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

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)(-k), where 0 < k < 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 O(N3/4-r) and O(N-r) in uniform and weighted L-2-norms, respectively. Whereas iterated Jacobi spectral Galerkin method converges with the order of convergence 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 O(N-3r logN) and O(N-3r) in uniform and weighted L-2-norms, respectively. Theoretical results are verified by numerical illustrations.