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

Authors

  • Chokri Chniti Umm Al-Qura University
  • Sharefa Eisa Ali Alhazmi Umm Al-Qura University College of Education for Girls at Al-Qunfudah Mathematics department, Macca, KSA

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 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.

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