Highly-accurate numerical scheme for a system of nonlinear Abel–Volterra integral equations

This study explores a system of second-kind nonlinear Abel–Volterra integral equations (SSNAVIEs) that involve a weakly singular kernel function. To effectively solve these integral equations, we propose Galerkin and Iterated Galerkin (IG) methods based on Jacobi polynomials. The existence and uniqueness of the solution are established using the Banach contraction theorem. Additionally, we develop numerical algorithms to obtain approximate solutions and conduct a comprehensive convergence and error analysis for the proposed methods. For the Galerkin method, we establish an order of convergence of $O\left(N^{-q}\right)$, while the IG method exhibits an improved convergence rate of $O\left(N^{-2q}\right)$. To the best of our knowledge, this is the first study to apply spectral methods for solving systems of second-kind nonlinear Volterra–Abel integral equations with weak singularities. To validate our theoretical results, we perform numerical experiments, confirming the accuracy and efficiency of the proposed approach.