Well-Posedness and Uniform Approximations of the Solution of a Boundary Value Problem for a Singular Integro-Differential Equation

Abstract

On a real segment, we consider a boundary value problem for a singular integro-differential equation of the first kind with the Cauchy kernel in the characteristic part. The well-posedness of this problem, established by the authors on a pair of specially selected spaces, allows to use approximate methods for its solving. We propose a general projection method, establish the conditions for its convergence in the chosen spaces and estimates the error of approximate solutions. As a result, uniform error estimates are obtained. A computational scheme of the wavelet collocation method is constructed, its theoretical substantiation is carried out, the results of a numerical experiment are presented on a model example.