A Wavelet-Based Approximation Scheme Involving Legendre Polynomial for Fredholm Integral Equations with Logarithmic Singular Kernel
Sudam Bin
Department of Mathematics, Visva-Bharati, Santiniketan-731235, West Bengal, India.
Sharda Kumari
Department of Mathematics, Visva-Bharati, Santiniketan-731235, West Bengal, India.
M M Panja *
Department of Mathematics, Visva-Bharati, Santiniketan-731235, West Bengal, India.
*Author to whom correspondence should be addressed.
Abstract
An approximation scheme based on orthonormal Legendre polynomial wavelets has been proposed here to obtain the approximate solution of the Fredholm integral equations of the second kind with a logarithmic singular kernel. In conjunction with double exponential quadrature, the Galerkin approach is exercised here to convert the integral equation to a system of linear algebraic equations of unknown coefficients of elements of the wavelet basis. An estimate of a posteriori error in the approximate solution has been provided. An exercise of the proposed scheme for some test examples exhibits that the method is (i) efficient, (ii) rapidly convergent, and (iii) stable for a variety of source terms for obtaining highly accurate approximate solutions of singular integral equations involving logarithmic kernels and precise estimates of their accuracies. An extension of this scheme to more subtle physical problems has been suggested.
Keywords: Orthonormal Legendre polynomial wavelets, Fredholm integral equation of second kind, Logarithmic singular kernel, Double exponential quadrature, Wavelet-Galerkin method for singular integral equations