular integrals. In this paper, the operators δij and εij are used to act on the derivative BIE. The boundary displacements, tractions and displacement derivatives are transformed into a set of new boundary tensors as boundary variables. A new BIE formulation termed natural boundary integral equation (NBIE) is obtained. The NBIE is applied to solving two-dimensional elasticity problems. In the NBIE only the strongly singular integrals are contained. The Cauchy principal value integrals occurring in the NBIE are evaluated. A combination of the NBIE and displacement BIE can be used to directly calculate the boundary stresses. The numerical results of several examples demonstrate the accuracy of the NBIE.
Computing the Casimir force using regularized boundary integral equations
Kilen, Isak; Jakobsen, Per Kristen
2014-11-01
In this paper we use a novel regularization procedure to reduce the calculation of the Casimir force for 2D scalar fields between compact objects to the solution of a classical integral equation defined on the boundaries of the objects. The scalar fields are subject to Dirichlet boundary conditions on the object boundaries. We test the integral equation by comparing with what we get for parallel plates, concentric circles and adjacent circles using mode summation and the functional integral method. We show how symmetries in the shapes and configuration of boundaries can easily be incorporated into our method and that it leads to fast evaluation of the Casimir force for symmetric situations.
APPLICATION OF BOUNDARY INTEGRAL EQUATION METHOD FOR THERMOELASTICITY PROBLEMS
Vorona Yu.V.; Kara I.D.
2015-01-01
Boundary Integral Equation Method is used for solving analytically the problems of coupled thermoelastic spherical wave propagation. The resulting mathematical expressions coincide with the solutions obtained in a conventional manner.
APPLICATION OF BOUNDARY INTEGRAL EQUATION METHOD FOR THERMOELASTICITY PROBLEMS
Vorona Yu.V.
2015-12-01
Full Text Available Boundary Integral Equation Method is used for solving analytically the problems of coupled thermoelastic spherical wave propagation. The resulting mathematical expressions coincide with the solutions obtained in a conventional manner.
On the Implementation of 3D Galerkin Boundary Integral Equations
Nintcheu Fata, Sylvain [ORNL; Gray, Leonard J [ORNL
2010-01-01
In this article, a reverse contribution technique is proposed to accelerate the construction of the dense influence matrices associated with a Galerkin approximation of singular and hypersingular boundary integral equations of mixed-type in potential theory. In addition, a general-purpose sparse preconditioner for boundary element methods has also been developed to successfully deal with ill-conditioned linear systems arising from the discretization of mixed boundary-value problems on non-smooth surfaces. The proposed preconditioner, which originates from the precorrected-FFT method, is sparse, easy to generate and apply in a Krylov subspace iterative solution of discretized boundary integral equations. Moreover, an approximate inverse of the preconditioner is implicitly built by employing an incomplete LU factorization. Numerical experiments involving mixed boundary-value problems for the Laplace equation are included to illustrate the performance and validity of the proposed techniques.
Boundary Integral Equations and A Posteriori Error Estimates
YU Dehao; ZHAO Longhua
2005-01-01
Adaptive methods have been rapidly developed and applied in many fields of scientific and engineering computing. Reliable and efficient a posteriori error estimates play key roles for both adaptive finite element and boundary element methods. The aim of this paper is to develop a posteriori error estimates for boundary element methods. The standard a posteriori error estimates for boundary element methods are obtained from the classical boundary integral equations. This paper presents hyper-singular a posteriori error estimates based on the hyper-singular integral equations. Three kinds of residuals are used as the estimates for boundary element errors. The theoretical analysis and numerical examples show that the hyper-singular residuals are good a posteriori error indicators in many adaptive boundary element computations.
Galerkin Boundary Integral Analysis for the 3D Helmholtz Equation
Swager, Melissa [Emporia State University; Gray, Leonard J [ORNL; Nintcheu Fata, Sylvain [ORNL
2010-01-01
A linear element Galerkin boundary integral analysis for the three-dimensional Helmholtz equation is presented. The emphasis is on solving acoustic scattering by an open (crack) surface, and to this end both a dual equation formulation and a symmetric hypersingular formulation have been developed. All singular integrals are defined and evaluated via a boundary limit process, facilitating the evaluation of the (finite) hypersingular Galerkin integral. This limit process is also the basis for the algorithm for post-processing of the surface gradient. The analytic integrations required by the limit process are carried out by employing a Taylor series expansion for the exponential factor in the Helmholtz fundamental solutions. For the open surface, the implementations are validated by comparing the numerical results obtained by using the two different methods.
Advanced applications of boundary-integral equation methods
The BIE (boundary integral equation) method is based on the numerical solution of a set of integral constraint equations which couple boundary tractions (stresses) to boundary displacements. Thus the dimensionality of the problem is reduced by one; only boundary geometry and data are discretized. Stresses at any set of selected interior points are computed following the boundary solution without any further numerical approximations. Thus, the BIE method has inherently greater resolution capability for stress gradients than does the finite element method. Conversely, the BIE method is not efficient for problems involving significant inhomogeneity such as in multi-thin-layered materials, or in elastoplasticity. Some progress in applyiing the BIE method to the latter problem has been made but much more work remains. Further, the BIE method is only optional for problems with significant stress risers, and only when boundary stresses are most important. Interior stress calculations are expensive, per point, and can drive the solution costs up rapidly. The current report summarizes some of the advanced elastic applications of fracture mechanics and three-dimensional stress analysis, while referencing some of the much broader developmental effort. Future emphasis is needed to exploit the BIE method in conjunction with other techniques such as the finite element method through the creation of hybrid stress analysis methods
Advanced applications of boundary-integral equation methods
Numerical analysis has become the basic tool for both design and research problems in solid mechanics. The boundary-integral equation (BIE) method is based on classical mathematical techniques but is finding new life as a basic stress analysis tool for engineering applications. The BIE method is based on the numerical solution of a set of integral constraint equations which couple boundary tractions (stresses) to boundary displacements. Thus the dimensionality of the problem is reduced by one; only boundary geometry and data are discretized. Stresses at any set of selected interior points are computed following the boundary solution without any further numerical approximations. Thus, the BIE method has inherently greater resolution capability for stress gradients than does the finite element method. Conversely, the BIE method is not efficient for problems involving significant inhomogeneity such as in multi-thin-layered materials, or in elastoplasticity. Some progress in applying the BIE method to the latter problem has been made but much more work remains. Further, the BIE method is only optional for problems with significant stress risers, and only when boundary stresses are more important. Interior stress calculations are expensive, per point, and can drive the solution costs up rapidly. The current report summarizes some of the advanced elastic applications of fracture mechanics and three-dimensional stress analysis, while referring some of the much broader developmental effort. (Auth.)
Tokamak plasma shape identification based on the boundary integral equations
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)
A spectral boundary integral equation method for the 2D Helmholtz equation
In this paper, we present a new numerical formulation of solving the boundary integral equations reformulated from the Helmholtz equation. The boundaries of the problems are assumed to be smooth closed contours. The solution on the boundary is treated as a periodic function, which is in turn approximated by a truncated Fourier series. A Fourier collocation method is followed in which the boundary integral equation is transformed into a system of algebraic equations. It is shown that in order to achieve spectral accuracy for the numerical formulation, the non-smoothness of the integral kernels, associated with the Helmholtz equation, must be carefully removed. The emphasis of the paper is on investigating the essential elements of removing the non-smoothness of the integral kernels in the spectral implementation. The present method is robust for a general smooth boundary contour. Aspects of efficient implementation of the method using FFT are also discussed. Numerical examples of wave scattering are given in which the exponential accuracy of the present numerical method is demonstrated. 15 refs., 3 figs., 4 tabs
Advanced applications of boundary-integral equation methods
Numerical analysis has become the basic tool for both design and research problems in solid mechanics. The need for accuracy and detail, plus the availablity of the high speed computer has led to the development of many new modeling methods ranging from general purpose structural analysis finite element programs to special purpose research programs. The boundary-integral equation (BIE) method is based on classical mathematical techniques but is finding new life as a basic stress analysis tool for engineering applications. The paper summarizes some advanced elastic applications of fracture mechanics and three-dimensional stress analysis, while referencing some of the much broader developmental effort. Future emphasis is needed to exploit the BIE method in conjunction with other techniques such as the finite element method through the creation of hybrid stress analysis methods. (Auth.)
The reflection equation algebra of Sklyanin is extended to the supersymmetric case. A graded reflection equation algebra is proposed and the corresponding graded (supersymmetric) boundary quantum inverse scattering method (QISM) is formulated. As an application, integrable open-boundary conditions for the doped spin-1 chain of the supersymmetric t-J model are studied in the framework of the boundary QISM. Diagonal boundary K-matrices are found and four classes of integrable boundary terms are determined. (author)
RESTRICTED NONLINEAR APPROXIMATION AND SINGULAR SOLUTIONS OF BOUNDARY INTEGRAL EQUATIONS
Reinhard Hochmuth
2002-01-01
This paper studies several problems, which are potentially relevant for the construction of adaptive numerical schemes. First, biorthogonal spline wavelets on [0,1 ] are chosen as a starting point for characterizations of functions in Besov spaces B , (0,1) with 0＜σ＜∞ and (1+σ)-1＜τ＜∞. Such function spaces are known to be related to nonlinear approximation. Then so called restricted nonlinear approximation procedures with respect to Sobolev space norms are considered. Besides characterization results Jackson type estimates for various tree-type and tresholding algorithms are investigated. Finally known approximation results for geometry induced singularity functions of boundary integeral equations are combined with the characterization results for restricted nonlinear approximation to show Besov space regularity results.
Moiseiwitsch, B L
2005-01-01
Two distinct but related approaches hold the solutions to many mathematical problems--the forms of expression known as differential and integral equations. The method employed by the integral equation approach specifically includes the boundary conditions, which confers a valuable advantage. In addition, the integral equation approach leads naturally to the solution of the problem--under suitable conditions--in the form of an infinite series.Geared toward upper-level undergraduate students, this text focuses chiefly upon linear integral equations. It begins with a straightforward account, acco
Ntouyas SotirisK
2011-01-01
Full Text Available This paper studies a boundary value problem of nonlinear fractional differential equations of order with three-point integral boundary conditions. Some new existence and uniqueness results are obtained by using standard fixed point theorems and Leray-Schauder degree theory. Our results are new in the sense that the nonlocal parameter in three-point integral boundary conditions appears in the integral part of the conditions in contrast to the available literature on three-point boundary value problems which deals with the three-point boundary conditions restrictions on the solution or gradient of the solution of the problem. Some illustrative examples are also discussed.
Mouffak Benchohra; Fatima-Zohra Mostefai
2012-01-01
The aim of this paper is to investigate a class of boundary value problems for fractional differential equations involving nonlinear integral conditions. The main tool used in our considerations is the technique associated with measures of weak noncompactness.
A well-conditioned boundary integral equation for transmission problems of electromagnetism
Levadoux, David; Millot, Florence; Pernet, Sébastien
2015-01-01
We propose a new well-conditioned boundary integral equation to solve transmission problems of electromagnetism. This equation is well posed and appears as a compact perturbation of the identity leading to fast iterative solutions without the help of any preconditioner. Some numerical experiments confirm this result.
Boundary integral equation Neumann-to-Dirichlet map method for gratings in conical diffraction.
Wu, Yumao; Lu, Ya Yan
2011-06-01
Boundary integral equation methods for diffraction gratings are particularly suitable for gratings with complicated material interfaces but are difficult to implement due to the quasi-periodic Green's function and the singular integrals at the corners. In this paper, the boundary integral equation Neumann-to-Dirichlet map method for in-plane diffraction problems of gratings [Y. Wu and Y. Y. Lu, J. Opt. Soc. Am. A26, 2444 (2009)] is extended to conical diffraction problems. The method uses boundary integral equations to calculate the so-called Neumann-to-Dirichlet maps for homogeneous subdomains of the grating, so that the quasi-periodic Green's functions can be avoided. Since wave field components are coupled on material interfaces with the involvement of tangential derivatives, a least squares polynomial approximation technique is developed to evaluate tangential derivatives along these interfaces for conical diffraction problems. Numerical examples indicate that the method performs equally well for dielectric or metallic gratings. PMID:21643404
Shih, H.R. [Jackson State Univ., MS (United States); Duffield, R.C.; Lin, J. [Univ. of Missouri, Columbia, MO (United States)
1996-10-01
An integral equation formulation and a numerical procedure for a boundary-finite element technique are developed for the static analysis of a stiffened plate with eccentric stiffeners. This formulation employs the fundamental solution associated with unstiffened plate bending and plane stress problems. With this approach, the resulting integral equation not only contained integrals along the perimeter of the stiffened but additional integrals along the stiffeners and the interface between the plate and its stiffeners. Thus the domain of the plate has to be divided into zones between the stiffeners. Each zone is modeled by boundary elements and stiffeners by finite elements. In this paper, the boundary element solution procedures for plate bending and in-plane problems are presented. The zone technique which permits coupling of unstiffened plate boundary element with stiffener finite elements is presented as well. Numerical example is given to demonstrate the effectiveness of this approach.
Analytical Solution of Boundary Integral Equations for 2-D Steady Linear Wave Problems
J.M. Chuang
2005-01-01
Based on the Fourier transform, the analytical solution of boundary integral equations formulated for the complex velocity of a 2-D steady linear surface flow is derived. It has been found that before the radiation condition is imposed,free waves appear both far upstream and downstream. In order to cancel the free waves in far upstream regions, the eigensolution of a specific eigenvalue, which satisfies the homogeneous boundary integral equation, is found and superposed to the analytical solution. An example, a submerged vortex, is used to demonstrate the derived analytical solution. Furthermore,an analytical approach to imposing the radiation condition in the numerical solution of boundary integral equations for 2-D steady linear wave problems is proposed.
Retarded potentials and time domain boundary integral equations a road map
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...
Yang Xiao-Jun
2015-01-01
Full Text Available In the present paper we investigate the fractal boundary value problems for the Fredholm\\Volterra integral equations, heat conduction and wave equations by using the local fractional decomposition method. The operator is described by the local fractional operators. The four illustrative examples are given to elaborate the accuracy and reliability of the obtained results. [Projekat Ministarstva nauke Republike Srbije, br. OI 174001, III41006 i br. TI 35006
Mouffak Benchohra
2012-01-01
Full Text Available The aim of this paper is to investigate a class of boundary value problems for fractional differential equations involving nonlinear integral conditions. The main tool used in our considerations is the technique associated with measures of weak noncompactness.
Brahim Tellab
2016-01-01
Full Text Available In this paper, we investigate the existence and uniqueness of solutions for second order nonlinear fractional differential equation with integral boundary conditions. Our result is an application of the Banach contraction principle and the Krasnoselskii fixed point theorem.
Boundary integral equation formulations for free-surface flow problems in two and three dimensions
Romate, J.E.; Zandbergen, P.J.
1989-01-01
To compute the transient solution of free-surface flow problems in two and three dimensions boundary integral equation formulations are considered. Consistent lower and higher order approximations based on small curvature expansions are compared and applied to a time-dependent, linear free-surface wave problem.
Chen, Ke [Univ. of Liverpool (United Kingdom)
1996-12-31
We study various preconditioning techniques for the iterative solution of boundary integral equations, and aim to provide a theory for a class of sparse preconditioners. Two related ideas are explored here: singularity separation and inverse approximation. Our preliminary conclusion is that singularity separation based preconditioners perform better than approximate inverse based while it is desirable to have both features.
Sareni, Bruno; Krähenbühl, Laurent; Beroual, Abderrahmane; Nicolas, Alain; Brosseau, C.
1997-01-01
We present a numerical method based upon the resolution of boundary integral equations for the calculation of the effective permittivity of a lossless composite structure consisting of a two component mixture, each with its own dielectric anti shape characteristics. The topological arrangements considered are periodic lattices inhomogeneities. Our numerical simulations are compared to the effective medium approach and with results of previous works.
Kitanine, N; Niccoli, G
2014-01-01
We solve the longstanding problem to define a functional characterization of the spectrum of the transfer matrix associated to the most general spin-1/2 representations of the 6-vertex reflection algebra for general inhomogeneous chains. The corresponding homogeneous limit reproduces the spectrum of the Hamiltonian of the spin-1/2 open XXZ and XXX quantum chains with the most general integrable boundaries. The spectrum is characterized by a second order finite difference functional equation of Baxter type with an inhomogeneous term which vanishes only for some special but yet interesting non-diagonal boundary conditions. This functional equation is shown to be equivalent to the known separation of variable (SOV) representation hence proving that it defines a complete characterization of the transfer matrix spectrum. The polynomial character of the Q-function allows us then to show that a finite system of equations of generalized Bethe type can be similarly used to describe the complete transfer matrix spectru...
Boundary integral equation methods in eigenvalue problems of elastodynamics and thin plates
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
Fatma Kanca
2013-01-01
This paper investigates the inverse problem of finding a time-dependent diffusion coefficient in a parabolic equation with the periodic boundary and integral overdetermination conditions. Under some assumption on the data, the existence, uniqueness, and continuous dependence on the data of the solution are shown by using the generalized Fourier method. The accuracy and computational efficiency of the proposed method are verified with the help of the numerical examples.
Solution of the Stokes system by boundary integral equations and fixed point iterative schemes
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
Tetervin, Neal; Lin, Chia Chiao
1951-01-01
A general integral form of the boundary-layer equation, valid for either laminar or turbulent incompressible boundary-layer flow, is derived. By using the experimental finding that all velocity profiles of the turbulent boundary layer form essentially a single-parameter family, the general equation is changed to an equation for the space rate of change of the velocity-profile shape parameter. The lack of precise knowledge concerning the surface shear and the distribution of the shearing stress across turbulent boundary layers prevented the attainment of a reliable method for calculating the behavior of turbulent boundary layers.
Hu, Fang Q.; Pizzo, Michelle E.; Nark, Douglas M.
2016-01-01
Based on the time domain boundary integral equation formulation of the linear convective wave equation, a computational tool dubbed Time Domain Fast Acoustic Scattering Toolkit (TD-FAST) has recently been under development. The time domain approach has a distinct advantage that the solutions at all frequencies are obtained in a single computation. In this paper, the formulation of the integral equation, as well as its stabilization by the Burton-Miller type reformulation, is extended to cases of a constant mean flow in an arbitrary direction. In addition, a "Source Surface" is also introduced in the formulation that can be employed to encapsulate regions of noise sources and to facilitate coupling with CFD simulations. This is particularly useful for applications where the noise sources are not easily described by analytical source terms. Numerical examples are presented to assess the accuracy of the formulation, including a computation of noise shielding by a thin barrier motivated by recent Historical Baseline F31A31 open rotor noise shielding experiments. Furthermore, spatial resolution requirements of the time domain boundary element method are also assessed using point per wavelength metrics. It is found that, using only constant basis functions and high-order quadrature for surface integration, relative errors of less than 2% may be obtained when the surface spatial resolution is 5 points-per-wavelength (PPW) or 25 points-per-wavelength squared (PPW2).
Boundary-integral equation in engineering stress and fracture mechanics. Technical report
A boundary-integral equation (BIE) method, as employed by a computer program called PESTIE (Plane Elastic Solution Technique of Integral Equations), is described for application to engineering stress and fracture mechanics problems of interest to the electric power generating industry. The paper describes a unique combination of several numerical analysis capabilities and user-oriented features that produce significant advantages over finite element programs and earlier BIE programs. These advantages are demonstrated by comparing numerical results and performance of PESTIE with those of other programs for a series of stress concentration and stress intensity factor problems. The examples include the effect of closely spaced notches (e.g. as encountered in the rim area of a turbine rotor spindle), and the stress intensity factor of a shallow wide crack (e.g. a longitudinal crack in a large diameter pipe)
Application of the boundary-integral-equation method to the three-dimensional thermoelastic problem
It is possible to solve three-dimensional steady state thermoelastic problems from data of temperature, fluxes, tractions, and displacements on the boundary alone. The numerical discretisation of these integral equations is based on a technique similar to that of finite elements. The boundary alone needs to be discretised. Two-dimensional examples are given to prove the accuracy and the feasibility. In the case of three-dimensional problems, the surface is represented by eight nodes quadrilateral elements and six nodes triangular elements. The unknown (temperature, flux, displacement, traction) may be considered to vary linearly, quadratically, or cubically, with respect to he intrinsic coordinates of each element. The integration is performed numerically using Gaussian quadrature formulas, for which the number of integration points is chosen automatically by the program, so that the upper bound of error in integration is minimized. In order to obtain a banded form matrix and also to be able to study elongated structures, the body is divided into subregions, for each of which the integral equations are written. The thermoelastic stress and deformation field are obtained from two successive calculations on the same mesh. The first gives the thermal field. The second, taking account of the thermal field, calculates the thermoelastic displacement and stress field. This type of approach is especially suited to complicated three-dimensional thick structures for which finite element procedures are very expensive. The data are simple to generate, for data on the boundary alone have to be given. Much time can be saved by the user of the program in data generations and data checks
A Fast Spectral Galerkin Method for Hypersingular Boundary Integral Equations in Potential Theory
Nintcheu Fata, Sylvain [ORNL; Gray, Leonard J [ORNL
2009-01-01
This research is focused on the development of a fast spectral method to accelerate the solution of three-dimensional hypersingular boundary integral equations of potential theory. Based on a Galerkin approximation, the Fast Fourier Transform and local interpolation operators, the proposed method is a generalization of the Precorrected-FFT technique to deal with double-layer potential kernels, hypersingular kernels and higher-order basis functions. Numerical examples utilizing piecewise linear shape functions are included to illustrate the performance of the method.
TIME–HARMONIC BEHAVIOUR OF CRACKED PIEZOELECTRIC SOLID BY BOUNDARY INTEGRAL EQUATION METHOD
Rangelov Tsviatko
2014-03-01
Full Text Available Anti-plane cracked functionally graded finite piezoelectric solid under time-harmonic elecromechanical load is studied by a non-hypersingular traction boundary integral equation method (BIEM. Exponentially varying material properties are considered. Numerical solutions are obtained by using Mathematica. The dependance of the intensity factors (IF - mechanical stress intensity factor (SIF and electrical field intensity factor (FIF on the inhomogeneous material parameters, on the type and frequency of the dynamic load and on the crack position are analyzed by numerical illustrative examples
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.
Boundary integral equation method for added mass in arrays of cylinders
The dynamic behavior of a group of cylinders in fluid, such as heat exchanger tubes and nuclear fuel assemblies, is strongly influenced by the surrounding fluid. Although, the added mass of such clusters of cylinders has been studied by many researchers with various analytical methods and numerical methods, no attempt has been made so far to analyze these problems by the Boundary Integral Equation Method (BIEM). This paper presents a BIEM model simulating the added mass arising when clusters of cylinders vibrate in an inviscid and incompressible fluid. In this model, perturbed fluid pressure is described by a two- dimensional Laplace equation. The primary advantage of this approach compared with other numerical methods, e.g., the finite element method (FEM), is that the integration and discretization of the model are only needed on boundary rather than in whole domain. Therefore, the proposed approach is much more economical than the finite element method. Various numerical examples are subsequently presented in this paper to illustrate the methodology and to demonstrate its accuracy
无
2010-01-01
This paper investigates the existence and multiplicity of nonnegative solutions to a singular nonlinear boundary value problem of second order differential equations with integral boundary conditions in a Banach space. The arguments are based on the construction of a nonempty bounded open convex set and fixed point index theory. Our nonlinearity possesses singularity and first derivative which makes it different with that in [10].
Boundary integral equation methods and numerical solutions thin plates on an elastic foundation
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...
We present a method to solve initial-boundary-value problems for linear and integrable nonlinear differential-difference evolution equations. The method is the discrete version of the one developed by A S Fokas to solve initial-boundary-value problems for linear and integrable nonlinear partial differential equations via an extension of the inverse scattering transform. The method takes advantage of the Lax pair formulation for both linear and nonlinear equations, and is based on the simultaneous spectral analysis of both parts of the Lax pair. A key role is also played by the global algebraic relation that couples all known and unknown boundary values. Even though additional technical complications arise in discrete problems compared to continuum ones, we show that a similar approach can also solve initial-boundary-value problems for linear and integrable nonlinear differential-difference equations. We demonstrate the method by solving initial-boundary-value problems for the discrete analogue of both the linear and the nonlinear Schrödinger equations, comparing the solution to those of the corresponding continuum problems. In the linear case we also explicitly discuss Robin-type boundary conditions not solvable by Fourier series. In the nonlinear case, we also identify the linearizable boundary conditions, we discuss the elimination of the unknown boundary datum, we obtain explicitly the linear and continuum limit of the solution, and we write the soliton solutions
Boundary integral equation method calculations of surface regression effects in flame spreading
Altenkirch, R. A.; Rezayat, M.; Eichhorn, R.; Rizzo, F. J.
1982-01-01
A solid-phase conduction problem that is a modified version of one that has been treated previously in the literature and is applicable to flame spreading over a pyrolyzing fuel is solved using a boundary integral equation (BIE) method. Results are compared to surface temperature measurements that can be found in the literature. In addition, the heat conducted through the solid forward of the flame, the heat transfer responsible for sustaining the flame, is also computed in terms of the Peclet number based on a heated layer depth using the BIE method and approximate methods based on asymptotic expansions. Agreement between computed and experimental results is quite good as is agreement between the BIE and the approximate results.
ZHAO Ming-hao; LI Dong-xia; SHEN Ya-peng
2005-01-01
The integral-differential equations for three-dimensional planar interfacial cracks of arbitrary shape in transversely isotropic bimaterials were derived by virtue of the Somigliana identity and the fundamental solutions, in which the displacement discontinuities across the crack faces are the unknowns to be determined. The interface is parallel to both the planes of isotropy. The singular behaviors of displacement and stress near the crack border were analyzed and the stress singularity indexes were obtained by integral equation method. The stress intensity factors were expressed in terms of the displacement discontinuities. In the non-oscillatory case, the hyper-singular boundary integral-differential equations were reduced to hyper-singular boundary integral equations similar to those of homogeneously isotropie materials.
Alsaedi Ahmed
2009-01-01
Full Text Available A generalized quasilinearization technique is developed to obtain a sequence of approximate solutions converging monotonically and quadratically to a unique solution of a boundary value problem involving Duffing type nonlinear integro-differential equation with integral boundary conditions. The convergence of order for the sequence of iterates is also established. It is found that the work presented in this paper not only produces new results but also yields several old results in certain limits.
Boundary transfer matrices and boundary quantum KZ equations
Vlaar, Bart, E-mail: Bart.Vlaar@nottingham.ac.uk [School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD (United Kingdom)
2015-07-15
A simple relation between inhomogeneous transfer matrices and boundary quantum Knizhnik-Zamolodchikov (KZ) equations is exhibited for quantum integrable systems with reflecting boundary conditions, analogous to an observation by Gaudin for periodic systems. Thus, the boundary quantum KZ equations receive a new motivation. We also derive the commutativity of Sklyanin’s boundary transfer matrices by merely imposing appropriate reflection equations, in particular without using the conditions of crossing symmetry and unitarity of the R-matrix.
Chatthai Thaiprayoon
2014-01-01
Full Text Available By developing a new comparison result and using the monotone iterative technique, we are able to obtain existence of minimal and maximal solutions of periodic boundary value problems for first-order impulsive functional integrodifferential equations with integral-jump conditions. An example is also given to illustrate our results.
Nittaya Pongarm
2013-01-01
Full Text Available This paper studies sufficient conditions for the existence of solutions to the problem of sequential derivatives of nonlinear q-difference equations with three-point q-integral boundary conditions. Our results are concerned with several quantum numbers of derivatives and integrals. By using Banach's contraction mapping, Krasnoselskii's fixed-point theorem, and Leray-Schauder degree theory, some new existence results are obtained. Two examples illustrate our results.
A Collocation Method for Volterra Integral Equations with Diagonal and Boundary Singularities
Kolk, Marek; Pedas, Arvet; Vainikko, Gennadi
2009-08-01
We propose a smoothing technique associated with piecewise polynomial collocation methods for solving linear weakly singular Volterra integral equations of the second kind with kernels which, in addition to a diagonal singularity, may have a singularity at the initial point of the interval of integration.
D. Diederen
2015-06-01
Full Text Available We present a new equation describing the hydrodynamics in infinitely long tidal channels (i.e., no reflection under the influence of oceanic forcing. The proposed equation is a simple relationship between partial derivatives of water level and velocity. It is formally derived for a progressive wave in a frictionless, prismatic, tidal channel with a horizontal bed. Assessment of a large number of numerical simulations, where an open boundary condition is posed at a certain distance landward, suggests that it can also be considered accurate in the more natural case of converging estuaries with nonlinear friction and a bed slope. The equation follows from the open boundary condition and is therefore a part of the problem formulation for an infinite tidal channel. This finding provides a practical tool for evaluating tidal wave dynamics, by reconstructing the temporal variation of the velocity based on local observations of the water level, providing a fully local open boundary condition and allowing for local friction calibration.
Tricomi, Francesco Giacomo
1957-01-01
This classic text on integral equations by the late Professor F. G. Tricomi, of the Mathematics Faculty of the University of Turin, Italy, presents an authoritative, well-written treatment of the subject at the graduate or advanced undergraduate level. To render the book accessible to as wide an audience as possible, the author has kept the mathematical knowledge required on the part of the reader to a minimum; a solid foundation in differential and integral calculus, together with some knowledge of the theory of functions is sufficient. The book is divided into four chapters, with two useful
We introduce the reader to an approximate method of solving the transport equation which was developed in the context of neutron thermalisation by Kladnik and Kuscer in 1962 [Kladnik R, Kuscer I. Velocity dependent Milne's problem. Nucl Sci Eng 1962;13:149]. Essentially the method is based upon two special weighted integrals of the one-dimensional transport equation which are valid regardless of the boundary conditions, and any solution must satisfy these integral relationships which are called the K-integrals. To obtain an approximate solution to the transport equation we turn the argument around and insist that any approximate solution must also satisfy the K-integrals. These integrals are particularly useful when the problem under consideration cannot be solved easily by analytic methods. It also has the marked advantage of being applicable to problems where there is energy exchange in a collision and anisotropy of scattering. To establish the feasibility of the method we obtain a number of approximate solutions using the K-integral method for problems to which we have exact analytical solutions. This enables us to validate the method. It is then applied to a new problem that has not yet been solved; namely the calculation of the discontinuity in the scalar intensity at the boundary between two optically dissimilar materials
Stenroos, Matti
2016-01-01
Boundary element methods (BEM) are used for forward computation of bioelectromagnetic fields in multi-compartment volume conductor models. Most BEM approaches assume that each compartment is in contact with at most one external compartment. In this work, I present a general surface integral equation and BEM discretization that remove this limitation and allow BEM modeling of general piecewise-homogeneous medium. The new integral equation allows positioning of field points at junctioned boundary of more than two compartments, enabling the use of linear collocation BEM in such a complex geometry. A modular BEM implementation is presented for linear collocation and Galerkin approaches, starting from standard formulation. The approach and resulting solver are verified in three ways, including comparison to finite element method (FEM). In a two-compartment split-sphere model with two spaced monopoles, the results obtained with high-resolution FEM and the BEMs were almost identical (relative difference < 0.003).
Abarbanel, Saul; Gottlieb, David; Carpenter, Mark H.
1994-01-01
It has been previously shown that the temporal integration of hyperbolic partial differential equations (PDE's) may, because of boundary conditions, lead to deterioration of accuracy of the solution. A procedure for removal of this error in the linear case has been established previously. In the present paper we consider hyperbolic (PDE's) (linear and non-linear) whose boundary treatment is done via the SAT-procedure. A methodology is present for recovery of the full order of accuracy, and has been applied to the case of a 4th order explicit finite difference scheme.
Adel A.K. Mohsen
2010-07-01
Full Text Available The problem of nonuniqueness (NU of the solution of exterior acoustic problems via boundary integral equations is discussed in this article. The efficient implementation of the CHIEF (Combined Helmholtz Integral Equations Formulation method to axisymmetric problems is studied. Interior axial fields are used to indicate the solution error and to select proper CHIEF points. The procedure makes full use of LU-decomposition as well as the forward solution derived in the solution. Implementations of the procedure for hard spheres are presented. Accurate results are obtained up to a normalised radius of ka = 20.983, using only one CHIEF point. The radiation from a uniformly vibrating sphere is also considered. Accurate results for ka up to 16.927 are obtained using two CHIEF points.
Boriskina, Svetlana V; Sewell, Phillip; Benson, Trevor M; Nosich, Alexander I
2004-03-01
A fast and accurate method is developed to compute the natural frequencies and scattering characteristics of arbitrary-shape two-dimensional dielectric resonators. The problem is formulated in terms of a uniquely solvable set of second-kind boundary integral equations and discretized by the Galerkin method with angular exponents as global test and trial functions. The log-singular term is extracted from one of the kernels, and closed-form expressions are derived for the main parts of all the integral operators. The resulting discrete scheme has a very high convergence rate. The method is used in the simulation of several optical microcavities for modern dense wavelength-division-multiplexed systems. PMID:15005404
Wang, Rui; Tan, Baolin
2013-01-01
A 3-D coronal magnetic field is reconstructed for NOAA 11158 on Feb 14, 2011. A GPU-accelerated direct boundary integral equation (DBIE) method is implemented. This is about 1000 times faster than the original DBIE used on solar NLFFF modeling. Using the SDO/HMI vector magnetogram as the bottom boundary condition, the reconstructed magnetic field lines are compared with the projected EUV loop structures from different views three-dimensionally by SDO/AIA and STEREO A/B spacecraft simultaneously for the first time. They show very good agreement so that the topological configurations of the magnetic fields can be analyzed, thus its role in the flare process of the active region can be better understood. A quantitative comparison with some stereoscopically reconstructed coronal loops shows that the present averaged misalignment angles are at the same order as the state-of-the-art results obtained with reconstructed coronal loops as prescribed conditions and better than other NLFFF methods. It is found that the o...
Rigorous electromagnetic analysis of two dimensional micro-axicon by boundary integral equations.
Lin, Jie; Tan, Jiubin; Liu, Jian; Liu, Shutian
2009-02-01
The focal performance of the micro-axicon and the Fresnel axicon (fraxicon) are investigated, for the first time, by the rigorous electromagnetic theory and boundary element method. The micro-axicon with different angle of apex and the fraxicon with various period and angle of apex are investigated. The dark segments of the fraxicon are explored numerically. Rigorous results of focal performance of the micro-axicon and the fraxicon are different from the results given by the approximation of geometrical optics and the scalar diffraction theory. The scattering effects are dominant in the fraxicon with small size of feature. It is expected that our study can provides very useful information in analyzing the axicon in optical trapping systems. PMID:19188975
Three-dimensional layerwise modeling of layered media with boundary integral equations
Kokkinos, Filis-Triantaphyllos T.
1995-01-01
A hybrid method is presented for the analysis of layers, plates, and multi-layered systems consisting of isotropic and linear elastic materials. The problem is formulated for the general case of a multi-layered system using a total potential energy formulation and employing the layerwise laminate theory of Reddy. A one-dimensional finite element model is used for the analysis of the multi-layered system through its thickness, and integral Fourier transforms are used to obtai...
LongShuyao; HuDe'an
2003-01-01
The meshless method is a new numerical technique presented in recent years .It uses the moving least square (MLS) approximation as a shape function . The smoothness of the MLS approximation is determined by that of the basic function and of the weight function, and is mainly determined by that of the weight function. Therefore, the weight function greatly affects the accuracy of results obtained. Different kinds of weight functions, such as the spline function, the Gauss function and so on, are proposed recently by many researchers. In the present work, the features of various weight functions are illustrated through solving elasto-static problems using the local boundary integral equation method. The effect of various weight functions on the accuracy, convergence and stability of results obtained is also discussed. Examples show that the weight function proposed by Zhou Weiyuan and Gauss and the quartic spline weight function are better than the others if parameters c and a in Gauss and exponential weight functions are in the range of reasonable values, respectively, and the higher the smoothness of the weight function, the better the features of the solutions.
Geophysical interpretation using integral equations
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...
Applied singular integral equations
Mandal, B N
2011-01-01
The book is devoted to varieties of linear singular integral equations, with special emphasis on their methods of solution. It introduces the singular integral equations and their applications to researchers as well as graduate students of this fascinating and growing branch of applied mathematics.
Nonlinear streak computation using boundary region equations
The boundary region equations (BREs) are applied for the simulation of the nonlinear evolution of a spanwise periodic array of streaks in a flat plate boundary layer. The well-known BRE formulation is obtained from the complete Navier–Stokes equations in the high Reynolds number limit, and provides the correct asymptotic description of three-dimensional boundary layer streaks. In this paper, a fast and robust streamwise marching scheme is introduced to perform their numerical integration. Typical streak computations present in the literature correspond to linear streaks or to small-amplitude nonlinear streaks computed using direct numerical simulation (DNS) or the nonlinear parabolized stability equations (PSEs). We use the BREs to numerically compute high-amplitude streaks, a method which requires much lower computational effort than DNS and does not have the consistency and convergence problems of the PSE. It is found that the flow configuration changes substantially as the amplitude of the streaks grows and the nonlinear effects come into play. The transversal motion (in the wall normal-streamwise plane) becomes more important and strongly distorts the streamwise velocity profiles, which end up being quite different from those of the linear case. We analyze in detail the resulting flow patterns for the nonlinearly saturated streaks and compare them with available experimental results. (paper)
Reaction diffusion equations with boundary degeneracy
Huashui Zhan
2016-03-01
Full Text Available In this article, we consider the reaction diffusion equation $$ \\frac{\\partial u}{\\partial t} = \\Delta A(u,\\quad (x,t\\in \\Omega \\times (0,T, $$ with the homogeneous boundary condition. Inspired by the Fichera-Oleinik theory, if the equation is not only strongly degenerate in the interior of $\\Omega$, but also degenerate on the boundary, we show that the solution of the equation is free from any limitation of the boundary condition.
In this work is shown the solution of the advection-diffusion equation to simulate a pollutant dispersion in the Planetary Boundary Layer. The solution is obtained through of the GILTT (Generalized Integral Laplace Transform Technique) analytic method and of the numerical inversion Gauss Quadrature. The validity of the solution is proved using concentration obtained from the model with concentration obtained for Copenhagen experiment. In this comparison was utilized potential and logarithmic wind profile and eddy diffusivity derived by Degrazia et al (1997) [17] and (2002) [19]. The best results was using the potential wind profile and the eddy diffusivity derived by Degrazia et al (1997). The vertical velocity influence is shown in the plume behavior of the pollutant concentration. Moreover, the vertical and longitudinal velocity provided by Large Eddy Simulation (LES) was stood in the model to simulate the turbulent boundary layer more realistic, the result was satisfactory when compared with contained in the literature. (author)
Soliton equations solved by the boundary CFT
SAITO, Satoru; Sato, Ryuichi
2003-01-01
Soliton equations are derived which characterize the boundary CFT a la Callan et al. Soliton fields of classical soliton equations are shown to appear as a neutral bound state of a pair of soliton fields of BCFT. One soliton amplitude under the influence of the boundary is calculated explicitly and is shown that it is frozen at the Dirichlet limit.
Integral equations and computation problems
Volterra's Integral Equations and Fredholm's Integral Equations of the second kind are discussed. Computational problems are found in the derivations and the computations. The theorem of the solution of the Fredholm's Integral Equation is discussed in detail. (author)
BOUNDARY CONTROL OF MKDV-BURGERS EQUATION
TIAN Li-xin; ZHAO Zhi-feng; WANG Jing-feng
2006-01-01
The boundary control of MKdV-Burgers equation was considered by feedback control on the domain [0,1]. The existence of the solution of MKdV-Burgers equation with the feedback control law was proved. On the base, priori estimates for the solution was given. At last, the existence of the weak solution of MKdV-Burgers equation was proved and the global-exponential and asymptotic stability of the solution of MKdV-Burgers equation was given.
On Certain Dual Integral Equations
R. S. Pathak
1974-01-01
Full Text Available Dual integral equations involving H-Functions have been solved by using the theory of Mellin transforms. The proof is analogous to that of Busbridge on solutions of dual integral equations involving Bessel functions.
Functional equations for Feynman integrals
New types of equations for Feynman integrals are found. It is shown that Feynman integrals satisfy functional equations connecting integrals with different kinematics. A regular method is proposed for obtaining such relations. The derivation of functional equations for one-loop two-, three- and four-point functions with arbitrary masses and external momenta is given. It is demonstrated that functional equations can be used for the analytic continuation of Feynman integrals to different kinematic domains
Boundary value problems and partial differential equations
Powers, David L
2005-01-01
Boundary Value Problems is the leading text on boundary value problems and Fourier series. The author, David Powers, (Clarkson) has written a thorough, theoretical overview of solving boundary value problems involving partial differential equations by the methods of separation of variables. Professors and students agree that the author is a master at creating linear problems that adroitly illustrate the techniques of separation of variables used to solve science and engineering.* CD with animations and graphics of solutions, additional exercises and chapter review questions* Nearly 900 exercises ranging in difficulty* Many fully worked examples
Boundary stabilization of wave equations with variable coefficients
FENG; Shaoji
2001-01-01
［1］Chen, G., Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain, J. Math. Pures. & Appl., 1979, 58: 249.［2］Komornik, V., Exact controllability and stabilization, Research in Applied Mathematics (Series Editors: Ciarlet, P. G., Lions, J.), New York: Masson/John Wiley copublication, 1994.［3］Komornik, V., Zuazua, E., A direct method for the boundary stabilization of the wave equation, J. Math. Pures. & Appl., 1990, 69: 33.［4］Lagnese, J., Decay of solutions of wave equations in a bounded region with boundary dissipation, J. Differential Equations, 1983, 50: 163.［5］Lasiecka, I., Triggiani, R., Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions, Appl. Math. & Optim., 1992, 25: 189.［6］Wyler, A., Stability of wave equations with dissipative boundary conditions in a bounded domain, Differential and Integral Equations,1994, 7: 345.［7］Yao, P. F., On the observability inequality for exact controllability of wave equations with variable coefficients, SIAM J. Control & Optimization, 1999, 37, 5: 1568.［8］Wu, H., Shen, C. L., Yu, Y. L., Introduction to Riemannian Geometry (in Chinese), Beijing: Peking University Press, 1989.
Integral equations and their applications
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 in
Radioisotope diffusion in grain textures by boundary integral method
Aim of this contribution is to deal with radioisotope diffusion in grain texture by Boundary Integral method (BIM). Governing partial integral equation is transformed to an equivalent boundary integral equation, which is written in a discrete form and a system of linear algebraic equations is thus obtained. Advantage of BIM is that the system of equations is solved only for unknown values on the boundary. values in the domain are calculated explicitly in a down stream procedure. A given example indicates a good agreement with analytical results. (author)
Ando, R.
2014-12-01
The boundary integral equation method formulated in the real space and time domain (BIEM-ST) has been used as a powerful tool to analyze the earthquake rupture dynamics on non-planar faults. Generally, BIEM is more accurate than volumetric methods such as the finite difference method and the finite difference method. With the recent development of the high performance computing environment, the earthquake rupture simulation studies have been conducted considering three dimensional realistic fault geometry models. However, the utility of BIEM-ST has been limited due to its heavy computational demanding increased depending on square of time steps (N2), which was needed to evaluate the historic integration. While BIEM can be efficient with the spectral domain formulation, the applications of such a method are limited to planar fault cases. In this study, we propose a new method to reduce the calculation time of BIEM-ST to linear of time step (N) without degrading the accuracy in the 3 dimensional modeling space. We extends the method proposed earlier for the case of the 2 dimensional framework, applying the asymptotic expressions of the elasto-dynamic Green's functions. This method uses the physical nature of the stress Green's function as dividing the causality cone according to the distances from the wave-fronts. The scalability of this method is shown on the parallel computing environment of the distributed memory. We demonstrate the applicability to analyses of subduction earthquake cases, suffering long time from the numerical limitations of previously available BIEMs. We analyze the dynamic rupture processes on dipping reverse faults embed in a three dimensional elastic half space.
Integral form of the radiation transfer equation
The integral form of the radiation transfer equation is given in a non-scattering medium for which the source and absorption terms are known explicitly. The problem is solved for an one-dimensional, inhomogeneous, non stationary, non isotropic configuration, in cartesian and spherical coordinates for arbitrary initial and boundary conditions. The same problem is solved for a boundary condition that is given on a moving surface, then the three-dimensional problem is examined in cartesian coordinates
Enclosing Solutions of Integral Equations
Madsen, Kaj; NA NA NA Caprani, Ole; Stauning, Ole
1996-01-01
We present a method for enclosing the solution of an integral equation. It is assumed that a solution exists and that the corresponding integral operator T is a contraction near y. When solving the integral equation by iteration we obtain a result which is normally different from y because of...
The Dirichlet problem for the two-dimensional Helmholtz equation for an open boundary
Hayashi, Y.
1973-01-01
Development of a complete theory of the two-dimensional Dirichlet problem for an open boundary. It is shown that the solution of the Dirichlet problem for an open boundary requires the solution of a Fredholm integral equation of the first kind. Although a Fredholm integral equation of the first kind usually has no solution if the kernel is continuous, owing to the logarithmic singularity of the kernel, the equation in this case is converted to a singular integral equation with a Cauchy kernel. It is proven that the homogeneous adjoint equation of the singular integral equation has no nonzero solution. By virtue of this result, and with the aid of an existence theorem known in the theory of singular integral equations, the existence of solutions of the singular integral equation, and then of the unique solution of the Fredholm integral equation of the first kind is proved.
马杭; 黄兴
2003-01-01
Based on the fact that the singular boundary integrals in the sense of Cauchy principal value can be represented approxi-mately by the mean values of two companion nearly singular boundary integrals, a vary general approach was developed in the paper.In the approach, the approximate formulation before discretization was constructed to cope with the difficulties encountered in the cor-ner treatment in the formulations of hypersingular boundary integral equations. This makes it possible to solve the hypersingularboundary integral equation numerically in a non-regularized form and in a local manner by using conforming C0 quadratic boundary ele-ments and standard Gaussian quadratures similar to those employed in the conventional displacement-BIE formulations. The approxi-mate formulation is very convenient to use because the corner information is comprised naturally in the representations of those ap-proximate integrals. Numerical examples in plane elasticity show that with the present approach, the compatible or better results canbe achieved in comparison with those of the conventional BIE formulations.
Singular-boundary reductions of type-Q ABS equations
Atkinson, James; Joshi, Nalini
2011-01-01
We study the fully discrete elliptic integrable model Q4 and its immediate trigonometric and rational counterparts (Q3, Q2 and Q1). Singular boundary problems for these equations are systematised in the framework of global singularity analysis. We introduce a technique to obtain solutions of such problems, in particular constructing the exact solution on a regular singularity-bounded strip. The solution technique is based on the multidimensional consistency and uses new insights into these eq...
王家玉; 陈金兰
2001-01-01
By using the concept of abstract cone,improved comparable theoremand fixed point theorem of nonlinear operator and by discussing the periodic boundary condition of nonlinear integral-differential equation in Banach spaces,this paper gave two monotonic iterative sequences,also verified the minimum and maximum theorem for nonlinear integral-differential equation with periodic boundary condition in Banach spaces.%本文考虑Banach空间中非线性积分－微分方程的周期边值问题，利用抽象锥、推广了的比较定理及非线性算子的不动点定理，构造出两个单调迭代序列，证明了Banach空间中非线性积分－微分方程具有周期边值的最小解、最大解存在定理。
Boundary differentiability for inhomogeneous infinity Laplace equations
Guanghao Hong
2014-03-01
Full Text Available We study the boundary regularity of the solutions to inhomogeneous infinity Laplace equations. We prove that if $u\\in C(\\bar{\\Omega}$ is a viscosity solution to $\\Delta_{\\infty}u:=\\sum_{i,j=1}^n u_{x_i}u_{x_j}u_{x_ix_j}=f$ with $f\\in C(\\Omega\\cap L^{\\infty}(\\Omega$ and for $x_0\\in \\partial\\Omega$ both $\\partial\\Omega$ and $g:=u|_{\\partial\\Omega}$ are differentiable at $x_0$, then u is differentiable at $x_0$.
Boundary conditions for hyperbolic formulations of the Einstein equations
Frittelli, Simonetta; Gomez, Roberto
2003-01-01
In regards to the initial-boundary value problem of the Einstein equations, we argue that the projection of the Einstein equations along the normal to the boundary yields necessary and appropriate boundary conditions for a wide class of equivalent formulations. We explicitly show that this is so for the Einstein-Christoffel formulation of the Einstein equations in the case of spherical symmetry.
Li Yongkun; Shu Jiangye
2011-01-01
Abstract In this paper, by making use of the coincidence degree theory of Mawhin, the existence of the nontrivial solution for the boundary value problem with Riemann-Stieltjes Δ-integral conditions on time scales at resonance x Δ Δ ( t ) = f ( t , x ( t ) , x Δ ( t ) ) + e ( t ) , a . e . t ∈ [ 0 , T ] T , x Δ ( 0 ) = 0 , x ( T ) = ∫ 0 T x σ ( s ) Δ g ...
Ardema, M. D.; Yang, L.
1985-01-01
A method of solving the boundary-layer equations that arise in singular-perturbation analysis of flightpath optimization problems is presented. The method is based on Picard iterations of the integrated form of the equations and does not require iteration to find unknown boundary conditions. As an example, the method is used to develop a solution algorithm for the zero-order boundary-layer equations of the aircraft minimum-time-to-climb problem.
Integral equation methods for electromagnetics
Volakis, John
2012-01-01
This text/reference is a detailed look at the development and use of integral equation methods for electromagnetic analysis, specifically for antennas and radar scattering. Developers and practitioners will appreciate the broad-based approach to understanding and utilizing integral equation methods and the unique coverage of historical developments that led to the current state-of-the-art. In contrast to existing books, Integral Equation Methods for Electromagnetics lays the groundwork in the initial chapters so students and basic users can solve simple problems and work their way up to the mo
Recovering an obstacle using integral equations
Rundell, William
2009-05-01
We consider the inverse problem of recovering the shape, location and surface properties of an object where the surrounding medium is both conductive and homogeneous and we measure Cauchy data on an accessible part of the exterior boundary. It is assumed that the physical situation is modelled by harmonic functions and the boundary condition on the obstacle is one of Dirichlet type. The purpose of this paper is to answer some of the questions raised in a recent paper that introduced a nonlinear integral equation approach for the solution of this type of problem.
Calculation of Turbulent Boundary Layers Using the Dissipation Integral Method
MatthiasBuschmann
1999-01-01
This paper gives an introduction into the dissipation integral method.The general integral equations for the three-dimensional case are derved.It is found that for a practical calculation algorithm the integral monentum equation and the integral energy equation are msot useful.Using Two different sets of mean velocity profiles the hyperbolical character of a dissipation integral method is shown.Test cases for two-and three-dimensional boundary layers are analysed and discussed.The paper concludes with a discussion of the advantages and limits of dissipation integral methods.
LINEAR SINGULAR INTEGRAL EQUATION ON DOMAINS COMPOSED BY BALLS
无
2006-01-01
For domains composed by balls in Cn, this paper studies the boundary behaviour of Cauchy type integrals with discrete holomorphic kernels and the corresponding linear singular integral equation on each piece of smooth lower dimensional edges on the boundary of the domain.
Free Boundary Value Problems for Abstract Elliptic Equations and Applications
Veli SHAKHMUROV
2011-01-01
The free boundary value problems for elliptic differential-operator equations are studied.Several conditions for the uniform maximal regularity with respect to boundary parameters and the Fredholmness in abstract Lp-spaces are given.In application,the nonlocal free boundary problems for finite or infinite systems of elliptic and anisotropic type equations are studied.
The free boundary Euler equations with large surface tension
Disconzi, Marcelo M.; Ebin, David G.
2015-01-01
We study the free boundary Euler equations with surface tension in three spatial dimensions, showing that the equations are well-posed if the coefficient of surface tension is positive. Then we prove that under natural assumptions, the solutions of the free boundary motion converge to solutions of the Euler equations in a domain with fixed boundary when the coefficient of surface tension tends to infinity.
The free boundary Euler equations with large surface tension
Disconzi, Marcelo M.; Ebin, David G.
2016-07-01
We study the free boundary Euler equations with surface tension in three spatial dimensions, showing that the equations are well-posed if the coefficient of surface tension is positive. Then we prove that under natural assumptions, the solutions of the free boundary motion converge to solutions of the Euler equations in a domain with fixed boundary when the coefficient of surface tension tends to infinity.
SOLVABILITY FOR NONLINEAR ELLIPTIC EQUATION WITH BOUNDARY PERTURBATION
无
2007-01-01
The solvability of nonlinear elliptic equation with boundary perturbation is considered. The perturbed solution of original problem is obtained and the uniformly valid expansion of solution is proved.
Treatment of domain integrals in boundary element methods
Nintcheu Fata, Sylvain [ORNL
2012-01-01
A systematic and rigorous technique to calculate domain integrals without a volume-fitted mesh has been developed and validated in the context of a boundary element approximation. In the proposed approach, a domain integral involving a continuous or weakly-singular integrand is first converted into a surface integral by means of straight-path integrals that intersect the underlying domain. Then, the resulting surface integral is carried out either via analytic integration over boundary elements or by use of standard quadrature rules. This domain-to-boundary integral transformation is derived from an extension of the fundamental theorem of calculus to higher dimension, and the divergence theorem. In establishing the method, it is shown that the higher-dimensional version of the first fundamental theorem of calculus corresponds to the well-known Poincare lemma. The proposed technique can be employed to evaluate integrals defined over simply- or multiply-connected domains with Lipschitz boundaries which are embedded in an Euclidean space of arbitrary but finite dimension. Combined with the singular treatment of surface integrals that is widely available in the literature, this approach can also be utilized to effectively deal with boundary-value problems involving non-homogeneous source terms by way of a collocation or a Galerkin boundary integral equation method using only the prescribed surface discretization. Sample problems associated with the three-dimensional Poisson equation and featuring the Newton potential are successfully solved by a constant element collocation method to validate this study.
Boundary Integral Solutions to Three-Dimensional Unconfined Darcy's Flow
Lennon, Gerard P.; Liu, Philip L.-F.; Liggett, James A.
1980-08-01
The boundary integral equation method (BIEM) is used to solve three-dimensional potential flow problems in porous media. The problems considered here are time dependent and have a nonlinear boundary condition on the free surface. The entire boundary, including the moving free surface, discretized into linear finite elements for the purpose of evaluating the boundary integrals. The technique allows transient, three-dimensional problems to be solved with reasonable computational costs. Numerical examples include recharge through rectangular and circular areas and seepage flow from a surface pond. The examples are used to illustrate the method and show the nonlinear effects.
Explicit Expressions for 3D Boundary Integrals in Potential Theory
Nintcheu Fata, Sylvain [ORNL
2009-01-01
On employing isoparametric, piecewise linear shape functions over a flat triangular domain, exact expressions are derived for all surface potentials involved in the numerical solution of three-dimensional singular and hyper-singular boundary integral equations of potential theory. These formulae, which are valid for an arbitrary source point in space, are represented as analytic expressions over the edges of the integration triangle. They can be used to solve integral equations defined on polygonal boundaries via the collocation method or may be utilized as analytic expressions for the inner integrals in the Galerkin technique. Also, the constant element approximation can be directly obtained with no extra effort. Sample problems solved by the collocation boundary element method for the Laplace equation are included to validate the proposed formulae.
Compatibility between shape equation and boundary conditions of lipid membranes with free edges.
Tu, Z C
2010-02-28
Only some special open surfaces satisfying the shape equation of lipid membranes can be compatible with the boundary conditions. As a result of this compatibility, the first integral of the shape equation should vanish for axisymmetric lipid membranes, from which two theorems of nonexistence are verified: (i) there is no axisymmetric open membrane being a part of torus satisfying the shape equation; (ii) there is no axisymmetric open membrane being a part of a biconcave discodal surface satisfying the shape equation. Additionally, the shape equation is reduced to a second-order differential equation while the boundary conditions are reduced to two equations due to this compatibility. Numerical solutions to the reduced shape equation and boundary conditions agree well with the experimental data [A. Saitoh et al., Proc. Natl. Acad. Sci. U.S.A. 95, 1026 (1998)]. PMID:20192294
The Pauli equation with complex boundary conditions
Kochan, D; Novak, R; Siegl, P
2012-01-01
We consider one-dimensional Pauli Hamiltonians in a bounded interval with possibly non-self-adjoint Robin-type boundary conditions. We study the influence of the spin-magnetic interaction on the interplay between the type of boundary conditions and the spectrum. A special attention is paid to PT-symmetric boundary conditions with the physical choice of the time-reversal operator T.
Absorbing boundary conditions for second-order hyperbolic equations
Jiang, Hong; Wong, Yau Shu
1990-01-01
A uniform approach to construct absorbing artificial boundary conditions for second-order linear hyperbolic equations is proposed. The nonlocal boundary condition is given by a pseudodifferential operator that annihilates travelling waves. It is obtained through the dispersion relation of the differential equation by requiring that the initial-boundary value problem admits the wave solutions travelling in one direction only. Local approximation of this global boundary condition yields an nth-order differential operator. It is shown that the best approximations must be in the canonical forms which can be factorized into first-order operators. These boundary conditions are perfectly absorbing for wave packets propagating at certain group velocities. A hierarchy of absorbing boundary conditions is derived for transonic small perturbation equations of unsteady flows. These examples illustrate that the absorbing boundary conditions are easy to derive, and the effectiveness is demonstrated by the numerical experiments.
Numerical integration of asymptotic solutions of ordinary differential equations
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.
A New Algorithm for System of Integral Equations
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.
Transmission problem for the Laplace equation and the integral equation method
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
The Pauli equation with complex boundary conditions
We consider one-dimensional Pauli Hamiltonians in a bounded interval with possibly non-self-adjoint Robin-type boundary conditions. We study the influence of the spin–magnetic interaction on the interplay between the type of boundary conditions and the spectrum. Special attention is paid to PT-symmetric boundary conditions with the physical choice of the time-reversal operator T. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Quantum physics with non-Hermitian operators’. (paper)
The Pauli equation with complex boundary conditions
Kochan, D.; Krejčiřík, D.; Novák, R.; Siegl, P.
2012-11-01
We consider one-dimensional Pauli Hamiltonians in a bounded interval with possibly non-self-adjoint Robin-type boundary conditions. We study the influence of the spin-magnetic interaction on the interplay between the type of boundary conditions and the spectrum. Special attention is paid to {PT}-symmetric boundary conditions with the physical choice of the time-reversal operator {T}. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Quantum physics with non-Hermitian operators’.
Integrability of equations for soliton's eigenfunctions
Eigenfunctions of the auxiliary linear problems for the soliton equations obey the nonlinear evolution equations. It is shown that these eigenfunction equations are integrable by the inverse spectral transform method. Eigenfunction equations are also the generating equations. Several (1+1) and (2+1) dimensional eigenfunction equations and their properties are considered. 11 refs
The open boundary equation (discussion paper)
Diederen, D.; Savenije, H.H.G.; Toffolon, M.
2015-01-01
We present a new equation describing the hydrodynamics in infinitely long tidal channels (i.e., no reflection) under the influence of oceanic forcing. The proposed equation is a simple relationship between partial derivatives of water level and velocity. It is formally derived for a progressive wave
A LOCAL BOUNDARY INTEGRAL EQUATION[1mm］METHOD FOR THE ELASTICITY PROBLEM%弹性力学问题的局部边界积分方程方法
龙述尧; 许敬晓
2000-01-01
The basic concept and numerical implementation of a local boundaryintegral equation formulation for solving the elasticity problem havebeen presented in the present paper. It is a new truly meshless method,because the numerical implementation of the method leads to an efficientmeshless discrete model. The concept of a companion solution isintroduced, such that the traction terms would not appear in theintegrals over the local boundary after the modified integral kernel isused for all nodes whose local boundary s falls within theglobal boundary of the given problem; it uses the moving leastsquare approximations, and involves only boundary integration over alocal boundary centered at the node in question. It poses nodifficulties in satisfying essential boundary conditions, and leads to abanded and sparse system matrix. The undependence of the solution onthe size of the integral local boundary provides a great flexibilityin dealing with the numerical model of the elastic plane problems undervarious boundary conditions with arbitrary shapes. Convergence studiesin the numerical examples show that the present method possesses anexcellent rate of convergence and reasonably accurate results for boththe unknown displacement and strain energy, as the originalapproximated trial solutions have nice continuity and smoothness. Thenumerical results also show that using both linear and quadratic basesas well as spline and Gaussian weight functions in approximationfunctions can give quite accurate numerical results. Compared with the conventional boundary element method based on the globalboundary integral equations, the present method is advantageous in thefollowing aspects: i) No boundary and domain element needed to be constructed in the presentmethod, while it is necessary to discretize both the entire domain andits boundary for the conventional boundary element method in general. Thevolume and boundary integrals in the present method are evaluated onlyover small
Integrable (2k)-Dimensional Hitchin Equations
Ward, R S
2016-01-01
This letter describes a completely-integrable system of Yang-Mills-Higgs equations which generalizes the Hitchin equations on a Riemann surface to arbitrary k-dimensional complex manifolds. The system arises as a dimensional reduction of a set of integrable Yang-Mills equations in 4k real dimensions. Our integrable system implies other generalizations such as the Simpson equations and the non-abelian Seiberg-Witten equations. Some simple solutions in the k=2 case are described.
From the Boltzmann Equation to the Euler Equations in the Presence of Boundaries
Golse, François
2011-01-01
The fluid dynamic limit of the Boltzmann equation leading to the Euler equations for an incompressible fluid with constant density in the presence of material boundaries shares some important features with the better known inviscid limit of the Navier-Stokes equations. The present paper slightly extends recent results from [C. Bardos, F. Golse, L. Paillard, Comm. Math. Sci., 10 (2012), 159--190] to the case of boundary conditions for the Boltzmann equation more general than Maxwell's accomodation condition.
From the Boltzmann Equation to the Euler Equations in the Presence of Boundaries
Golse, François
2011-01-01
The fluid dynamic limit of the Boltzmann equation leading to the Euler equations for an incompressible fluid with constant density in the presence of material boundaries shares some important features with the better known inviscid limit of the Navier-Stokes equations. The present paper slightly extends recent results from [C. Bardos, F. Golse, L. Paillard, Comm. Math. Sci., 10 (2012), 159--190] to the case of boundary conditions for the Boltzmann equation more general than Maxwell's accomoda...
Boundary value problemfor multidimensional fractional advection-dispersion equation
Khasambiev Mokhammad Vakhaevich
2015-05-01
Full Text Available In recent time there is a very great interest in the study of differential equations of fractional order, in which the unknown function is under the symbol of fractional derivative. It is due to the development of the theory of fractional integro-differential theory and application of it in different fields.The fractional integrals and derivatives of fractional integro-differential equations are widely used in modern investigations of theoretical physics, mechanics, and applied mathematics. The fractional calculus is a very powerful tool for describing physical systems, which have a memory and are non-local. Many processes in complex systems have nonlocality and long-time memory. Fractional integral operators and fractional differential operators allow describing some of these properties. The use of the fractional calculus will be helpful for obtaining the dynamical models, in which integro-differential operators describe power long-time memory by time and coordinates, and three-dimensional nonlocality for complex medium and processes.Differential equations of fractional order appear when we use fractal conception in physics of the condensed medium. The transfer, described by the operator with fractional derivatives at a long distance from the sources, leads to other behavior of relatively small concentrations as compared with classic diffusion. This fact redefines the existing ideas about safety, based on the ideas on exponential velocity of damping. Fractional calculus in the fractal theory and the systems with memory have the same importance as the classic analysis in mechanics of continuous medium.In recent years, the application of fractional derivatives for describing and studying the physical processes of stochastic transfer is very popular too. Many problems of filtration of liquids in fractal (high porous medium lead to the need to study boundary value problems for partial differential equations in fractional order.In this paper the
Stochastic integration and differential equations
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...
Boundary Element Method with Non—overlapping Domain Decomposition for Diffusion Equation
ZHUJialin; ZHANGTaiping
2002-01-01
A boundary element method based on non-overlapping domain decomposition method to solve the time-dependent diffusion equations is presented.The time-dependent fundamental solution is used in the formulation of boundary integrals and the time integratioin process always restarts from the initial time condition.The process of replacing the interface values,which needs a summation of boundary integrals related to the boundary values at previous time steps can be treated in parallel parallel iterative procedure,Numerical experiments demonstrate that the implementation of the present alogrithm is efficient.
Analysis of Laminar Boundary Layer Equations
R. Yesman
2012-01-01
Full Text Available The paper proposes methodology for analysis and calculation of laminar fluid flow processes in a boundary layer.The presented dependences can be used for practical calculations while power carriers of various application are moving in the channels of heat and power devices.
Bessaih, Hakima
2015-04-01
The evolution Stokes equation in a domain containing periodically distributed obstacles subject to Fourier boundary condition on the boundaries is considered. We assume that the dynamic is driven by a stochastic perturbation on the interior of the domain and another stochastic perturbation on the boundaries of the obstacles. We represent the solid obstacles by holes in the fluid domain. The macroscopic (homogenized) equation is derived as another stochastic partial differential equation, defined in the whole non perforated domain. Here, the initial stochastic perturbation on the boundary becomes part of the homogenized equation as another stochastic force. We use the twoscale convergence method after extending the solution with 0 in the holes to pass to the limit. By Itô stochastic calculus, we get uniform estimates on the solution in appropriate spaces. In order to pass to the limit on the boundary integrals, we rewrite them in terms of integrals in the whole domain. In particular, for the stochastic integral on the boundary, we combine the previous idea of rewriting it on the whole domain with the assumption that the Brownian motion is of trace class. Due to the particular boundary condition dealt with, we get that the solution of the stochastic homogenized equation is not divergence free. However, it is coupled with the cell problem that has a divergence free solution. This paper represents an extension of the results of Duan and Wang (Comm. Math. Phys. 275:1508-1527, 2007), where a reaction diffusion equation with a dynamical boundary condition with a noise source term on both the interior of the domain and on the boundary was studied, and through a tightness argument and a pointwise two scale convergence method the homogenized equation was derived. © American Institute of Mathematical Sciences.
Boundary value problems of discrete generalized Emden-Fowler equation
YU; Jianshe; GUO; Zhiming
2006-01-01
By using the critical point theory, some sufficient conditions for the existence of the solutions to the boundary value problems of a discrete generalized Emden-Fowler equation are obtained. In a special case, a sharp condition is obtained for the existence of the boundary value problems of the above equation. For a linear case, by the discrete variational theory, a necessary and sufficient condition for the existence, uniqueness and multiplicity of the solutions is also established.
Integral Transform Approach to Generalized Tricomi Equations
Yagdjian, Karen
2014-01-01
We present some integral transform that allows to obtain solutions of the generalized Tricomi equation from solutions of a simpler equation. We used in [13,14],[41]-[46] the particular version of this transform in order to investigate in a unified way several equations such as the linear and semilinear Tricomi equations, Gellerstedt equation, the wave equation in Einstein-de Sitter spacetime, the wave and the Klein-Gordon equations in the de Sitter and anti-de Sitter spacetimes.
Linear integral equations and soliton systems
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.)
Multiple integral representation for the trigonometric SOS model with domain wall boundaries
Using the dynamical Yang-Baxter algebra we derive a functional equation for the partition function of the trigonometric SOS model with domain wall boundary conditions. The solution of the equation is given in terms of a multiple contour integral.
Multiple integral representation for the trigonometric SOS model with domain wall boundaries
Galleas, W.
2012-05-01
Using the dynamical Yang-Baxter algebra we derive a functional equation for the partition function of the trigonometric SOS model with domain wall boundary conditions. The solution of the equation is given in terms of a multiple contour integral.
On a stochastic Burgers equation with Dirichlet boundary conditions
Ekaterina T. Kolkovska
2003-01-01
Full Text Available We consider the one-dimensional Burgers equation perturbed by a white noise term with Dirichlet boundary conditions and a non-Lipschitz coefficient. We obtain existence of a weak solution proving tightness for a sequence of polygonal approximations for the equation and solving a martingale problem for the weak limit.
Thermodynamically admissible boundary conditions for the regularized 13 moment equations
A phenomenological approach to the boundary conditions for linearized R13 equations is derived using the second law of thermodynamics. The phenomenological coefficients appearing in the boundary conditions are calculated by comparing the slip, jump, and thermal creep coefficients with linearized Boltzmann solutions for Maxwell’s accommodation model for different values of the accommodation coefficient. For this, the linearized R13 equations are solved for viscous slip, thermal creep, and temperature jump problems and the results are compared to the solutions of the linearized Boltzmann equation. The influence of different collision models (hard-sphere, Bhatnagar–Gross–Krook, and Maxwell molecules) and accommodation coefficients on the phenomenological coefficients is studied
Thermodynamically admissible boundary conditions for the regularized 13 moment equations
Rana, Anirudh Singh; Struchtrup, Henning
2016-02-01
A phenomenological approach to the boundary conditions for linearized R13 equations is derived using the second law of thermodynamics. The phenomenological coefficients appearing in the boundary conditions are calculated by comparing the slip, jump, and thermal creep coefficients with linearized Boltzmann solutions for Maxwell's accommodation model for different values of the accommodation coefficient. For this, the linearized R13 equations are solved for viscous slip, thermal creep, and temperature jump problems and the results are compared to the solutions of the linearized Boltzmann equation. The influence of different collision models (hard-sphere, Bhatnagar-Gross-Krook, and Maxwell molecules) and accommodation coefficients on the phenomenological coefficients is studied.
Thermodynamically admissible boundary conditions for the regularized 13 moment equations
Rana, Anirudh Singh, E-mail: anirudh@uvic.ca [Department of Mechanical and Aerospace Engineering, Gyeongsang National University, Jinju, Gyeongnam 52828 (Korea, Republic of); Struchtrup, Henning, E-mail: struchtr@uvic.ca [Department of Mechanical Engineering, University of Victoria, Victoria, British Columbia V8W 2Y2 (Canada)
2016-02-15
A phenomenological approach to the boundary conditions for linearized R13 equations is derived using the second law of thermodynamics. The phenomenological coefficients appearing in the boundary conditions are calculated by comparing the slip, jump, and thermal creep coefficients with linearized Boltzmann solutions for Maxwell’s accommodation model for different values of the accommodation coefficient. For this, the linearized R13 equations are solved for viscous slip, thermal creep, and temperature jump problems and the results are compared to the solutions of the linearized Boltzmann equation. The influence of different collision models (hard-sphere, Bhatnagar–Gross–Krook, and Maxwell molecules) and accommodation coefficients on the phenomenological coefficients is studied.
LaChapelle, J.
2004-01-01
A path integral is presented that solves a general class of linear second order partial differential equations with Dirichlet/Neumann boundary conditions. Elementary kernels are constructed for both Dirichlet and Neumann boundary conditions. The general solution can be specialized to solve elliptic, parabolic, and hyperbolic partial differential equations with boundary conditions. This extends the well-known path integral solution of the Schr\\"{o}dinger/diffusion equation in unbounded space. ...
First integrals of difference Hamiltonian equations
In the present paper, the well-known Noether's identity, which represents the connection between symmetries and first integrals of Euler-Lagrange equations, is rewritten in terms of the Hamiltonian function. This approach, based on the Hamiltonian identity, provides a simple and clear way to find first integrals of canonical Hamiltonian equations without integration. A discrete analog of the Hamiltonian identity is developed. It leads to a connection between symmetries and first integrals of difference Hamiltonian equations that can be used to conserve the structural properties of Hamiltonian equations under discretizaton. The results are illustrated by a number of examples for both continuous and difference Hamiltonian equations.
Guotao Wang
2012-01-01
Full Text Available We study nonlinear impulsive differential equations of fractional order with irregular boundary conditions. Some existence and uniqueness results are obtained by applying standard fixed-point theorems. For illustration of the results, some examples are discussed.
Solitons induced by boundary conditions from the Boussinesq equation
Chou, Ru Ling; Chu, C. K.
1990-01-01
The behavior of solitons induced by boundary excitation is investigated at various time-dependent conditions and different unperturbed water depths, using the Korteweg-de Vries (KdV) equation. Then, solitons induced from Boussinesq equations under similar conditions were studied, making it possible to remove the restriction in the KdV equation and to treat soliton head-on collisions (as well as overtaking collisions) and reflections. It is found that the results obtained from the KdV and the Boussinesq equations are in good agreement.
任景莉; 葛渭高
2003-01-01
A boundary value problems f or functional differenatial equations, with nonlinear boundary condition, is studied by the theorem of differential inequality. Using new method to construct the upper solution and lower solution, sufficient conditions for the existence of the problems' solution are established. A uniformly valid asymptotic expansions of the solution is also given.
Periodic solutions to nonlinear equations with oblique boundary conditions
Allergretto, Walter; Papini, Duccio
2012-01-01
We study the existence of positive periodic solutions to nonlinear elliptic and parabolic equations with oblique and dynamical boundary conditions and non-local terms. The results are obtained through fixed point theory, topological degree methods and properties of related linear elliptic problems with natural boundary conditions and possibly non-symmetric principal part. As immediate consequences, we also obtain estimates on the principal eigenvalue for non-symmetric elliptic ...
Distribution theory for Schrödinger’s integral equation
Lange, Rutger-Jan, E-mail: rutger-jan.lange@cantab.net [VU University Amsterdam, 1081 HV Amsterdam (Netherlands)
2015-12-15
Much of the literature on point interactions in quantum mechanics has focused on the differential form of Schrödinger’s equation. This paper, in contrast, investigates the integral form of Schrödinger’s equation. While both forms are known to be equivalent for smooth potentials, this is not true for distributional potentials. Here, we assume that the potential is given by a distribution defined on the space of discontinuous test functions. First, by using Schrödinger’s integral equation, we confirm a seminal result by Kurasov, which was originally obtained in the context of Schrödinger’s differential equation. This hints at a possible deeper connection between both forms of the equation. We also sketch a generalisation of Kurasov’s [J. Math. Anal. Appl. 201(1), 297–323 (1996)] result to hypersurfaces. Second, we derive a new closed-form solution to Schrödinger’s integral equation with a delta prime potential. This potential has attracted considerable attention, including some controversy. Interestingly, the derived propagator satisfies boundary conditions that were previously derived using Schrödinger’s differential equation. Third, we derive boundary conditions for “super-singular” potentials given by higher-order derivatives of the delta potential. These boundary conditions cannot be incorporated into the normal framework of self-adjoint extensions. We show that the boundary conditions depend on the energy of the solution and that probability is conserved. This paper thereby confirms several seminal results and derives some new ones. In sum, it shows that Schrödinger’s integral equation is a viable tool for studying singular interactions in quantum mechanics.
Distribution theory for Schrödinger’s integral equation
Much of the literature on point interactions in quantum mechanics has focused on the differential form of Schrödinger’s equation. This paper, in contrast, investigates the integral form of Schrödinger’s equation. While both forms are known to be equivalent for smooth potentials, this is not true for distributional potentials. Here, we assume that the potential is given by a distribution defined on the space of discontinuous test functions. First, by using Schrödinger’s integral equation, we confirm a seminal result by Kurasov, which was originally obtained in the context of Schrödinger’s differential equation. This hints at a possible deeper connection between both forms of the equation. We also sketch a generalisation of Kurasov’s [J. Math. Anal. Appl. 201(1), 297–323 (1996)] result to hypersurfaces. Second, we derive a new closed-form solution to Schrödinger’s integral equation with a delta prime potential. This potential has attracted considerable attention, including some controversy. Interestingly, the derived propagator satisfies boundary conditions that were previously derived using Schrödinger’s differential equation. Third, we derive boundary conditions for “super-singular” potentials given by higher-order derivatives of the delta potential. These boundary conditions cannot be incorporated into the normal framework of self-adjoint extensions. We show that the boundary conditions depend on the energy of the solution and that probability is conserved. This paper thereby confirms several seminal results and derives some new ones. In sum, it shows that Schrödinger’s integral equation is a viable tool for studying singular interactions in quantum mechanics
Numerical Quadrature of Periodic Singular Integral Equations
Krenk, Steen
1978-01-01
This paper presents quadrature formulae for the numerical integration of a singular integral equation with Hilbert kernel. The formulae are based on trigonometric interpolation. By integration a quadrature formula for an integral with a logarithmic singularity is obtained. Finally it is demonstra......This paper presents quadrature formulae for the numerical integration of a singular integral equation with Hilbert kernel. The formulae are based on trigonometric interpolation. By integration a quadrature formula for an integral with a logarithmic singularity is obtained. Finally it is...
Traveling waves for a boundary reaction-diffusion equation
Caffarelli, L; Sire, Y
2011-01-01
We prove the existence of a traveling wave solution for a boundary reaction diffusion equation when the reaction term is the combustion nonlinearity with ignition temperature. A key role in the proof is plaid by an explicit formula for traveling wave solutions of a free boundary problem obtained as singular limit for the reaction-diffusion equation (the so-called high energy activation energy limit). This explicit formula, which is interesting in itself, also allows us to get an estimate on the decay at infinity of the traveling wave (which turns out to be faster than the usual exponential decay).
Linear boundary value problems for differential algebraic equations
Balla, Katalin; März, Roswitha
2003-01-01
By the use of the corresponding shift matrix, the paper gives a criterion for the unique solvability of linear boundary value problems posed for linear differential algebraic equations up to index 2 with well-matched leading coefficients. The solution is constructed by a proper Green function. Another characterization of the solutions is based upon the description of arbitrary affine linear subspaces of solutions to linear differential algebraic equations in terms of solutions to the adjoint ...
The Einstein Constraint Equations on Compact Manifolds with Boundary
Dilts, James
2013-01-01
We continue the study of the Einstein constraint equations on compact manifolds with boundary initiated by Holst and Tsogtgerel in their 2013 paper. In particular, we consider the full system and prove existence of solutions in both the near-CMC and far-from-CMC (for Yamabe positive metrics) cases. We also prove analogues many of the useful inequalities and results in previous "limit equation" papers by Dahl, Gicquaud, Humbert and others.
Numerical solutions of telegraph equations with the Dirichlet boundary condition
Ashyralyev, Allaberen; Turkcan, Kadriye Tuba; Koksal, Mehmet Emir
2016-08-01
In this study, the Cauchy problem for telegraph equations in a Hilbert space is considered. Stability estimates for the solution of this problem are presented. The third order of accuracy difference scheme is constructed for approximate solutions of the problem. Stability estimates for the solution of this difference scheme are established. As a test problem to support theoretical results, one-dimensional telegraph equation with the Dirichlet boundary condition is considered. Numerical solutions of this equation are obtained by first, second and third order of accuracy difference schemes.
Navier-Stokes equation with slip-like boundary condition
Ken'ichi Hashizume; Tetsuya Koyama; Mitsuharu Otani
2009-01-01
The aim of this note is to investigate a time-discretized 2-dimensional Navier-Stokes equation with a slip-like boundary condition, which arises in the melting ice problem. We prove the existence and uniqueness of a weak solution.
Positive Solutions for Nonlinear Differential Equations with Periodic Boundary Condition
Shengjun Li
2012-01-01
Full Text Available We study the existence of positive solutions for second-order nonlinear differential equations with nonseparated boundary conditions. Our nonlinearity may be singular in its dependent variable. The proof of the main result relies on a nonlinear alternative principle of Leray-Schauder. Recent results in the literature are generalized and significantly improved.
QUENCHING ON BOUNDARY TO THE NEWTON FILTRATION EQUATION (Ⅰ)
段志文; 谢春红; 卢伟明
2003-01-01
This paper discusses the global existence and quenching of the solution to the Newton filtration equation with the nonlinear boundary condition.The authors also discuss the profile of the quenching solution in the quenching time and obtain the quenching rate of the quenching solution.
Algebraic Bethe Ansatz for O(2N) sigma models with integrable diagonal boundaries
Gombor, Tamas
2015-01-01
The finite volume problem of O(2N) sigma models with integrable diagonal boundaries on a finite interval is investigated. The double row transfer matrix is diagonalized by Algebraic Bethe Ansatz. The boundary Bethe Yang equations for the particle rapidities and the accompanying Bethe Ansatz equations are derived.
An integral equation representation approach for valuing Russian options with a finite time horizon
Jeon, Junkee; Han, Heejae; Kim, Hyeonuk; Kang, Myungjoo
2016-07-01
In this paper, we first describe a general solution for the inhomogeneous Black-Scholes partial differential equation with mixed boundary conditions using Mellin transform techniques. Since Russian options with a finite time horizon are usually formulated into the inhomogeneous free-boundary Black-Scholes partial differential equation with a mixed boundary condition, we apply our method to Russian options and derive an integral equation satisfied by Russian options with a finite time horizon. Furthermore, we present some numerical solutions and plots of the integral equation using recursive integration methods and demonstrate the computational accuracy and efficiency of our method compared to other competing approaches.
Mo Jiaqi
2007-01-01
A class of nonlinear initial boundary value problems for reaction diffusion equations with boundary perturbation is considered. Under suitable conditions and using the theory of differential inequalities the asymptotic solution of the initial boundary value problems is studied.
On the wave equation with semilinear porous acoustic boundary conditions
Graber, Philip Jameson
2012-05-01
The goal of this work is to study a model of the wave equation with semilinear porous acoustic boundary conditions with nonlinear boundary/interior sources and a nonlinear boundary/interior damping. First, applying the nonlinear semigroup theory, we show the existence and uniqueness of local in time solutions. The main difficulty in proving the local existence result is that the Neumann boundary conditions experience loss of regularity due to boundary sources. Using an approximation method involving truncated sources and adapting the ideas in Lasiecka and Tataru (1993) [28], we show that the existence of solutions can still be obtained. Second, we prove that under some restrictions on the source terms, then the local solution can be extended to be global in time. In addition, it has been shown that the decay rates of the solution are given implicitly as solutions to a first order ODE and depends on the behavior of the damping terms. In several situations, the obtained ODE can be easily solved and the decay rates can be given explicitly. Third, we show that under some restrictions on the initial data and if the interior source dominates the interior damping term and if the boundary source dominates the boundary damping, then the solution ceases to exists and blows up in finite time. Moreover, in either the absence of the interior source or the boundary source, then we prove that the solution is unbounded and grows as an exponential function. © 2012 Elsevier Inc.
Gravity actions, boundary terms and second-order field equations
The Lovelock action is a natural generalization of the Einstein-Hilbert gravity action in higher (more than four) dimensions. Certain boundary terms have to be added to this action because of various reasons which we briefly review. Dimensional reduction of the Lovelock action leads to a generalized Einstein-Yang-Mills (+ scalar field or σ-models) action in lower dimensions. Since the field equations derived from the Lovelock action are of second order only, the same holds for the dimensionally reduced action. The necessary boundary terms for the latter action are then obtained by dimensional reduction of the boundary terms in the Lovelock action. We consider reduction from n to n-1 dimensions in some detail and obtain generalizations of Horndeski's non-minimally coupled Einstein-Maxwell action with second-order field equations in more than four dimensions. (orig./HSI)
Solving wave equation with spectral methods and nonreflecting boundary conditions
Novák, J; Novak, Jerome; Bonazzola, Silvano
2002-01-01
A multidomain spectral method for solving wave equations is presented. This method relies on the expansion of functions on basis of spherical harmonics $(Y_l^m(\\theta, \\phi))$ for the angular dependence and of Chebyshev polynomials $T_n(x)$ for the radial part. The spherical domains consist of shells surrounding a nucleus and cover the space up to a finite radius $R$ at which boundary conditions are imposed. Time derivatives are estimated using standard finite-differences second order schemes, which are chosen to be implicit to allow for (almost) any size of time-step. Emphasis is put on the implementation of absorbing boundary conditions that allow for the numerical boundary to be completely transparent to the physical wave. This is done using a multipolar expansion of an exact boundary condition for outgoing waves, which is truncated at some point. Using an auxiliary function, which is solution of a wave equation on the sphere defining the outer boundary of the numerical grid, the absorbing boundary conditi...
Partial differential equations & boundary value problems with Maple
Articolo, George A
2009-01-01
Partial Differential Equations and Boundary Value Problems with Maple presents all of the material normally covered in a standard course on partial differential equations, while focusing on the natural union between this material and the powerful computational software, Maple. The Maple commands are so intuitive and easy to learn, students can learn what they need to know about the software in a matter of hours- an investment that provides substantial returns. Maple''s animation capabilities allow students and practitioners to see real-time displays of the solutions of partial differential equations. Maple files can be found on the books website. Ancillary list: Maple files- http://www.elsevierdirect.com/companion.jsp?ISBN=9780123747327 Provides a quick overview of the software w/simple commands needed to get startedIncludes review material on linear algebra and Ordinary Differential equations, and their contribution in solving partial differential equationsIncorporates an early introduction to Sturm-L...
Integral equations for the electromagnetic field in dielectrics
Mostowski, Jan; Załuska-Kotur, Magdalena A.
2016-09-01
We study static the electric field and electromagnetic waves in dielectric media. In contrast to the standard approach, we use, formulate and solve integral equations for the field. We discuss the case of an electrostatic field of a point charge placed inside a dielectric; the integral equation approach allows us to find and interpret the dielectric constant in terms of molecular polarizability. Next we discuss propagation of electromagnetic waves using the same integral equation approach. We derive the dispersion relation and find the reflection and transmission coefficients at the boundary between the vacuum and the dielectric. The present approach supplements the standard approach based on macroscopic Maxwell equations and contributes to better a understanding of some electromagnetic effects.
A kernel-free boundary integral method for elliptic boundary value problems
Ying, Wenjun; Henriquez, Craig S.
2007-12-01
This paper presents a class of kernel-free boundary integral (KFBI) methods for general elliptic boundary value problems (BVPs). The boundary integral equations reformulated from the BVPs are solved iteratively with the GMRES method. During the iteration, the boundary and volume integrals involving Green's functions are approximated by structured grid-based numerical solutions, which avoids the need to know the analytical expressions of Green's functions. The KFBI method assumes that the larger regular domain, which embeds the original complex domain, can be easily partitioned into a hierarchy of structured grids so that fast elliptic solvers such as the fast Fourier transform (FFT) based Poisson/Helmholtz solvers or those based on geometric multigrid iterations are applicable. The structured grid-based solutions are obtained with standard finite difference method (FDM) or finite element method (FEM), where the right hand side of the resulting linear system is appropriately modified at irregular grid nodes to recover the formal accuracy of the underlying numerical scheme. Numerical results demonstrating the efficiency and accuracy of the KFBI methods are presented. It is observed that the number of GMRES iterations used by the method for solving isotropic and moderately anisotropic BVPs is independent of the sizes of the grids that are employed to approximate the boundary and volume integrals. With the standard second-order FEMs and FDMs, the KFBI method shows a second-order convergence rate in accuracy for all of the tested Dirichlet/Neumann BVPs when the anisotropy of the diffusion tensor is not too strong.
The determination of an unknown boundary condition in a fractional diffusion equation
Rundell, William
2013-07-01
In this article we consider an inverse boundary problem, in which the unknown boundary function ∂u/∂v = f(u) is to be determined from overposed data in a time-fractional diffusion equation. Based upon the free space fundamental solution, we derive a representation for the solution f as a nonlinear Volterra integral equation of second kind with a weakly singular kernel. Uniqueness and reconstructibility by iteration is an immediate result of a priori assumption on f and applying the fixed point theorem. Numerical examples are presented to illustrate the validity and effectiveness of the proposed method. © 2013 Copyright Taylor and Francis Group, LLC.
Optimal control problems for impulsive systems with integral boundary conditions
Allaberen Ashyralyev
2013-03-01
Full Text Available In this article, the optimal control problem is considered when the state of the system is described by the impulsive differential equations with integral boundary conditions. Applying the Banach contraction principle the existence and uniqueness of the solution is proved for the corresponding boundary problem by the fixed admissible control. The first and second variation of the functional is calculated. Various necessary conditions of optimality of the first and second order are obtained by the help of the variation of the controls.
Dual integral equations with Fox's H-function kernel
R. N. Kalia
1984-01-01
Full Text Available The dual integral equations involving Bessel function kernels were first considered by Weber in 1873. The problem comprised of finding potential of an electrified disc which belongs to a general category of mixed boundary value problems. Titchmarsh gave the formal solution using Wiener-Hopf procedure. We use this direct method as improvised by Busbridge to solve a class of dual integral equations which can be reduced to other known kernels by particularizing the parameters in the Fox's H-function.
Maxwell boundary conditions impose non-Lindblad master equation
Bamba, Motoaki
2016-01-01
From the Hamiltonian connecting the inside and outside of an Fabry-Perot cavity, which is derived from the Maxwell boundary conditions at a mirror of the cavity, a master equation of a non-Lindblad form is derived when the cavity embeds matters, although we can transform it to the Lindblad form by performing the rotating-wave approximation to that Hamiltonian. We calculate absorption spectra by these Lindblad and non-Lindblad master equations and also by the Maxwell boundary conditions in framework of the classical electrodynamics, which we consider the most reliable approach. We found that, compared to the Lindblad master equation, the absorption spectra by the non-Lindblad one agree better with those by the Maxwell boundary conditions. Although the discrepancy is highlighted only in the ultra-strong light-matter interaction regime with a relatively large broadening, the master equation of the non-Lindblad form is preferable rather than of the Lindblad one for pursuing the consistency with the classical elec...
ON A PARABOLIC FREE BOUNDARY EQUATION MODELING PRICE FORMATION
MARKOWICH, P. A.
2009-10-01
We discuss existence and uniqueness of solutions for a one-dimensional parabolic evolution equation with a free boundary. This problem was introduced by Lasry and Lions as description of the dynamical formation of the price of a trading good. Short time existence and uniqueness is established by a contraction argument. Then we discuss the issue of global-in-time-extension of the local solution which is closely related to the regularity of the free boundary. We also present numerical results. © 2009 World Scientific Publishing Company.
Counting master integrals. Integration by parts vs. functional equations
Kniehl, Bernd A.; Tarasov, Oleg V. [Hamburg Univ. (Germany). II. Inst. fuer Theoretische Physik
2016-01-15
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.
Counting master integrals. Integration by parts vs. functional equations
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.
Counting master integrals: Integration by parts vs. functional equations
Kniehl, Bernd A
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.
Optimal Control of a Parabolic Equation with Dynamic Boundary Condition
We investigate a control problem for the heat equation. The goal is to find an optimal heat transfer coefficient in the dynamic boundary condition such that a desired temperature distribution at the boundary is adhered. To this end we consider a function space setting in which the heat flux across the boundary is forced to be an Lp function with respect to the surface measure, which in turn implies higher regularity for the time derivative of temperature. We show that the corresponding elliptic operator generates a strongly continuous semigroup of contractions and apply the concept of maximal parabolic regularity. This allows to show the existence of an optimal control and the derivation of necessary and sufficient optimality conditions.
Engineering integrable nonautonomous nonlinear Schroedinger equations
We investigate Painleve integrability of a generalized nonautonomous one-dimensional nonlinear Schroedinger (NLS) equation with time- and space-dependent dispersion, nonlinearity, and external potentials. Through the Painleve analysis some explicit requirements on the dispersion, nonlinearity, dissipation/gain, and the external potential as well as the constraint conditions are identified. It provides an explicit way to engineer integrable nonautonomous NLS equations at least in the sense of Painleve integrability. Furthermore analytical solutions of this class of integrable nonautonomous NLS equations can be obtained explicitly from the solutions of the standard NLS equation by a general transformation. The result provides a significant way to control coherently the soliton dynamics in the corresponding nonlinear systems, as that in Bose-Einstein condensate experiments. We analyze explicitly the soliton dynamics under the nonlinearity management and the external potentials and discuss its application in the matter-wave dynamics. Some comparisons with the previous works have also been discussed.
Coupling Integrable Couplings of an Equation Hierarchy
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)
Integrability of doubly-periodic Riccati equation
Guan Ke-Ying
1998-12-01
Full Text Available By the structure of solvable subgroup of SL(2,Ã¢Â„Â‚ (see [1], the integrability and properties of solutions of a Riccati equation with an elliptic function coefficient, which is related to a Fuchsian equation on the torus T2,is studied.
THE COUPLING OF NATURAL BOUNDARY ELEMENT AND FINITE ELEMENT METHOD FOR 2D HYPERBOLIC EQUATIONS
De-hao Yu; Qi-kui Du
2003-01-01
In this paper, we investigate the coupling of natural boundary element and finite ele-ment methods of exterior initial boundary value problems for hyperbolic equations. Thegoverning equation is first discretized in time, leading to a time-step scheme, where anexterior elliptic problem has to be solved in each time step. Second, a circular artifi-in an unbounded domain is transformed into the nonlocal boundary value problem in abounded subdomain. And the natural integral equation and the Poisson integral formulaare obtained in the infinite domain Ω2 outside circle of radius R. The coupled variationalformulation is given. Only the function itself, not its normal derivative at artificial bound-and the boundary element stiffness matrix has a few different elements. Such a coupledmethod is superior to the one based on direct boundary element method. This paper dis-cusses finite element discretization for variational problem and its corresponding numericaltechnique, and the convergence for the numerical solutions. Finally, the numerical exampleis presented to illustrate feasibility and efficiency of this method.
Nahed S. Hussein
2014-01-01
Full Text Available A numerical boundary integral scheme is proposed for the solution to the system of eld equations of plane. The stresses are prescribed on one-half of the circle, while the displacements are given. The considered problem with mixed boundary conditions in the circle is replaced by two problems with homogeneous boundary conditions, one of each type, having a common solution. The equations are reduced to a system of boundary integral equations, which is then discretized in the usual way, and the problem at this stage is reduced to the solution to a rectangular linear system of algebraic equations. The unknowns in this system of equations are the boundary values of four harmonic functions which define the full elastic solution and the unknown boundary values of stresses or displacements on proper parts of the boundary. On the basis of the obtained results, it is inferred that a stress component has a singularity at each of the two separation points, thought to be of logarithmic type. The results are discussed and boundary plots are given. We have also calculated the unknown functions in the bulk directly from the given boundary conditions using the boundary collocation method. The obtained results in the bulk are discussed and three-dimensional plots are given. A tentative form for the singular solution is proposed and the corresponding singular stresses and displacements are plotted in the bulk. The form of the singular tangential stress is seen to be compatible with the boundary values obtained earlier. The efficiency of the used numerical schemes is discussed.
Boundary value problemfor multidimensional fractional advection-dispersion equation
Khasambiev Mokhammad Vakhaevich
2015-01-01
In recent time there is a very great interest in the study of differential equations of fractional order, in which the unknown function is under the symbol of fractional derivative. It is due to the development of the theory of fractional integro-differential theory and application of it in different fields.The fractional integrals and derivatives of fractional integro-differential equations are widely used in modern investigations of theoretical physics, mechanics, and applied mathematics. T...
High Accuracy Method for Integral Equations with Discontinuous Kernels
Kang, Sheon-Young; Koltracht, Israel; Rawitscher, George
1999-01-01
A new highly accurate numerical approximation scheme based on a Gauss type Clenshaw-Curtis Quadrature for Fredholm integral equations of the second kind, whose kernel is either discontinuous or not smooth along the main diagonal, is presented. This scheme is of spectral accuracy when the kernel is infinitely differentiable away from the main diagonal, and is also applicable when the kernel is singular along the boundary, and at isolated points on the main diagonal. The corresponding composite...
On Approximate Asymptotic Solution of Integral Equations
Jikia, Vagner
2013-01-01
It is well known that multi-particle integral equations of collision theory, in general, are not compact. At the same time it has been shown that the motion of three and four particles is described with consistent integral equations. In particular, by using identical transformations of the kernel of the Lipman-Schwinger equation for certain classes of potentials Faddeev obtained Fredholm type integral equations for three-particle problems $[1]$. The motion of for bodies is described by equations of Yakubovsky and Alt-Grassberger-Sandhas-Khelashvili $[2.3]$, which are obtained as a result of two subsequent transpormations of the kernel of Lipman-Schwinger equation. in the case of $N>4$ the compactness of multi-particle equations has not been proven yet. In turn out that for sufficiently high energies the $N$-particle $\\left( {N \\ge 3} \\right)$ dynamic equations have correct asymptotic solutions satisfying unitary condition $[4]$. In present paper by using the Heitler formalism we obtain the results briefly sum...
Path Integral for the Dirac Equation
Polonyi, Janos
1998-01-01
A c-number path integral representation is constructed for the solution of the Dirac equation. The integration is over the real trajectories in the continuous three-space and other two canonical pairs of compact variables controlling the spin and the chirality flips.
Path Integral for Relativistic Equations of Motion
Gosselin, Pierre; Polonyi, Janos
1997-01-01
A non-Grassmanian path integral representation is given for the solution of the Klein-Gordon and the Dirac equations. The trajectories of the path integral are rendered differentiable by the relativistic corrections. The nonrelativistic limit is briefly discussed from the point of view of the renormalization group.
A Collocation Method for Volterra Integral Equations
Kolk, Marek
2010-09-01
We propose a piecewise polynomial collocation method for solving linear Volterra integral equations of the second kind with logarithmic kernels which, in addition to a diagonal singularity, may have a singularity at the initial point of the interval of integration. An attainable order of the convergence of the method is studied. We illustrate our results with a numerical example.
A Nonlinear Singularly Perturbed Problem for Reaction Diffusion Equations with Boundary Perturbation
Jia-qi Mo; Wan-tao Lin
2005-01-01
A nonlinear singularly perturbed problems for reaction diffusion equation with boundary perturbation is considered. Under suitable conditions, the asymptotic behavior of solution for the initial boundary value problems of reaction diffusion equations is studied using the theory of differential inequalities.
A Numerical Method for 1-D Parabolic Equation with Nonlocal Boundary Conditions
M. Tadi
2014-01-01
Full Text Available This paper is concerned with a local method for the solution of one-dimensional parabolic equation with nonlocal boundary conditions. The method uses a coordinate transformation. After the coordinate transformation, it is then possible to obtain exact solutions for the resulting equations in terms of the local variables. These exact solutions are in terms of constants of integration that are unknown. By imposing the given boundary conditions and smoothness requirements for the solution, it is possible to furnish a set of linearly independent conditions that can be used to solve for the constants of integration. A number of examples are used to study the applicability of the method. In particular, three nonlinear problems are used to show the novelty of the method.
Spectral integration of linear boundary value problems
Viswanath, Divakar
2012-01-01
Spectral integration is a method for solving linear boundary value problems which uses the Chebyshev series representation of functions to avoid the numerical discretization of derivatives. It is occasionally attributed to Zebib (J. of Computational Physics vol. 53 (1984), p. 443-455) and more often to Greengard (SIAM J. on Numerical Analysis vol. 28 (1991), p. 1071-1080). Its advantage is believed to be its relative immunity to errors that arise when nearby grid points are used to approximate derivatives. In this paper, we reformulate the method of spectral integration by changing it in four different ways. The changes consist of a more convenient integral formulation, a different way to treat and interpret boundary conditions, treatment of higher order problems in factored form, and the use of piecewise Chebyshev grid points. Our formulation of spectral integration is more flexible and powerful as show by its ability to solve a problem that would otherwise take 8192 grid points using only 96 grid points. So...
Boričić Zoran
2009-01-01
Full Text Available This paper concerns with unsteady two-dimensional temperature laminar magnetohydrodynamic (MHD boundary layer of incompressible fluid. It is assumed that induction of outer magnetic field is function of longitudinal coordinate with force lines perpendicular to the body surface on which boundary layer forms. Outer electric filed is neglected and magnetic Reynolds number is significantly lower then one i.e. considered problem is in inductionless approximation. Characteristic properties of fluid are constant because velocity of flow is much lower than speed of light and temperature difference is small enough (under 50ºC . Introduced assumptions simplify considered problem in sake of mathematical solving, but adopted physical model is interesting from practical point of view, because its relation with large number of technically significant MHD flows. Obtained partial differential equations can be solved with modern numerical methods for every particular problem. Conclusions based on these solutions are related only with specific temperature MHD boundary layer problem. In this paper, quite different approach is used. First new variables are introduced and then sets of similarity parameters which transform equations on the form which don't contain inside and in corresponding boundary conditions characteristics of particular problems and in that sense equations are considered as universal. Obtained universal equations in appropriate approximation can be solved numerically once for all. So-called universal solutions of equations can be used to carry out general conclusions about temperature MHD boundary layer and for calculation of arbitrary particular problems. To calculate any particular problem it is necessary also to solve corresponding momentum integral equation.
About potential of double layer and boundary value problems for Laplace equation
An integral operator raisen by a kernel of the double layer's potential is investigated. The kernel is defined on S (S - two-digit variety of C2 class presented by a boundary of the finite domain in R3). The operator is considered on C(S). Following results are received: the operator's spectrum belongs to [-1,1]; it's eigenvalues and eigenfunctions may be found by Kellog's method; knowledge of the operator's spectrum is enough to construct it's resolvent. These properties permit to point out the determined interation processes, solving boundary value problems for Laplace equation. One of such processes - solving of Roben problem - is generalized on electrostatic problems. 6 refs
Chen, Gui-Qiang; Osborne, Dan; Qian, Zhongmin
2008-01-01
We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a general domain in $\\R^n$ with compact and smooth boundary, subject to the kinematic and vorticity boundary conditions on the non-flat boundary. We observe that, under the nonhomogeneous boundary conditions, the pressure $p$ can be still recovered by solving the Neumann problem for the Poisson equation. Then we establish the well-posedness of the unsteady Stokes equations and employ the so...
Asymptotic behavior of solutions to wave equations with a memory condition at the boundary
Mauro de Lima Santos
2001-11-01
Full Text Available In this paper, we study the stability of solutions for wave equations whose boundary condition includes a integral that represents the memory effect. We show that the dissipation is strong enough to produce exponential decay of the solution, provided the relaxation function also decays exponentially. When the relaxation function decays polynomially, we show that the solution decays polynomially and with the same rate.
Periodic and Boundary Value Problems for Second Order Differential Equations
Nikolaos S Papageorgiou; Francesca Papalini
2001-02-01
In this paper we study second order scalar differential equations with Sturm–Liouville and periodic boundary conditions. The vector field (, , ) is Caratheodory and in some instances the continuity condition on or is replaced by a monotonicity type hypothesis. Using the method of upper and lower solutions as well as truncation and penalization techniques, we show the existence of solutions and extremal solutions in the order interval determined by the upper and lower solutions. Also we establish some properties of the solutions and of the set they form.
Adaptive integral equation methods in transport theory
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
Inequalities for differential and integral equations
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
Quadratic integral solutions to double Pell equations
Veneziano, Francesco
2011-01-01
We study the quadratic integral points-that is, (S-)integral points defined over any extension of degree two of the base field-on a curve defined in P_3 by a system of two Pell equations. Such points belong to three families explicitly described, or belong to a finite set whose cardinality may be explicitly bounded in terms of the base field, the equations defining the curve and the set S. We exploit the peculiar geometry of the curve to adapt the proof of a theorem of Vojta, which in this case does not apply.
Algorithms For Integrating Nonlinear Differential Equations
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.
Boundary integral method applied in chaotic quantum billiards
Li, B; Li, Baowen; Robnik, Marko
1995-01-01
The boundary integral method (BIM) is a formulation of Helmholtz equation in the form of an integral equation suitable for numerical discretization to solve the quantum billiard. This paper is an extensive numerical survey of BIM in a variety of quantum billiards, integrable (circle, rectangle), KAM systems (Robnik billiard) and fully chaotic (ergodic, such as stadium, Sinai billiard and cardioid billiard). On the theoretical side we point out some serious flaws in the derivation of BIM in the literature and show how the final formula (which nevertheless was correct) should be derived in a sound way and we also argue that a simple minded application of BIM in nonconvex geometries presents serious difficulties or even fails. On the numerical side we have analyzed the scaling of the averaged absolute value of the systematic error \\Delta E of the eigenenergy in units of mean level spacing with the density of discretization (b = number of numerical nodes on the boundary within one de Broglie wavelength), and we f...
PREFACE: Symmetries and Integrability of Difference Equations
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
Gui-Qiang Chen; Dan Osborne; Zhongmin Qian
2009-01-01
We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a general domain in RN with compact and smooth boundary, subject to the kinematic and vorticity boundary conditions on the non-fiat boundary. We observe that, under the nonhomogeneons boundary conditions, the pressure p can be still recovered by solving the Neumann problem for the Poisson equation. Then we establish the well-posedness of the unsteady Stokes equations and employ the solution to reduce our initial-boundary value problem into an initial-boundary value problem with absolute boundary conditions. Based on this, we first establish the well-posedness for an appropriate local linearized problem with the absolute boundary conditions and the initial condition (without the incompressibility condition), which establishes a velocity mapping. Then we develop apriori estimates for the velocity mapping, especially involving the Sobolev norm for the time-derivative of the mapping to deal with the complicated boundary conditions, which leads to the existence of the fixed point of the mapping and the existence of solutions to our initial-boundary value problem. Finally, we establish that, when the viscosity coefficient tends zero, the strong solutions of the initial-boundary value problem in RN(n≥3) with nonhomogeneous vorticity boundary condition converge in L2 to the corresponding Euler equations satisfying the kinematic condition.
Path Integral Methods for Stochastic Differential Equations
Chow, Carson C.; Buice, Michael A.
2015-01-01
Stochastic differential equations (SDEs) have multiple applications in mathematical neuroscience and are notoriously difficult. Here, we give a self-contained pedagogical review of perturbative field theoretic and path integral methods to calculate moments of the probability density function of SDEs. The methods can be extended to high dimensional systems such as networks of coupled neurons and even deterministic systems with quenched disorder.
Nonparametric (smoothed) likelihood and integral equations
Groeneboom, Piet
2012-01-01
We show that there is an intimate connection between the theory of nonparametric (smoothed) maximum likelihood estimators for certain inverse problems and integral equations. This is illustrated by estimators for interval censoring and deconvolution problems. We also discuss the asymptotic efficiency of the MLE for smooth functionals in these models.
Simplified differential equations approach for Master Integrals
Papadopoulos, Costas G.
2014-01-01
A simplified differential equations approach for Master Integrals is presented. It allows to express them, straightforwardly, in terms of Goncharov Polylogarithms. As a proof-of-concept of the proposed method, results at one and two loops are presented, including the massless one-loop pentagon with up to one off-shell leg at order epsilon.
Bethe ansatz for the Temperley–Lieb spin chain with integrable open boundaries
In this paper we study the spectrum of the spin-1 Temperley–Lieb spin chain with integrable open boundary conditions. We obtain the eigenvalue expressions as well as its associated Bethe ansatz equations by means of the coordinate Bethe ansatz. These equations provide the complete description of the spectrum of the model. (paper)
Three New Integrable Hierarchies of Equations
A general Lie algebra Vs and the corresponding loop algebra V-tildes are constructed, from which the linear isospectral Lax pairs are established, whose compatibility presents the zero curvature equation. As its application, a new Lax integrable hierarchy containing two parameters is worked out. It is not Liouville-integrable, however, its two reduced systems are Liouville-integrable, whose Hamiltonian structures are derived by making use of the quadratic-form identity and the γ formula (i.e. the computational formula on the constant γ appeared in the trace identity and the quadratic-form identity).
Three New Integrable Hierarchies of Equations
无
2007-01-01
A general Lie algebra Vs and the corresponding loop algebra ～Vs are constructed, from which the linear isospectral Lax pairs are established, whose compatibility presents the zero curvature equation. As its application, a new Lax integrable hierarchy containing two parameters is worked out. It is not Liouville-integrable, however, its two reduced systems are Liouville-integrable, whose Hamiltonian structures are derived by making use of the quadratic-form identity and the γ formula (i.e. the computational formula on the constant γ appeared in the trace identity and the quadratic-form identity).
Lectures on differential equations for Feynman integrals
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)
Obidjon Kh. Abdullaev
2016-06-01
Full Text Available In this work, we study the existence and uniqueness of solutions to non-local boundary value problems with integral gluing condition. Mixed type equations (parabolic-hyperbolic involving the Caputo fractional derivative have loaded parts in Riemann-Liouville integrals. Thus we use the method of integral energy to prove uniqueness, and the method of integral equations to prove existence.
A direct solver with O(N) complexity for integral equations on one-dimensional domains
Gillman, Adrianna; Young, Patrick; Martinsson, Per-Gunnar
2011-01-01
An algorithm for the direct inversion of the linear systems arising from Nystrom discretization of integral equations on one-dimensional domains is described. The method typically has O(N) complexity when applied to boundary integral equations (BIEs) in the plane with non-oscillatory kernels such as those associated with the Laplace and Stokes' equations. The scaling coefficient suppressed by the "big-O" notation depends logarithmically on the requested accuracy. The method can also be applie...
Nonlocal singular problem with integral condition for a second-order parabolic equation
Ahmed Lakhdar Marhoune
2015-03-01
Full Text Available We prove the existence and uniqueness of a strong solution for a parabolic singular equation in which we combine Dirichlet with integral boundary conditions given only on parts of the boundary. The proof uses a priori estimate and the density of the range of the operator generated by the problem considered.
Reducing differential equations for multiloop master integrals
Lee, Roman
2014-01-01
We present an algorithm of the reduction of the differential equations for master integrals the Fuchsian form with the right-hand side matrix linearly depending on dimensional regularization parameter $\\epsilon$. We consider linear transformations of the functions column which are rational in the variable and in $\\epsilon$. Apart from some degenerate cases described below, the algorithm allows one to obtain the required transformation or to ascertain irreducibility to the form required. Degen...
Boundary conditions for the Pauli equation: application to photodetachment of Cs-.
Bahrim, C; Fabrikant, I I; Thumm, U
2001-09-17
We formulate the boundary conditions near the atomic nucleus for solving the Pauli equation, based on the analytic solution of the Dirac equation for a Coulomb potential. We then integrate the Pauli equation using an effective potential that is adjusted to reproduce Dirac R-matrix scattering phase shifts, and find the (3)P(o)(1) resonance contribution to the photodetachment cross section of Cs-. Our photodetachment cross sections agree with recent experiments by Scheer et al. [Phys. Rev. Lett. 80, 684 (1998)] after tuning the resonance position by 2.4 meV. We also provide angle-differential photodetachment cross sections and the corresponding asymmetry parameter beta near the Cs(6s) threshold. PMID:11580504
Recursion operators, conservation laws, and integrability conditions for difference equations
Mikhailov, A. V.; Wang, Jing Ping; Xenitidis, P.
2011-04-01
We attempt to propose an algebraic approach to the theory of integrable difference equations. We define the concept of a recursion operator for difference equations and show that it generates an infinite sequence of symmetries and canonical conservation laws for a difference equation. As in the case of partial differential equations, these canonical densities can serve as integrability conditions for difference equations. We obtain the recursion operators for the Viallet equation and all the Adler-Bobenko-Suris equations.
SOLVING INTEGRAL EQUATIONS WITH LOGARITHMIC KERNEL BY USING PERIODIC QUASI-WAVELET
Han-lin Chen; Si-long Peng
2000-01-01
In solving integral equations with logarithmic kernel which arises from the boundary integral equation reformulation of some boundary value problems for the two dimensional Helmholtz equation, we combine the Galerkin method with Beylkin's ([2]) approach, series of dense and nonsymmetric matrices may appear if we use traditional method. By appealing the so-called periodic quasi-wavelet (PQW in abbr.) ([5]), some of these matrices become diagonal, therefore we can find a algorithm with only O(K(m)2) arithmetic operations where m is the highest level. The Galerkin approximation has a polynomial rate of convergence.
Jia-qi Mo; Wan-tao Lin
2006-01-01
A class of nonlinear singularly perturbed problems for reaction diffusion equations with boundary perturbation are considered. Under suitable conditions, the asymptotic behavior of solution for the initial boundary value problems is studied using the theory of differential inequalities.
OuyangCheng; MoJiaqi
2005-01-01
The nonlinear singularly perturbed problems for reaction diffusion equations with boundary perturbation are considered. Under suitable conditions, using the theory of differential inequalities the asymptotic behavior of solution for the initial boundary value problems is studied.
A Priori Estimates for Solutions of Boundary Value Problems for Fractional-Order Equations
Alikhanov, A A
2011-01-01
We consider boundary value problems of the first and third kind for the diffusionwave equation. By using the method of energy inequalities, we find a priori estimates for the solutions of these boundary value problems.
An initial-boundary value problem for three-dimensional Zakharov-Kuznetsov equation
Faminskii, Andrei V.
2016-02-01
An initial-boundary value problem with homogeneous Dirichlet boundary conditions for three-dimensional Zakharov-Kuznetsov equation is considered. Results on global existence, uniqueness and large-time decay of weak solutions in certain weighted spaces are established.
LIMIT BEHAVIOUR OF SOLUTIONS TO EQUIVALUED SURFACE BOUNDARY VALUE PROBLEM FOR PARABOLIC EQUATIONS
LI Fengquan
2002-01-01
In this paper, we discuss the limit behaviour of solutions to equivalued surface boundary value problem for parabolic equations when the equivalued surface boundary shrinks to a point and the space dimension of the domain is two or more.
SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS FOR ELLIPTIC EQUATION WITH A CURVE OF TURNING POINT
MoJiaqi
2002-01-01
The singularly perturbed elliptic equation boundary value problem with a curve of turning point is considered. Using the method of multiple scales and the comparison theorem,the asymptotic behavior of solution for the boundary value problem is studied.
Integrated watershed planning across jurisdictional boundaries
Watts, A. W.; Roseen, R.; Stacey, P.; Bourdeau, R.
2014-12-01
We will present the foundation for an Coastal Watershed Integrated Plan for three communities in southern New Hampshire. Small communities are often challenged by complex regulatory requirements and limited resources, but are wary of perceived risks in engaging in collaborative projects with other communities. Potential concerns include loss of control, lack of resources to engage in collaboration, technical complexity, and unclear benefits. This project explores a multi-town subwatershed application of integrated planning across jurisdictional boundaries that addresses some of today's highest priority water quality issues: wastewater treatment plant upgrades for nutrient removal; green infrastructure stormwater management for developing and re-developing areas; and regional monitoring of ecosystem indicators in support of adaptive management to achieve nutrient reduction and other water quality goals in local and downstream waters. The project outcome is a collaboratively-developed inter-municipal integrated plan, and a monitoring framework to support cross jurisdictional planning and assess attainment of water quality management goals. This research project has several primary components: 1) assessment of initial conditions, including both the pollutant load inputs and the political, economic and regulatory status within each community, 2) a pollutant load model for point and non-point sources, 3) multi-criteria evaluation of load reduction alternatives 4) a watershed management plan optimized for each community, and for Subwatersheds combining multiple communities. The final plan will quantify the financial and other benefits/drawbacks to each community for both inter municipal and individual pollution control approaches. We will discuss both the technical and collaborative aspects of the work, with lessons learned regarding science to action, incorporation of social, economic and water quality assessment parameters, and stakeholder/researcher interaction.
XU Xi-Xiang; ZHANG Yu-Feng
2004-01-01
A discrete matrix spectral problem and the associated hierarchy of Lax integrable lattice equations are presented, and it is shown that the resulting Lax integrable lattice equations are all Liouville integrable discrete Hamiltonian systems. A new integrable symplectic map is given by binary Bargmann .constraint of the resulting hierarchy.Finally, an infinite set of conservation laws is given for the resulting hierarchy.
Monotone iterative technique for fractional differential equations with periodic boundary conditions
J. D. Ramírez
2009-01-01
Full Text Available In this paper we develop Monotone Method using upper and lower solutions for fractional differential equations with periodic boundary conditions. Initially we develop a comparison result and prove that the solution of the linear fractional differential equation with periodic boundary condition exists and is unique. Using this we develop iterates which converge uniformly monotonically to minimal and maximal solutions of the nonlinear fractional differential equations with periodic boundary conditions in the weighted norm.
Monotone iterative technique for fractional differential equations with periodic boundary conditions
J. D. Ramírez; A. S. Vatsala
2009-01-01
In this paper we develop Monotone Method using upper and lower solutions for fractional differential equations with periodic boundary conditions. Initially we develop a comparison result and prove that the solution of the linear fractional differential equation with periodic boundary condition exists and is unique. Using this we develop iterates which converge uniformly monotonically to minimal and maximal solutions of the nonlinear fractional differential equations with periodic boundary con...
Quano, Y H
2001-01-01
The $A^{(1)}_{n-1}$-face model with boundary reflection is considered on the basis of the boundary CTM bootstrap. We construct the fused boundary Boltzmann weights to determine the normalization factor. We derive difference equations of the quantum Knizhnik-Zamolodchikov type for correlation functions of the present model. The simplest difference equations are solved the in the case of the free boundary condition.
The boundary value problem of axisymmetrical slow motion of viscous incompressible fluid in the upper half-space or in the whole space surrounding a fixed infinite circular cylinder is considered. The solution of the linearized non-homogeneous Navier-Stokes equations is obtained in each case in quadratures by using Abel integral equations, which transform the axisymmetrical problem to solvable two dimensional plane problem. (author)
Pierluigi Colli
2015-07-01
Full Text Available In this paper we establish second-order sufficient optimality conditions for a boundary control problem that has been introduced and studied by three of the authors in the preprint arXiv:1407.3916. This control problem regards the viscous Cahn–Hilliard equation with possibly singular potentials and dynamic boundary conditions.
A.S. BERDYSHEV; A. CABADA; B.Kh. TURMETOV
2014-01-01
This paper is concerned with the solvability of a boundary value problem for a nonhomogeneous biharmonic equation. The boundary data is determined by a differential operator of fractional order in the Riemann-Liouville sense. The considered problem is a generalization of the known Dirichlet and Neumann problems.
Bounded solutions for fuzzy differential and integral equations
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.
A Study of the Navier-Stokes Equations with the Kinematic and Navier Boundary Conditions
Chen, Gui-Qiang; Qian, Zhongmin
2008-01-01
We study the initial-boundary value problem of the Navier-Stokes equations for incompressible fluids in a domain in $\\R^3$ with compact and smooth boundary, subject to the kinematic and Navier boundary conditions. We first reformulate the Navier boundary condition in terms of the vorticity, which is motivated by the Hodge theory on manifolds with boundary from the viewpoint of differential geometry, and establish basic elliptic estimates for vector fields subject to the kinematic and Navier b...
Path Integral and Solutions of the Constraint Equations: The Case of Reducible Gauge Theories
Ferraro, R.; Henneaux, M.; Puchin, M.
1994-01-01
It is shown that the BRST path integral for reducible gauge theories, with appropriate boundary conditions on the ghosts, is a solution of the constraint equations. This is done by relating the BRST path integral to the kernel of the evolution operator projected on the physical subspace.
Explicit expressions for three-dimensional boundary integrals in linear elasticity
Nintcheu Fata, Sylvain [ORNL
2011-01-01
On employing isoparametric, piecewise linear shape functions over a flat triangle, exact formulae are derived for all surface potentials involved in the numerical treatment of three-dimensional singular and hyper-singular boundary integral equations in linear elasticity. These formulae are valid for an arbitrary source point in space and are represented as analytical expressions along the edges of the integration triangle. They can be employed to solve integral equations defined on triangulated surfaces via a collocation method or may be utilized as analytical expressions for the inner integrals in a Galerkin technique. A numerical example involving a unit triangle and a source point located at various distances above it, as well as sample problems solved by a collocation boundary element method for the Lame equation are included to validate the proposed formulae.
Funnel Equations and Multivalued Integration Problems for Control Synthesis
Kurzhanski, A.B.; Nikonov, O.I.
1989-01-01
This paper indicates a sequence of evolutionary "funnel equations" with set-valued solutions which are crucial for the construction of respective feed-back control strategies along the schemes introduced by N.N. Krasovskii. The integration of these funnel equations leads to a sequence of multivalued integrals that generalize some of those introduced earlier (Auman's integral, the Convolution integral, Pontryagin's Alternated integral).
Evolution equations: Frobenius integrability, conservation laws and travelling waves
We give new results concerning the Frobenius integrability and solution of evolution equations admitting travelling wave solutions. In particular, we give a powerful result which explains the extraordinary integrability of some of these equations. We also discuss ‘local’ conservations laws for evolution equations in general and demonstrate all the results for the Korteweg–de Vries equation. (paper)
Shu, Jian-Jun
2014-01-01
A very simple and accurate numerical method which is applicable to systems of differentio-integral equations with quite general boundary conditions has been devised. Although the basic idea of this method stems from the Keller Box method, it solves the problem of systems of differential equations involving integral operators not previously considered by the Keller Box method. Two main preparatory stages are required: (i) a merging procedure for differential equations and conditions without integral operators and; (ii) a reduction procedure for differential equations and conditions with integral operators. The differencing processes are effectively simplified by means of the unit-step function. The nonlinear difference equations are solved by Newton method using an efficient block arrow-like matrix factorization technique. As an example of the application of this method, the systems of equations for combined gravity body force and forced convection in laminar film condensation can be solved for prescribed valu...
Rosnitskiy, P.; Yuldashev, P.; Khokhlova, V.
2015-10-01
An equivalent source model was proposed as a boundary condition to the nonlinear parabolic Khokhlov-Zabolotskaya (KZ) equation to simulate high intensity focused ultrasound (HIFU) fields generated by medical ultrasound transducers with the shape of a spherical shell. The boundary condition was set in the initial plane; the aperture, the focal distance, and the initial pressure of the source were chosen based on the best match of the axial pressure amplitude and phase distributions in the Rayleigh integral analytic solution for a spherical transducer and the linear parabolic approximation solution for the equivalent source. Analytic expressions for the equivalent source parameters were derived. It was shown that the proposed approach allowed us to transfer the boundary condition from the spherical surface to the plane and to achieve a very good match between the linear field solutions of the parabolic and full diffraction models even for highly focused sources with F-number less than unity. The proposed method can be further used to expand the capabilities of the KZ nonlinear parabolic equation for efficient modeling of HIFU fields generated by strongly focused sources.
Adomian Method for Solving Fuzzy Fredholm-Volterra Integral Equations
M. Barkhordari Ahmadi
2013-09-01
Full Text Available In this paper, Adomian method has been applied to approximate the solution of fuzzy volterra-fredholm integral equation. That, by using parametric form of fuzzy numbers, a fuzzy volterra-fredholm integral equation has been converted to a system of volterra-fredholm integral equation in crisp case. Finally, the method is explained with illustrative examples.
BOUNDARY INTEGRAL FORMULA OF ELASTIC PROBLEMS IN CIRCLE PLANE
DONG Zheng-zhu; LI Shun-cai; YU De-hao
2005-01-01
By bianalytic functions, the boundary integral formula of the stress function for the elastic problem in a circle plane is developed. But this integral formula includes a strongly singular integral and can not be directly calculated. After the stress function is expounded to Fourier series, making use of some formulas in generalized functions to the convolutions, the boundary integral formula which does not include strongly singular integral is derived further. Then the stress function can be got simply by the integration of the values of the stress function and its derivative on the boundary. Some examples are given. It shows that the boundary integral formula of the stress function for the elastic problem is convenient.
One-way spatial integration of hyperbolic equations
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.
Well-posed initial-boundary value problems for the Zakharov-Kuznetsov equation
Andrei V. Faminskii
2008-09-01
Full Text Available This paper deals with non-homogeneous initial-boundary value problems for the Zakharov-Kuznetsov equation, which is one of the variants of multidimensional generalizations of the Korteweg-de Vries equation. Results on local and global well-posedness are established in a scale of Sobolev-type spaces under natural assumptions on initial and boundary data.
REGULARITY THEORY FOR SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS WITH NEUMANN BOUNDARY CONDITIONS
无
2002-01-01
The objective of this paper is to consider the theory of regularity of systems of partial differential equations with Neumann boundary conditions. It complements previous works of the authors for the Dirichlet case. This type of problem is motivated by stochastic differential games. The Neumann case corresponds to stochastic differential equations with reflection on boundary of the domain.
A Cartesian Embedded Boundary Method for the Compressible Navier-Stokes Equations
Kupiainen, M; Sjogreen, B
2008-03-21
We here generalize the embedded boundary method that was developed for boundary discretizations of the wave equation in second order formulation in [6] and for the Euler equations of compressible fluid flow in [11], to the compressible Navier-Stokes equations. We describe the method and we implement it on a parallel computer. The implementation is tested for accuracy and correctness. The ability of the embedded boundary technique to resolve boundary layers is investigated by computing skin-friction profiles along the surfaces of the embedded objects. The accuracy is assessed by comparing the computed skin-friction profiles with those obtained by a body fitted discretization.
Ivanyshyn Yaman, Olha; Le Louër, Frédérique
2016-09-01
This paper deals with the material derivative analysis of the boundary integral operators arising from the scattering theory of time-harmonic electromagnetic waves and its application to inverse problems. We present new results using the Piola transform of the boundary parametrisation to transport the integral operators on a fixed reference boundary. The transported integral operators are infinitely differentiable with respect to the parametrisations and simplified expressions of the material derivatives are obtained. Using these results, we extend a nonlinear integral equations approach developed for solving acoustic inverse obstacle scattering problems to electromagnetism. The inverse problem is formulated as a pair of nonlinear and ill-posed integral equations for the unknown boundary representing the boundary condition and the measurements, for which the iteratively regularized Gauss-Newton method can be applied. The algorithm has the interesting feature that it avoids the numerous numerical solution of boundary value problems at each iteration step. Numerical experiments are presented in the special case of star-shaped obstacles.
Partial differential equations IX elliptic boundary value problems
Egorov, Yu; Shubin, M
1997-01-01
This EMS volume gives an overview of the modern theory of elliptic boundary value problems. The contribution by M.S. Agranovich is devoted to differential elliptic boundary problems, mainly in smooth bounded domains, and their spectral properties. This article continues his contribution to EMS 63. The contribution by A. Brenner and E. Shargorodsky concerns the theory of boundary value problems for elliptic pseudodifferential operators. Problems both with and without the transmission property, as well as parameter-dependent problems are considered. The article by B. Plamenevskij deals with general differential elliptic boundary value problems in domains with singularities.
Local existence of solution to free boundary value problem for compressible Navier-Stokes equations
Liu, Jian
2015-01-01
This paper is concerned with the free boundary value problem for multi-dimensional Navier-Stokes equations with density-dependent viscosity where the flow density vanishes continuously across the free boundary. A local (in time) existence of weak solution is established, in particular, the density is positive and the solution is regular away from the free boundary.
Boundary conditions for the Einstein-Christoffel formulation of Einstein's equations
Arnold, Douglas N.; Nicolae Tarfulea
2007-01-01
Specifying boundary conditions continues to be a challenge in numerical relativity in order to obtain a long time convergent numerical simulation of Einstein's equations in domains with artificial boundaries. In this paper, we address this problem for the Einstein--Christoffel (EC) symmetric hyperbolic formulation of Einstein's equations linearized around flat spacetime. First, we prescribe simple boundary conditions that make the problem well posed and preserve the constraints. Next, we indi...
A Kind of Discrete Non-Reflecting Boundary Conditions for Varieties of Wave Equations
Xiu-min Shao; Zhi-ling Lan
2002-01-01
In this paper, a new kind of discrete non-reflecting boundary conditions is developed. It can be used for a variety of wave equations such as the acoustic wave equation, the isotropic and anisotropic elastic wave equations and the equations for wave propagation in multi-phase media and so on. In this kind of boundary conditions, the composition of all artificial reflected waves, but not the individual reflected ones, is considered and eliminated. Thus, it has a uniform formula for different wave equations. The velocity CA of the composed reflected wave is determined in the way to make the reflection coefficients minimal, the value of which depends on equations. In this paper, the construction of the boundary conditions is illustrated and CA is found, numerical results are presented to illustrate the effectiveness of the boundary conditions.
Boundary singularities and characteristics of Hamilton–Jacobi equation
Melikyan, A.; Olsder, G.J.
2010-01-01
Boundary-value problems for first order PDEs are locally considered, when the classical sufficient condition for the solution existence does not hold, but a solution still exists, possibly defined on one or both sides of the boundary surface. We note three situations when such a surface (locally) ar
The 3D MHD free boundary equilibrium NEMEC code combines the fixed boundary VMEC code with the vacuum NESTOR code. In NESTOR a Neumann problem is solved for a single valued potential Φ by an integral equation technique satisfying the condition rvec BV· rvec n≡(rvec B0+∇Φ)· rvec n=0 on the plasma-vacuum boundary (rvec B0=field produced by external currents and the total toroidal plasma current It, rvec n=exterior normal to the boundary). The only quantity required for the free boundary equilibrium computation is the value of BV2/2 at the boundary. Here, an equivalent method for computing the vacuum field rvec BV is considered which is particularly suited for the case It≠0 and makes use of the fact that this field can be generated by the external current jext and by a surface current rvec j on the plasma vacuum boundary. The difficulty inherent to the Green's function method is the numerical integration of the singular Green's function. This is solved by regularizing the kernel: an analytically integrable function with the same singularity is subtracted and its integral is added to the integral equation
Numerical strategies to solve coupled, singular, nonlinear integral equations
Full text: In QFT, the Dyson-Schwinger equations constitute a set of nonlinear integral equations, whose solutions are the Green's functions of the theory. Implementing consistent pictures of confinement, these integral equations become highly infrared singular. The latter property especially slows down convergence in any iterative algorithm. Different numerical strategies to solve such integral equations are compared and benchmarked results for the efficiency of different algorithms are presented. (author)
Edwards, S.; Reuther, J.; Chattot, J. J.
The objective of this paper is to present a control theory approach for the design of airfoils in the presence of viscous compressible flows. A coupled system of the integral boundary layer and the Euler equations is solved to provide rapid flow simulations. An adjoint approach consistent with the complete coupled state equations is employed to obtain the sensitivities needed to drive a numerical optimization algorithm. Design to a target pressure distribution is demonstrated on an RAE 2822 airfoil at transonic speeds.
Solution of Moving Boundary Space-Time Fractional Burger’s Equation
E. A.-B. Abdel-Salam
2014-01-01
Full Text Available The fractional Riccati expansion method is used to solve fractional differential equations with variable coefficients. To illustrate the effectiveness of the method, the moving boundary space-time fractional Burger’s equation is studied. The obtained solutions include generalized trigonometric and hyperbolic function solutions. Among these solutions, some are found for the first time. The linear and periodic moving boundaries for the kink solution of the Burger’s equation are presented graphically and discussed.
Porosity of Free Boundaries in the Obstacle Problem for Quasilinear Elliptic Equations
Jun Zheng; Zhihua Zhang; Peihao Zhao
2013-08-01
In this paper, we establish growth rate of solutions near free boundaries in the identical zero obstacle problem for quasilinear elliptic equations. As a result, we obtain porosity of free boundaries, which is naturally an extension of the previous works by Karp et al. (J. Diff. Equ. 164 (2000) 110–117) for -Laplacian equations, and by Zheng and Zhang (J. Shaanxi Normal Univ. 40(2) (2012) 11–13, 18) for -Laplacian type equations.
Hyers-Ulam stability for second-order linear differential equations with boundary conditions
Pasc Gavruta
2011-06-01
Full Text Available We prove the Hyers-Ulam stability of linear differential equations of second-order with boundary conditions or with initial conditions. That is, if y is an approximate solution of the differential equation $y''+ eta (x y = 0$ with $y(a = y(b =0$, then there exists an exact solution of the differential equation, near y.
KdV equation under periodic boundary conditions and its perturbations
Huang, Guan; Kuksin, Sergei
2013-01-01
In this paper we discuss properties of the KdV equation under periodic boundary conditions, especially those which are important to study perturbations of the equation. Next we review what is known now about long-time behaviour of solutions for perturbed KdV equations.
The KdV equation under periodic boundary conditions and its perturbations
Guan, Huang; Kuksin, Sergei
2014-09-01
In this paper we discuss properties of the Korteweg-de Vries (KdV) equation under periodic boundary conditions, especially those which are important for studying perturbations of the equation. We then review what is known about the long-time behaviour of solutions for perturbed KdV equations.
Boundary Layer Equations and Lie Group Analysis of a Sisko Fluid
Gözde Sarı
2012-01-01
Full Text Available Boundary layer equations are derived for the Sisko fluid. Using Lie group theory, a symmetry analysis of the equations is performed. A partial differential system is transferred to an ordinary differential system via symmetries. Resulting equations are numerically solved. Effects of non-Newtonian parameters on the solutions are discussed.
Dimension reduction for periodic boundary value problems of functional differential equations
Sieber, Jan
2010-01-01
Periodic boundary-value problems for functional differential equations can be reduced to finite-dimensional algebraic systems of equations. The smoothness assumptions on the right-hand side follow those of the review by Hartung et al. (2006) and are set up such that the result can be applied to differential equations with state-dependent delays.
The adjoint equation method for constructing first integrals of difference equations
A new method for finding first integrals of discrete equations is presented. It can be used for discrete equations which do not possess a variational (Lagrangian or Hamiltonian) formulation. The method is based on a newly established identity which links symmetries of the underlying discrete equations, solutions of the discrete adjoint equations and first integrals. The method is applied to an invariant mapping and to discretizations of second order and third order ordinary differential equations. In examples the set of independent first integrals makes it possible to find the general solution of the discrete equations. (paper)
Integrability of two coupled Kadomtsev–Petviashvili equations
Abdul-Majid Wazwaz
2011-08-01
The integrability of two coupled KP equations is studied. The simpliﬁed Hereman form of Hirota’s bilinear method is used to examine the integrability of each coupled equation. Multiplesoliton solutions and multiple singular soliton solutions are formally derived for each coupled KdV equation.
BOUNDARY VALUE PROBLEM TO DYNAMIC EQUATION ON TIME SCALE
无
2011-01-01
In this paper we consider a nonlinear first-order boundary value problem on a time scale. The existence results of three positive solutions are obtained using fixed point theorems. Finally,examples are presented to illustrate the main results.
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
Toward a theory of integrable hyperbolic equations of third order
Examples are considered of integrable third order hyperbolic equations with two independent variables. In particular, an equation is found which admits as evolutionary symmetries the Krichever–Novikov equation and the modified Landau–Lifshitz system. The problem of the choice of dynamical variables for the hyperbolic equations is discussed. (paper)