Convergence and stability of a collocation method for the generalized airfoil equation

摘要

We establish the mean-square convergence and numerical stability of a collocation method for solving a class of Cauchy-singular integral equations. The method uses the Jacobi polynomials {P12, -12n} as basis elements and the zeros of the polynomial P-12,12n+1 as collocation points. Uniform convergence is shown to hold for the first iterate. Convergence is proved via Vainikko's embedding method, and stability is demonstrated by generalizing the approach of Atkinson for integral equations of the second kind.