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
We introduce multilevel augmentation methods for solving operator equations based on direct sum decompositions of the range space of the operator and the solution space of the operator equation and a matrix splitting ...We introduce multilevel augmentation methods for solving operator equations based on direct sum decompositions of the range space of the operator and the solution space of the operator equation and a matrix splitting scheme. We establish a general setting for the analysis of these methods, showing that the methods yield approximate solutions of the same convergence order as the best approximation from the subspace. These augmentation methods allow us to develop fast, accurate and stable nonconventional numerical algorithms for solving operator equations. In particular, for second kind equations, special splitting techniques are proposed to develop such algorithms. These algorithms are then applied to solve the linear systems resulting from matrix compression schemes using wavelet-like functions for solving Fredholm integral equations of the second kind. For this special case, a complete analysis for computational complexity and convergence order is presented. Numerical examples are included to demonstrate the efficiency and accuracy of the methods. In these examples we use the proposed augmentation method to solve large scale linear systems resulting from the recently developed wavelet Galerkin methods and fast collocation methods applied to integral equations of the secondkind. Our numerical results confirm that this augmentation method is particularly efficient for solving large scale linear systems induced from wavelet compression schemes.展开更多
We consider solving integral equations of the second kind defined on the half-line [0, infinity) by the preconditioned conjugate gradient method. Convergence is known to be slow due to the non-compactness of the assoc...We consider solving integral equations of the second kind defined on the half-line [0, infinity) by the preconditioned conjugate gradient method. Convergence is known to be slow due to the non-compactness of the associated integral operator. In this paper, we construct two different circulant integral operators to be used as preconditioners for the method to speed up its convergence rate. We prove that if the given integral operator is close to a convolution-type integral operator, then the preconditioned systems will have spectrum clustered around 1 and hence the preconditioned conjugate gradient method will converge superlinearly. Numerical examples are given to illustrate the fast convergence.展开更多
In this work,we propose a Jacobi-collocation method to solve the second kind linear Fredholm integral equations with weakly singular kernels.Particularly,we consider the case when the underlying solutions are sufficie...In this work,we propose a Jacobi-collocation method to solve the second kind linear Fredholm integral equations with weakly singular kernels.Particularly,we consider the case when the underlying solutions are sufficiently smooth.In this case,the proposed method leads to a fully discrete linear system.We show that the fully discrete integral operator is stable in both infinite and weighted square norms.Furthermore,we establish that the approximate solution arrives at an optimal convergence order under the two norms.Finally,we give some numerical examples,which confirm the theoretical prediction of the exponential rate of convergence.展开更多
While the numerical solution of one-dimensional Volterra integral equations of the second kind with regular kernels is well understood, there exist no systematic studies of asymptotic error expansion for the approxima...While the numerical solution of one-dimensional Volterra integral equations of the second kind with regular kernels is well understood, there exist no systematic studies of asymptotic error expansion for the approximate solution. In this paper,we analyse the Nystrom solution of one-dimensional nonlinear Volterra integral equation of the second kind and show that approkimate solution admits an asymptotic error expansion in even powers of the step-size h, beginning with a term in h2. So that the Richardson's extrapolation can be done. This will increase the accuracy of numerical solution greatly.展开更多
The main purpose of this work is to provide a novel numerical approach for the Volterra integral equations based on a spectral approach. A Legendre-collocation method is proposed to solve the Volterra integral equatio...The main purpose of this work is to provide a novel numerical approach for the Volterra integral equations based on a spectral approach. A Legendre-collocation method is proposed to solve the Volterra integral equations of the second kind. We provide a rigorous error analysis for the proposed method, which indicates that the numerical errors decay exponentially provided that the kernel function and the source function are sufficiently smooth. Numerical results confirm the theoretical prediction of the exponential rate of convergence. The result in this work seems to be the first successful spectral approach （with theoretical justification） for the Volterra type equations.展开更多
We propose two error control techniques for numerical integrations in fast multiscale collocation methods for solving Fredholm integral equations of the second kind with weakly singular kernels. Both techniques utiliz...We propose two error control techniques for numerical integrations in fast multiscale collocation methods for solving Fredholm integral equations of the second kind with weakly singular kernels. Both techniques utilize quadratures for singular integrals using graded points. One has a polynomial order of accuracy if the integrand has a polynomial order of smoothness except at the singular point and the other has exponential order of accuracy if the integrand has an infinite order of smoothness except at the singular point. We estimate the order of convergence and computational complexity of the corresponding approximate solutions of the equation. We prove that the second technique preserves the order of convergence and computational complexity of the original collocation method. Numerical experiments are presented to illustrate the theoretical estimates.展开更多
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.
基金Supported in part by the Natural Science Foundation of China under grants 10371137and 10201034Foundation of Doctoral Program of National Higher Education of China under under grant 20030558008Guangdong Provincial Natural Science Foundation of China u
文摘We introduce multilevel augmentation methods for solving operator equations based on direct sum decompositions of the range space of the operator and the solution space of the operator equation and a matrix splitting scheme. We establish a general setting for the analysis of these methods, showing that the methods yield approximate solutions of the same convergence order as the best approximation from the subspace. These augmentation methods allow us to develop fast, accurate and stable nonconventional numerical algorithms for solving operator equations. In particular, for second kind equations, special splitting techniques are proposed to develop such algorithms. These algorithms are then applied to solve the linear systems resulting from matrix compression schemes using wavelet-like functions for solving Fredholm integral equations of the second kind. For this special case, a complete analysis for computational complexity and convergence order is presented. Numerical examples are included to demonstrate the efficiency and accuracy of the methods. In these examples we use the proposed augmentation method to solve large scale linear systems resulting from the recently developed wavelet Galerkin methods and fast collocation methods applied to integral equations of the secondkind. Our numerical results confirm that this augmentation method is particularly efficient for solving large scale linear systems induced from wavelet compression schemes.
文摘We consider solving integral equations of the second kind defined on the half-line [0, infinity) by the preconditioned conjugate gradient method. Convergence is known to be slow due to the non-compactness of the associated integral operator. In this paper, we construct two different circulant integral operators to be used as preconditioners for the method to speed up its convergence rate. We prove that if the given integral operator is close to a convolution-type integral operator, then the preconditioned systems will have spectrum clustered around 1 and hence the preconditioned conjugate gradient method will converge superlinearly. Numerical examples are given to illustrate the fast convergence.
基金supported by National Natural Science Foundation of China(Grant No.10901093)National Science Foundation of Shandong Province(Grant No.ZR2013AM006)
文摘In this work,we propose a Jacobi-collocation method to solve the second kind linear Fredholm integral equations with weakly singular kernels.Particularly,we consider the case when the underlying solutions are sufficiently smooth.In this case,the proposed method leads to a fully discrete linear system.We show that the fully discrete integral operator is stable in both infinite and weighted square norms.Furthermore,we establish that the approximate solution arrives at an optimal convergence order under the two norms.Finally,we give some numerical examples,which confirm the theoretical prediction of the exponential rate of convergence.
文摘While the numerical solution of one-dimensional Volterra integral equations of the second kind with regular kernels is well understood, there exist no systematic studies of asymptotic error expansion for the approximate solution. In this paper,we analyse the Nystrom solution of one-dimensional nonlinear Volterra integral equation of the second kind and show that approkimate solution admits an asymptotic error expansion in even powers of the step-size h, beginning with a term in h2. So that the Richardson's extrapolation can be done. This will increase the accuracy of numerical solution greatly.
基金supported by CERG Grants of Hong Kong Research Grant CouncilFRG grants of Hong Kong Baptist University
文摘The main purpose of this work is to provide a novel numerical approach for the Volterra integral equations based on a spectral approach. A Legendre-collocation method is proposed to solve the Volterra integral equations of the second kind. We provide a rigorous error analysis for the proposed method, which indicates that the numerical errors decay exponentially provided that the kernel function and the source function are sufficiently smooth. Numerical results confirm the theoretical prediction of the exponential rate of convergence. The result in this work seems to be the first successful spectral approach （with theoretical justification） for the Volterra type equations.
基金The NNSF (10371137 and 10201034) of Chinathe Foundation (20030558008) of Doctoral Program of National Higher Education, Guangdong Provincial Natural Science Foundation (1011170) of China and the Advanced Research Foundation of Zhongshan UniversityThe US National Science Foundation (9973427 and 0312113)NSF (10371122) of China and the Chinese Academy of Sciences under the program of "Hundred Distinguished Young Chinese Scientists."
文摘We propose two error control techniques for numerical integrations in fast multiscale collocation methods for solving Fredholm integral equations of the second kind with weakly singular kernels. Both techniques utilize quadratures for singular integrals using graded points. One has a polynomial order of accuracy if the integrand has a polynomial order of smoothness except at the singular point and the other has exponential order of accuracy if the integrand has an infinite order of smoothness except at the singular point. We estimate the order of convergence and computational complexity of the corresponding approximate solutions of the equation. We prove that the second technique preserves the order of convergence and computational complexity of the original collocation method. Numerical experiments are presented to illustrate the theoretical estimates.
