In this paper, we suggest a method for solving Fredholm integral equation of the first kind based on wavelet basis. The continuous Legendre and Chebyshev wavelets of the first, second, third and fourth kind on [0,1] a...In this paper, we suggest a method for solving Fredholm integral equation of the first kind based on wavelet basis. The continuous Legendre and Chebyshev wavelets of the first, second, third and fourth kind on [0,1] are used and are utilized as a basis in Galerkin method to approximate the solution of integral equations. Then, in some examples the mentioned wavelets are compared with each other.展开更多
The regularized integrodifferential equation for the first kind of Fredholm, integral equation with a complex kernel is derived by generalizing the Tikhonov regularization method and the convergence of approximate reg...The regularized integrodifferential equation for the first kind of Fredholm, integral equation with a complex kernel is derived by generalizing the Tikhonov regularization method and the convergence of approximate regularized solutions is discussed. As an application of the method, an inverse problem in the two-dimensional wave-making problem of a flat plate is solved numerically, and a practical approach of choosing optimal regularization parameter is given.展开更多
In recent papers, Babolian & Delves [2] and Belward[3] described a Chebyshev series method for the solution of first kind integral equations. The expansion coefficients of the solution are determined as the soluti...In recent papers, Babolian & Delves [2] and Belward[3] described a Chebyshev series method for the solution of first kind integral equations. The expansion coefficients of the solution are determined as the solution of a mathematical programming problem.The method involves two regularization parameters, Cf and r, but values assigned to these parameters are heuristic in nature. Essah & Delves[7] described an algorithm for setting these parameters automatically, but it has some difficulties. In this paper we describe three iterative algorithms for computing these parameters for singular and non-singular first kind integral equations. We give also error estimates which are cheap to compute. Finally, we give a number of numerical examples showing that these algorithms work well in practice.展开更多
The dual integral equations of vertical forced vibration of elastic plate on an elastic half space subject to harmonic uniform distribution loading are established according to the mixed boundary-value condition. By a...The dual integral equations of vertical forced vibration of elastic plate on an elastic half space subject to harmonic uniform distribution loading are established according to the mixed boundary-value condition. By applying Abel transformation the dual integral equations are reduced to Fredholm integral equation of the second kind which is solved numerically.展开更多
In this paper, we will use the successive approximation method for solving Fredholm integral equation of the second kind using Maple18. By means of this method, an algorithm is successfully established for solving the...In this paper, we will use the successive approximation method for solving Fredholm integral equation of the second kind using Maple18. By means of this method, an algorithm is successfully established for solving the non-linear Fredholm integral equation of the second kind. Finally, several examples are presented to illustrate the application of the algorithm and results appear that this method is very effective and convenient to solve these equations.展开更多
This paper discusses the numerical solutions for the nonlinear Fredholm integral equations of thesecond kind. On the basis of the Galerkin method, the author establishes a Galerkin algorithm, a Wavelet-Galerkinalgorit...This paper discusses the numerical solutions for the nonlinear Fredholm integral equations of thesecond kind. On the basis of the Galerkin method, the author establishes a Galerkin algorithm, a Wavelet-Galerkinalgorithm and their corresponding iterated correction schemes for this kind of equations.The superconvergemceof the numerical solutions of these two algorithms is proved. Not only are the results concerning the Hammersteinintegral equations generalized to nonlinear Fredilolm equations of the second kind, but also more precise resultsare obtained by tising the wavelet method.展开更多
This paper solves the two dimensional linear Fredholm integral equations of the second kind by combining the meshless barycentric Lagrange interpolation functions and the Gauss-Legendre quadrature formula. Inspired by...This paper solves the two dimensional linear Fredholm integral equations of the second kind by combining the meshless barycentric Lagrange interpolation functions and the Gauss-Legendre quadrature formula. Inspired by this thought, we convert the equations into the associated algebraic equations. The results of the numerical examples are given to illustrate that the approximated method is feasible and efficient.展开更多
This work mainly focuses on the numerical simulation of the Fredholm integral equation of the second kind. Applying the idea of Gauss-Lobatto quadrature formula, a numerical method is developed. For the integral item,...This work mainly focuses on the numerical simulation of the Fredholm integral equation of the second kind. Applying the idea of Gauss-Lobatto quadrature formula, a numerical method is developed. For the integral item, we give an approximation with high precision. The existence condition of the solution for the Fredholm equation is given. Furthermore, the error analyses are presented. Finally, the numerical examples verify the theoretical analysis, and show the efficiency of the algorithm we discussed.展开更多
1. Introduction It is known that the following Cauchy problem for a parabolic partial differential equation (where the values at the right boundary, u.(1, t)=v(t) are unknown and sought for) is ill-posed: the solution...1. Introduction It is known that the following Cauchy problem for a parabolic partial differential equation (where the values at the right boundary, u.(1, t)=v(t) are unknown and sought for) is ill-posed: the solution (v) does not depend continuously on the data (g). In order to treat the ill-posedness and develop the numerical method, one reformulates the problem as a Volterra integral equation of the first kind wish a convolution type kernel (see Sneddon [1], Carslaw and Jaeger [2])展开更多
Integral equations theoretical parts and applications have been studied and investigated in previous works. In this work, results on investigations of the uniqueness of the Fredholm-Stiltjes linear integral equations ...Integral equations theoretical parts and applications have been studied and investigated in previous works. In this work, results on investigations of the uniqueness of the Fredholm-Stiltjes linear integral equations solutions of the third kind were considered. Volterra integral equations of the first and third kind with smooth kernels were studied, and proof of the existence of a multiparameter family of solutions is described. Additionally, linear Fredholm integral equations of the first kind were investigated, for which Lavrent’ev regularizing operators were constructed.展开更多
We present a fast algorithm based on polynomial interpolation to approximate matrices arising from the discretization of second-kind integral equations where the kernel function is either smooth, non-oscillatory and p...We present a fast algorithm based on polynomial interpolation to approximate matrices arising from the discretization of second-kind integral equations where the kernel function is either smooth, non-oscillatory and possessing only a finite number of singularities or a product of such function with a highly oscillatory coefficient function. Contrast to wavelet-like approximations, ourapproximation matrix is not sparse. However, the approximation can be construced in O(n) operations and requires O(n) storage, where n is the number of quadrature points used in the discretization. Moreover, the matrix-vector multiplication cost is of order O(nlogn). Thus our scheme is well suitable for conjugate gradient type methods. Our numerical results indicate that the algorithm is very accurate and stable for high degree polynomial interpolation.展开更多
The article is considering the third kind of nonlinear Volterra-Stieltjes integral equations with the solution by Lavrentyev regularizing operator. A uniqueness theorem was proved, and a regularization parameter was c...The article is considering the third kind of nonlinear Volterra-Stieltjes integral equations with the solution by Lavrentyev regularizing operator. A uniqueness theorem was proved, and a regularization parameter was chosen. This can be used in further development of the theory of the integral equations in non-standard problems, classes in the numerical solution of third kind Volterra-Stieltjes integral equations, and when solving specific problems that lead to equations of the third kind.展开更多
In this article, we use scaling function interpolation method to solve linear Fredholm integral equations, and we prove a convergence theorem for the solution of Fredholm integral equations. We present two examples wh...In this article, we use scaling function interpolation method to solve linear Fredholm integral equations, and we prove a convergence theorem for the solution of Fredholm integral equations. We present two examples which have better results than others.展开更多
In this paper, we use the Sinc Function to solve the Fredholme-Volterra Integral Equations. By using collocation method we estimate a solution for Fredholme-Volterra Integral Equations. Finally convergence of this met...In this paper, we use the Sinc Function to solve the Fredholme-Volterra Integral Equations. By using collocation method we estimate a solution for Fredholme-Volterra Integral Equations. Finally convergence of this method will be discussed and efficiency of this method is shown by some examples. Numerical examples show that the approximate solutions have a good degree of accuracy.展开更多
In this paper, Adomian decomposition method (ADM) is used to solve the Volterra-Fredholm integral equation. A number of examples have been presented to explain the numerical results, which is the comparison between th...In this paper, Adomian decomposition method (ADM) is used to solve the Volterra-Fredholm integral equation. A number of examples have been presented to explain the numerical results, which is the comparison between the exact solution and the numerical solution, and it is found through the tables and the amount of error between the exact solution and the numerical solution, it is very small and almost non-existent and is also illustrated through the graph how the exact solution of completely applies to the numerical solution This proves the accuracy of the method, which is the Adomian decomposition method (ADM) for solving the Volterra Fredholm integral equation using Maple 18. And that this method is characterized by ease, speed and great accuracy in obtaining numerical results.展开更多
The Rayleigh-Ritz and the inverse iteration methods are used in order to compute the eigenvalues of 3D Fredholm-Stieltjes integral equations, i.e. 3D Fredholm equations with respect to suitable Stieltjes-type measures...The Rayleigh-Ritz and the inverse iteration methods are used in order to compute the eigenvalues of 3D Fredholm-Stieltjes integral equations, i.e. 3D Fredholm equations with respect to suitable Stieltjes-type measures. Some applications are shown, relevant to the problem of computing the eigenvalues of a body charged by a finite number of masses concentrated on points, curves or surfaces lying in.展开更多
In recent papers the solution of nonlinear Fredholm integral equations was discussed using Adomian decomposition method (ADM). For case in which the integrals are analytically impossible, ADM can not be applied. In th...In recent papers the solution of nonlinear Fredholm integral equations was discussed using Adomian decomposition method (ADM). For case in which the integrals are analytically impossible, ADM can not be applied. In this paper a discretized version of the ADM is introduced and the proposed version will be called discrete Adomian decomposition method (DADM). An accelerated formula of Adomian polynomials is used in calculations. Based on this formula, a new convergence approach of ADM is introduced. Convergence approach is reliable enough to obtain an explicit formula for the maximum absolute truncated error of the Adomian’s series solution. Also, we prove that the solution of nonlinear Fredholm integral equation by DADM converges to ADM solution. Finally, some numerical examples were introduced.展开更多
We consider a kind of scattering problem by a crack F that is buried in a bounded domain D, and we put a point source inside the domain D. This leads to a mixed boundary value problem to the Helmholtz equation in the ...We consider a kind of scattering problem by a crack F that is buried in a bounded domain D, and we put a point source inside the domain D. This leads to a mixed boundary value problem to the Helmholtz equation in the domain D with a crack Г. Both sides of the crack F are given Dirichlet-impedance boundary conditions, and different boundary condition （Dirichlet, Neumann or Impedance boundary condition） is set on the boundary of D. Applying potential theory, the problem can be reformulated as a system of boundary integral equations. We establish the existence and uniqueness of the solution to the system by using the Fredholm theory.展开更多
文摘In this paper, we suggest a method for solving Fredholm integral equation of the first kind based on wavelet basis. The continuous Legendre and Chebyshev wavelets of the first, second, third and fourth kind on [0,1] are used and are utilized as a basis in Galerkin method to approximate the solution of integral equations. Then, in some examples the mentioned wavelets are compared with each other.
文摘The regularized integrodifferential equation for the first kind of Fredholm, integral equation with a complex kernel is derived by generalizing the Tikhonov regularization method and the convergence of approximate regularized solutions is discussed. As an application of the method, an inverse problem in the two-dimensional wave-making problem of a flat plate is solved numerically, and a practical approach of choosing optimal regularization parameter is given.
文摘In recent papers, Babolian & Delves [2] and Belward[3] described a Chebyshev series method for the solution of first kind integral equations. The expansion coefficients of the solution are determined as the solution of a mathematical programming problem.The method involves two regularization parameters, Cf and r, but values assigned to these parameters are heuristic in nature. Essah & Delves[7] described an algorithm for setting these parameters automatically, but it has some difficulties. In this paper we describe three iterative algorithms for computing these parameters for singular and non-singular first kind integral equations. We give also error estimates which are cheap to compute. Finally, we give a number of numerical examples showing that these algorithms work well in practice.
文摘The dual integral equations of vertical forced vibration of elastic plate on an elastic half space subject to harmonic uniform distribution loading are established according to the mixed boundary-value condition. By applying Abel transformation the dual integral equations are reduced to Fredholm integral equation of the second kind which is solved numerically.
文摘In this paper, we will use the successive approximation method for solving Fredholm integral equation of the second kind using Maple18. By means of this method, an algorithm is successfully established for solving the non-linear Fredholm integral equation of the second kind. Finally, several examples are presented to illustrate the application of the algorithm and results appear that this method is very effective and convenient to solve these equations.
文摘This paper discusses the numerical solutions for the nonlinear Fredholm integral equations of thesecond kind. On the basis of the Galerkin method, the author establishes a Galerkin algorithm, a Wavelet-Galerkinalgorithm and their corresponding iterated correction schemes for this kind of equations.The superconvergemceof the numerical solutions of these two algorithms is proved. Not only are the results concerning the Hammersteinintegral equations generalized to nonlinear Fredilolm equations of the second kind, but also more precise resultsare obtained by tising the wavelet method.
文摘This paper solves the two dimensional linear Fredholm integral equations of the second kind by combining the meshless barycentric Lagrange interpolation functions and the Gauss-Legendre quadrature formula. Inspired by this thought, we convert the equations into the associated algebraic equations. The results of the numerical examples are given to illustrate that the approximated method is feasible and efficient.
文摘This work mainly focuses on the numerical simulation of the Fredholm integral equation of the second kind. Applying the idea of Gauss-Lobatto quadrature formula, a numerical method is developed. For the integral item, we give an approximation with high precision. The existence condition of the solution for the Fredholm equation is given. Furthermore, the error analyses are presented. Finally, the numerical examples verify the theoretical analysis, and show the efficiency of the algorithm we discussed.
文摘1. Introduction It is known that the following Cauchy problem for a parabolic partial differential equation (where the values at the right boundary, u.(1, t)=v(t) are unknown and sought for) is ill-posed: the solution (v) does not depend continuously on the data (g). In order to treat the ill-posedness and develop the numerical method, one reformulates the problem as a Volterra integral equation of the first kind wish a convolution type kernel (see Sneddon [1], Carslaw and Jaeger [2])
文摘Integral equations theoretical parts and applications have been studied and investigated in previous works. In this work, results on investigations of the uniqueness of the Fredholm-Stiltjes linear integral equations solutions of the third kind were considered. Volterra integral equations of the first and third kind with smooth kernels were studied, and proof of the existence of a multiparameter family of solutions is described. Additionally, linear Fredholm integral equations of the first kind were investigated, for which Lavrent’ev regularizing operators were constructed.
基金Research supported in part by Hong Kong Research Grant Council grats no.CUHK178/83E
文摘We present a fast algorithm based on polynomial interpolation to approximate matrices arising from the discretization of second-kind integral equations where the kernel function is either smooth, non-oscillatory and possessing only a finite number of singularities or a product of such function with a highly oscillatory coefficient function. Contrast to wavelet-like approximations, ourapproximation matrix is not sparse. However, the approximation can be construced in O(n) operations and requires O(n) storage, where n is the number of quadrature points used in the discretization. Moreover, the matrix-vector multiplication cost is of order O(nlogn). Thus our scheme is well suitable for conjugate gradient type methods. Our numerical results indicate that the algorithm is very accurate and stable for high degree polynomial interpolation.
文摘The article is considering the third kind of nonlinear Volterra-Stieltjes integral equations with the solution by Lavrentyev regularizing operator. A uniqueness theorem was proved, and a regularization parameter was chosen. This can be used in further development of the theory of the integral equations in non-standard problems, classes in the numerical solution of third kind Volterra-Stieltjes integral equations, and when solving specific problems that lead to equations of the third kind.
文摘In this article, we use scaling function interpolation method to solve linear Fredholm integral equations, and we prove a convergence theorem for the solution of Fredholm integral equations. We present two examples which have better results than others.
文摘In this paper, we use the Sinc Function to solve the Fredholme-Volterra Integral Equations. By using collocation method we estimate a solution for Fredholme-Volterra Integral Equations. Finally convergence of this method will be discussed and efficiency of this method is shown by some examples. Numerical examples show that the approximate solutions have a good degree of accuracy.
文摘In this paper, Adomian decomposition method (ADM) is used to solve the Volterra-Fredholm integral equation. A number of examples have been presented to explain the numerical results, which is the comparison between the exact solution and the numerical solution, and it is found through the tables and the amount of error between the exact solution and the numerical solution, it is very small and almost non-existent and is also illustrated through the graph how the exact solution of completely applies to the numerical solution This proves the accuracy of the method, which is the Adomian decomposition method (ADM) for solving the Volterra Fredholm integral equation using Maple 18. And that this method is characterized by ease, speed and great accuracy in obtaining numerical results.
文摘The Rayleigh-Ritz and the inverse iteration methods are used in order to compute the eigenvalues of 3D Fredholm-Stieltjes integral equations, i.e. 3D Fredholm equations with respect to suitable Stieltjes-type measures. Some applications are shown, relevant to the problem of computing the eigenvalues of a body charged by a finite number of masses concentrated on points, curves or surfaces lying in.
文摘In recent papers the solution of nonlinear Fredholm integral equations was discussed using Adomian decomposition method (ADM). For case in which the integrals are analytically impossible, ADM can not be applied. In this paper a discretized version of the ADM is introduced and the proposed version will be called discrete Adomian decomposition method (DADM). An accelerated formula of Adomian polynomials is used in calculations. Based on this formula, a new convergence approach of ADM is introduced. Convergence approach is reliable enough to obtain an explicit formula for the maximum absolute truncated error of the Adomian’s series solution. Also, we prove that the solution of nonlinear Fredholm integral equation by DADM converges to ADM solution. Finally, some numerical examples were introduced.
基金supported by the grant from the National Natural Science Foundation of China(11301405)supported by the grants from the National Natural Science Foundation of China(11171127 and 10871080)
文摘We consider a kind of scattering problem by a crack F that is buried in a bounded domain D, and we put a point source inside the domain D. This leads to a mixed boundary value problem to the Helmholtz equation in the domain D with a crack Г. Both sides of the crack F are given Dirichlet-impedance boundary conditions, and different boundary condition （Dirichlet, Neumann or Impedance boundary condition） is set on the boundary of D. Applying potential theory, the problem can be reformulated as a system of boundary integral equations. We establish the existence and uniqueness of the solution to the system by using the Fredholm theory.
