A computational approach for the non-smooth solution of non-linear weakly singular Volterra integral equation with proportional delay

This paper develops a well-conditioned Jacobi spectral Galerkin method for the analysis of Volterra-Hammerstein integral equations with weakly singular kernels and proportional delay. A recursive formula reduces the computational load when approximating the solutions of badly conditioned and complex non-linear algebraic systems. Additionally, the convergence properties of the method are also investigated. The spectral accuracy is obtained regardless of the discontinuities in the derivatives solution. Three examples illustrate the performance of the new approach.