Optimizing Computational Techniques for Cauchy and Integral Equations Through Advanced Polynomial Methods

This study introduces a numerical approach for solving Cauchy and integral equations using the Chebyshev pseudospectral method. The approach involves approximating the solution with an Nth-degree interpolating polynomial based on Chebyshev nodes, followed by problem discretization through a cell-averaging technique. The main properties of the Chebyshev pseudospectral method are discussed and explained to simplify the computation of Cauchy and integral equations into a system of algebraic equations. Several examples are presented to validate the method and to show how the method is computationally efficient, and to prove its effectiveness.