Jacobi Spectral Galerkin Methods for a Class of Nonlinear Weakly Singular Volterra Integral Equations

Abstract

We propose the Jacobi spectral Galerkin and Jacobi spectral multi Galerkin methods with their iterated versions for obtaining the superconvergence results of a general class of nonlinear Volterra integral equations with a kernel $x^{\beta}(z-x)^{-\kappa},$ where $0<\kappa<1$, $\beta>0$, which have an Abel-type and an endpoint singularity. The exact solutions for these types of integral equations are singular at the initial point of integration. First, we apply a transformation of independent variables to find a new integral equation with a sufficiently smooth solution. Then we discuss the superconvergence rates for the transformed equation in both uniform and weighted $L^2$-norms. We obtain the order of convergence in Jacobi spectral Galerkin method $\mathcal{O}(N^{\frac{3}{4}-r})$ and $\mathcal{O}(N^{-r})$ in uniform and weighted $L^2$-norms, respectively. Whereas iterated Jacobi spectral Galerkin method converges with the order of convergence $\mathcal{O}(N^{-2r})$ in both uniform and weighted $L^2$-norms. We also show that iterated Jacobi spectral multi Galerkin method converges with the orders $\mathcal{O}(N^{-3r}\log{N})$ and $\mathcal{O}(N^{-3r})$ in uniform and weighted $L^2$-norms, respectively. Theoretical results are verified by numerical illustrations.