Solutions of integral equations via shifted Legendre polynomials

Abstract

The unique property of the convolution integral of the shifted Legendre polynomials is used to solve convolution integral equations. Three important types of integral equations: (i) first-order of the first kind, (ii) second-order of the first kind, and (iii) the integral equation of the second kind, are studied in the present paper. The basic idea in solving the integral equation is that the state variables are expressed in terms of the shifted Legendre polynomial series. Series of the algebraic equations of the expansion coefficients are obtained and are calculated recursively by the powerful proposed computational algorithm. Examples are given for illustration. Very satisfactory computational results are obtained.