Modified homotopy perturbation method for solving singular integral equations of the first kind

The Homotopy Perturbation Method (HPM) stands out as a robust tool for addressing linear and nonlinear problems across diverse domains of science, engineering, and technology. This paper explores the application of both the standard HPM and modified Homotopy Perturbation Method (MHPM) as semi-analytical solutions for first-kind Cauchy-type singular integral equations (CSIEs). The proposed method effectively transforms singular integral equations into an iterative sequence of algebraic integral equations. The initial iteration introduces unknown coefficients, and a judicious selection of these coefficients often results in the precise solution to the problem. Several examples are presented to display the validity as well as accuracy of the MHPM. Lastly, the obtained results are compared with those derived from HPM, Jacobi polynomials approximation and Taylor approximation. The paper concludes by establishing the stability analysis as well as convergence of the proposed method within a suitable class of functions. It is proven that if characteristic SIEs is considered then the solution obtained by HPM as well as MHPM will coincides with exact solution for any choice of the initial guess.