Optimal formulas for the approximate-analytical solution of the general Abel integral equation in the Sobolev space

Abstract

This article discusses the development of a new algorithm, which is based on optimal quadrature formulas for obtaining solutions to the generalized Abel integral equations. Based on the study, it was found that the proposed scheme is very effective, extremely accurate and can be extended to other special tasks. The quadrature formulas presented in this paper are optimal in the Sobolev space of functions that have square integrable derivatives of order m. Using the quadrature formula and the Maple computer algebra system, exact and approximate values of the Abel integral equations are found, illustrating the effectiveness of the proposed approach.