Uniform Wavelet-Approximation of Singular Integral Equation Solutions

Abstract

In this article we consider a singular integral equation of the first kind with a Cauchy kernel on a segment of the real axis, which is a mathematical model of many applied problems. It is known that such an equation is exactly solved only in rare cases, therefore, the problem of its approximate solution with obtaining uniform error estimates is very actual. This equation is considered on a pair of weighted spaces that are constrictions of the space of continuous functions. The correctness of the problem of solving this equation on a chosen pair of spaces of the desired elements and right-hand sides gives the possibility of its approximate solution with a theoretical justification. The numerical method proposed in this article is based on the approximation of the unknown function by Chebyshev wavelets of the second kind. Uniform error estimates are established depending on the structural properties of the initial data. The numerical experiment in the Wolfram Mathematica package showed a good convergence rate of the approximate solution to the exact one.