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

## DOI:

https://doi.org/10.14419/ijamr.v4i1.4081## Published:

2015-02-25## Keywords:

Volterra-Fredholm Integral Equations, Logarithmic Kernels, Integral Equation, Collocation Matrix Method, Legendre and Chebychev Polynomials.## Abstract

A smoothing transformation, Legendre and Chebyshev collocation method areÂ presented to solve numerically the Voltterra-Fredholm Integral Equations withÂ Logarithmic Kernel. We transform the Volterra Fredholm integral equationsÂ 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.

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