Approximate solution to solve nonlinear Volterra integral equations with discontinuous kernels using shifted alternative Legendre polynomials

Abstract

We consider nonlinear Volterra integral equations of the second kind with discontinuous kernels, which present significant analytical and numerical challenges due to the combined presence of nonlinearity and kernel discontinuity. To address these difficulties, we develop a new method based on shifted alternative Legendre polynomials and associated operational matrices. The proposed approach approximates the unknown solution via truncated polynomial expansions and systematically transforms the original integral equation into a system of nonlinear algebraic equations through matrix-based discretization. We establish several theoretical results concerning the convergence, stability, and error bounds of the method. Numerical experiments are conducted to validate the proposed approach, demonstrating its accuracy, efficiency, and capability in handling these equations .