On the numerical solution of Volterra-Fredholm integral equations with logarithmic kernel using smoothing transformation

Abstract

A smoothing transformation, Legendre and Chebyshev collocation method are presented to solve numerically the Voltterra-Fredholm Integral Equations with Logarithmic Kernel. We transform the Volterra Fredholm integral equations to a system of Fredholm integral equations of the second kind, using a smoothing transformation to cancel the singularities in the kernel, a system Fredholm integral equation with smooth kernel is obtained and will be solved using Legendre and Chebyshev polynomials. This lead to a system of algebraic equations with Legendre or Chebychev coefficients. Thus, by solving the matrix equation, Legendre and Chebychev coefficients are obtained. Some numerical examples are included to demonstrate the validity and applicability of the proposed technique.

On the Numerical Solution of Singular Integral Equation with Degenerate Kernel Using Laguerre Polynomials

“…Sequences of orthogonal polynomials appear frequently used as applications in mathematics, mathematical physics, engineering and computer science, in particular during the resolution of partial differential equations (Laplace, Schrödinger) by the method of separation of variables, also these polynomials can be used to solve integral equations of first and second kind [1] [11]. One of the most common set of orthogonal polynomials is the Laguerre polynomials.…”