Discrete Galerkin method for Fredholm integro-differential equations with weakly singular kernels

Approximations to a solution and its derivatives of a boundary value problem of an nth order linear Fredholm integro-differential equation with weakly singular or other nonsmooth kernels are determined. These approximations are piecewise polynomial functions on special graded grids. For their finding a discrete Galerkin method and an integral equation reformulation of the boundary value problem are used. Optimal global convergence estimates are derived and an improvement of the convergence rate of the method for a special choice of parameters is obtained. To illustrate the theoretical results a collection of numerical results of a test problem is presented.     

Thin plate spline Galerkin scheme for numerically solving nonlinear weakly singular Fredholm integral equations

The present work proposes a numerical method to obtain an approximate solution of non-linear weakly singular Fredholm integral equations. The discrete Galerkin method in addition to thin-plate splines established on scattered points is utilized to estimate the solution of these integral equations. The thin-plate splines can be regarded as a type of free shape parameter radial basis functions which create an efficient and stable technique to approximate a function. The discrete Galerkin method for the approximate solution of integral equations results from the numerical integration of all integrals in the method. We utilize a special accurate quadrature formula via the non-uniform composite Gauss-Legendre integration rule and employ it to compute the singular integrals appeared in the scheme. Since the approach does not need any background meshes, it can be identified as a meshless method. Error analysis is also given for the method. Illustrative examples are shown clearly the reliability and efficiency of the new scheme and confirm the theoretical error estimates.     

Numerical solution for the weakly singular Fredholm integro-differential equations using Legendre multiwavelets

An effective method based upon Legendre multiwavelets is proposed for the solution of Fredholm weakly singular integro-differential equations. The properties of Legendre multiwavelets are first given and their operational matrices of integral are constructed. These wavelets are utilized to reduce the solution of the given integro-differential equation to the solution of a sparse linear system of algebraic equations. In order to save memory requirement and computational time, a threshold procedure is applied to obtain the solution to this system of algebraic equations. Through numerical examples, performance of the present method is investigated concerning the convergence and the sparseness of the resulted matrix equation.     

Iterated fast multiscale Galerkin methods for Fredholm integral equations of second kind with weakly singular kernels

We propose iterated fast multiscale Galerkin methods for the second kind Fredholm integral equations with mildly weakly singular kernel by combining the advantages of fast methods and iteration post-processing methods. To study the super-convergence of these methods, we develop a theoretical framework for iterated fast multiscale schemes, and apply the scheme to integral equations with weakly singular kernels. We show theoretically that even the computational complexity is almost optimal, our schemes improve the accuracy of numerical solutions greatly, and exhibit the global super-convergence. Numerical examples are presented to illustrate the theoretical results and the efficiency of the methods.     

Using Sinc-collocation method for solving weakly singular Fredholm integral equations of the first kind

In this paper, Sinc-collocation method is used to approximate the solution of weakly singular nonlinear Fredholm integral equations of the first kind. Some of the important advantages of this method are rate of convergence of an approximate solution and simplicity for performing even in the presence of singularities. The convergence analysis of the proposed method is proved by preparing the theorems which show the errors decay exponentially and guarantee the applicability of that. Finally, several numerical examples are considered to show the capabilities, validity, and accuracy of the numerical scheme.     

A Jacobi-collocation method for solving second kind Fredholm integral equations with weakly singular kernels

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.     

Sinc-collocation methods for weakly singular Fredholm integral equations of the second kind

In this paper we propose new numerical methods for linear Fredholm integral equations of the second kind with weakly singular kernels. The methods are developed by means of the Sinc approximation with smoothing transformations, which is an effective technique against the singularities of the equations. Numerical examples show that the methods achieve exponential convergence, and in this sense the methods improve conventional results where only polynomial convergence have been reported so far.     

Legendre spectral Galerkin method for second-kind Volterra integral equations

The Legendre spectral Galerkin method for the Volterra integral equations of the second kind is proposed in this paper. We provide a rigorous error analysis for the proposed method, which indicates that the numerical errors (in the L ₂ norm) will decay exponentially provided that the kernel function and the source function are sufficiently smooth. Numerical examples are given to illustrate the theoretical results.      

A note on the convergence of product integration and Galerkin method for weakly singular integral equations

We consider weakly singular integral equations of Fredholm-type whose kernels satisfy certain algebraic estimates with their derivatives. In particular, we establish optimal convergence order estimates for product integration and Galerkin method applied on suitable grading mesh for the solution of such equations. Some superconvergence results are also derived.     

A discrete collocation method for Fredholm integro-differential equations with weakly singular kernels

A fully discrete version of a piecewise polynomial collocation method is constructed to solve initial or boundary value problems of linear Fredholm integro-differential equations with weakly singular kernels. Using an integral equation reformulation and special graded grids, optimal global convergence estimates are derived. For special values of parameters an improvement of the convergence rate of elaborated numerical schemes is established. Some of our theoretical results are illustrated by numerical experiments.     

A fast multiscale Kantorovich method for weakly singular integral equations

In this paper, we use the idea of Kantorovich regularization to develop the fast multiscale Kantorovich method and the fast iterated multiscale Kantorovich method. For some kinds of weakly singular integral equations with nonsmooth inhomogeneous terms, we show that our two proposed methods can still obtain the optimal order of convergence and superconvergence order, respectively. Numerical examples are given to demonstrate the efficiency of the methods.     

Wavelet Galerkin method for eigenvalue problem of a compact integral operator

We consider the approximation of eigenfunctions of a compact integral operator with a smooth kernel by the Galerkin method using wavelet bases. By truncating the Galerkin operator, we obtain a sparse representation of a matrix eigenvalue problem. We prove that the error bounds for the eigenvalues and for the distance between the spectral subspaces are of the orders O(nμ^-2nr) and O(μ^-nr), respectively, where μ⁻ⁿ denotes the norm of the partition and r denotes the order of the wavelet basis functions. By iterating the eigenvectors, we show that the error bounds for the eigenvectors are of the order O(nμ^-2nr). We illustrate our results with numerical results.     

Solution of nonlinear weakly singular Volterra integral equations using the fractional‐order Legendre functions and pseudospectral method

In this article, our main goal is to render an idea to convert a nonlinear weakly singular Volterra integral equation to a non‐singular one by new fractional‐order Legendre functions. The fractional‐order Legendre functions are generated by change of variable on well‐known shifted Legendre polynomials. We consider a general form of singular Volterra integral equation of the second kind. Then the fractional Legendre–Gauss–Lobatto quadratures formula eliminates the singularity of the kernel of the integral equation. Finally, the Legendre pseudospectral method reduces the solution of this problem to the solution of a system of algebraic equations. This method also can be utilized on fractional differential equations as well. The comparison of results of the presented method and other numerical solutions shows the efficiency and accuracy of this method. Also, the obtained maximum error between the results and exact solutions shows that using the present method leads to accurate results and fast convergence for solving nonlinear weakly singular Volterra integral equations. Copyright © 2015 John Wiley & Sons, Ltd.     

The approximate solution of a class of Fredholm integral equations with a weakly singular kernel

A method for finding the numerical solution of a weakly singular Fredholm integral equation of the second kind is presented. The Taylor series is used to remove singularity and Legendre polynomials are used as a basis. Furthermore, the Legendre function of the second kind is used to remove singularity in the Cauchy type integral equation. The integrals that appear in this method are computed in terms of gamma and beta functions and some of these integrals are computed in the Cauchy principal value sense without using numerical quadratures. Four examples are given to show the accuracy of the method.     

Solution of certain weakly singular integral equations

^** Email: alok{at}math.iisc.ernet.in Direct function theoretic methods are developed to handle twoweakly singular integral equations with their kernels havinglogarithmic singularity. The present methods avoid the occurrenceof higher-order (or strong) singularities, like the Cauchy typesingularity in the representation of the solutions of such integralequations.     

Numerical methods for Fredholm integral equations with singular right-hand sides

Fredholm integral equations with the right-hand side having singularities at the endpoints are considered. The singularities are moved into the kernel that is subsequently regularized by a suitable one-to-one map. The Nyström method is applied to the regularized equation. The convergence, stability and well conditioning of the method is proved in spaces of weighted continuous functions. The special case of the weakly singular and symmetric kernel is also investigated. Several numerical tests are included.     

An SQP method for optimal control of weakly singular Hammerstein integral equations

We investigate local convergence of an SQP method for nonlinear optimal control of weakly singular Hammerstein integral equations. Sufficient conditions for local quadratic convergence of the method are discussed.     

Information complexity of weakly singular integral equations

We establish the exact power order of information complexity for integral equations whose kernels have power singularities and free terms belong to the corresponding Hölder space.Translated from Ukrainskii Matematicheskii Zhurnal, Vol.46, No. 11, pp. 1527–1533, November, 1994.This work was supported by the Foundation for Fundamental Researches of the Ukrainian State Committee on Science and Technology.