Galerkin solution of a singular integral equation with constant coefficients

Galerkin methods are used to approximate the singular integral equation a phi(x) + b/pi integral(1)(-1)phi(t)/t-x dt + lambda integral(1)(-1)k(t, x)phi(t)dt = f (x), -1 < x < 1. with solution phi having weak singularity at the endpoint -1, where a, b not equal 0 are constants. In this case phi is decomposed as phi(x) = (1- x)(alpha) (1 + x)(beta)u(x), where beta = -alpha, 0 < alpha < 1. Jacobi polynomials are used in the discussions. Under the conditions f is an element of H-mu [-1, 1] and k(t, x) is an element of H-mu,H-mu [-1, 1] x [-1, 1], 0 < mu < 1, the error estimate under a weighted L-2 norm is O(n(-mu)). Under the strengthened conditions f '' is an element of H-mu [-1, 1] and partial derivative(2)k/partial derivative x(2)(t, x) is an element of H-mu,H-mu [-1, 1] x [-1, 1], 2 alpha - rho < mu < 1, the error estimate under maximum norm is proved to be O(n(2-alpha-rho-mu+is an element of)), where rho = min{alpha, 1/2}, is an element of > 0 is a small enough constant. (C) 2008 Elsevier B.V. All rights reserved.