Superconvergent Jacobi spectral Galerkin method for nonlinear fractional Volterra-Fredholm integro-differential equations

This article presents a new approach to the Jacobi spectral Galerkin method for approximating solutions of nonlinear fractional Volterra-Fredholm integro-differential equations with smooth and weakly singular kernel functions. We determine the regularity properties of the solution and find that it lacks smoothness at the origin but exhibits Hölder continuity. The given equation is first reformulated to a Volterra-Fredholm integral equation (V-FIE) with kernel functions having distinct regularity behavior. To achieve better convergence, we transform V-FIE to an equivalent Fredholm integral equation via an appropriate variable transformation. Despite the singularity in the solution and the distinct nature of the regularity of kernel functions, we achieve enhanced convergence rates by including the less regular parts of kernel functions into corresponding weight functions and taking advantage of the Hölder continuity of the solution. Further, we employ a regularization technique to improve the solution’s regularity, leading to superconvergence results. Finally, numerical examples, along with comparisons to existing methods, are presented to validate the theoretical results. © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2025.