Comparison of Selected Numerical Methods for Solving Integro-Differential Equations with the Cauchy Kernel

Abstract

The integro-differential equation with the Cauchy kernel is used in many different technical problems, such as in circuit analysis or gas infrared radiation studies. Therefore, it is important to be able to solve this type of equation, even in an approximate way. This article compares two approaches for solving this type of equation. One of the considered methods is based on the application of the differential Taylor series, while the second approach uses selected heuristic algorithms inspired by the behavior of animals. Due to the problem domain, which is symmetric, and taking into account the form of the function appearing in this equation, we can use this symmetry in some cases. The paper also presents numerical examples illustrating how each method works and comparing the discussed approaches.