The numerical solution of integral equations using Chebyshev polynomials

Abstract

An investigation has been made into the numerical solution of non-singular linear integral equations by the direct expansion of the unknown function f(x) into a series of Chebyshev polynomials of the first kind. The use of polynomial expansions is not new, and was first described by Crout [1]. He writes f(x) as a Lagrangian-type polynomial over the range in x, and determines the unknown coefficients in this expansion by evaluating the functions and integral arising in the equation at chosen points x_i. A similar method (known as collocation) is used here for cases where the kernel is not separable. From the properties of expansion of functions in Chebyshev series (see, for example, [2]), one expects greater accuracy in this case when compared with other polynomial expansions of the same order. This is well borne out in comparison with one of Crout's examples.