Piecewise Fractional Jacobi Polynomial Approximations for Volterra Integro-Differential Equations with Weakly Singular Kernels

This paper is concerned with numerical solutions to Volterra integro-differential equations with weakly singular kernels. Making use of the transformed fractional Jacobi polynomials, we develop a class of piecewise fractional Galerkin methods for solving this kind of Volterra equation. Then, we study the existence, uniqueness and convergence properties of Galerkin solutions by exploiting the decaying rate of the coefficients of the transformed fractional Jacobi series. Finally, numerical experiments are carried out to illustrate the performance of the piecewise Galerkin solution.