Numerical solution of integro-differential equations via pertubed-Gegenbauer, Jacobi polynomials and Galerkin method

Abstract

In this paper, we proposed perturbed Galerkin method for solving integro-diffrential equations via shifted Gegenbauer and shifted Jacobi polynomials as approximating polynomials. We use Galerkin method to transform the perturbed integro-differential equation to system of linear algebraic equations and obtained N + 1 linear equations with N + m + 2 unknowns. Moreover, with m+1 boundary conditions we obtained N+m+2 algebraic equations which was then solved to obtain the approximate solutions at various values of α and β depending on the orthogonal polynomials, that's shifted Gegenbauer or shifted Jacobi polynomials. The proposed method was implemented on some selected problems in the literature to validate the effectiveness and the accuracy of the proposed method.