COLLOCATION METHODS FOR SYSTEMS OF CAUCHY SINGULAR INTEGRAL EQUATIONS ON AN INTERVAL

Abstract

Recently a collocation method, which is based on the Chebyshev nodes of second kind as collocation points and on approximating the solution by polynomials multiplied with the Chebyshev weight of second kind, was studied for both linear and nonlinear Cauchy singular integral equations (CSIE’s) on the interval [−1, 1] (see [11, 12, 22] for the linear case and [9] for the nonlinear case). There are several reasons for choosing Chebyshev nodes as collocation points independently from the asymptotic of the solution of the CSIE. At first we get a very cheap preprocessing for the construction of the matrix of the discretized equation, which is especially important in case of approximating the solution of a nonlinear CSIE by a sequence of solutions of linear equations (cf. [9]). A second reason is the possibility to apply such collocation methods to systems of CSIE’s, which is, in some sense, the main topic of the present paper. Indeed, in [11] there are only given necessary and sufficient conditions for the stability of the mentioned collocation method in the case of scalar CSIE’s of the form