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

Abstract

MSC:Keywords: Cauchy kernel Weakly singular Taylor series Galerkin method Legendre functions a b s t r a c t 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.

On integral equations with Weakly Singular kernel by using Taylor series and Legendre polynomials

“…It is assumed that the function ϕ(y) has the Taylor series expansion. The main purpose of this paper is to present a numerical method for solving integral Equation (1), which is generalization of the paper [9] for k(x, y) = .…”

“…In [9] Babolian et al solved (1) with k(x, y) = , they removed singularity with Taylor expansion of ϕ(y) at point x, and then used Legendre functions as basis and computed all de nite integrals involved without numerical quadratures. In [10] Lakestani et al utilized Legendre multiwavelets as basis to reduce the solution of Fredholm integro-di erential equation to the solution of sparse linear system of algebraic equations.…”

Some Bounded Linear Integral Operators and Linear Fredholm Integral Equations in the SpacesHα,δ,γ((Özdemır1, Akhmedov2, Temizer3 2013Abstract and Applied Analysis0000View full textAdd to dashboardCite

“…see [5]. One of the weakly singular integral and integrodifferential equations with this kernel was given in [6][7][8].…”

A novel computing multi-parametric homotopy approach for system of linear and nonlinear Fredholm integral equations

“…The mathematical description of many problems of engineering interest contains Fredholm integral equations. A variety of approximate analytical and numerical methods are introduced to deal with Fredholm integral equations, such as Adomian decomposition method [1], Fourier functions [2], homotopy perturbation method [3], discrete Adomian decomposition method [4], triangular functions (TF) method [5], Sinc-collocation method [6], harmonic wavelet method [7], Taylor series and Legendre function method [8], Half-Sweep arithmetic mean method [9], quasi-interpolation method [10], etc. The literature on the topic is quite broad and hence cannot be described here in detail.…”