Wavelet-Collocation Method of Solving Singular Integral Equation

Objectives: This article describes one of the approaches to the approximate solution of the singular integral equation of the first kind with the Cauchy kernel on material axis interval, based on the approximation of the desired function by Chebyshev’s wavelets of the II-nd kind. Methods: The theory of such equations states that the solution in a closed form can be obtained only in rare cases. Therefore, various approximation methods are used, followed by their theoretical basis. The Uniform error estimates of obtained approximate solutions are very relevant ones for practice. Results: However, the incorrect problem of this equation solution caused primarily by universal values of first and reversible singular operators, on the pair of continuous function spaces, leads to particular difficulties at a numerical solution. The work as the area of desired elements and right sides considers weighted spaces, some of which are the restrictions of continuous functions on which a correct task is set. A computational scheme of wavelet collocation method is developed. The theorem on the unique solvability of obtained linear algebraic equation system is proved. Uniform estimates for the relative solution errors are set depending on the structural properties of original data. Conclusion: The performed numerical experiment in Wolfram Mathematical package showed a real convergence of the approximate solution, obtained by the method of wavelet collocation, with the exact one.