Convergence analysis and numerical implementation of projection methods for solving classical and fractional Volterra integro-differential equations

In this article, we discuss the convergence analysis of the classical first-order and fractional-order Volterra integro-differential equations of the second kind with a smooth kernel by reducing them into a system of fractional Fredholm integro-differential equations (FFIDEs). For that, we first reformulate the given equation into a system of fractional Volterra integro-differential equations and then transform it into the system of FFIDEs using a simple transformation. We develop a general framework of the newly defined iterated Galerkin method for the reduced system of equations and investigate the existence and uniqueness of the approximate solutions in the given Banach space. We provide the error estimates and convergence analysis for the iterated Galerkin approximate solutions in the supremum norm without any limiting conditions. Further, we provide the superconvergence results for classical first-order and fractional-order Volterra integro-differential equations by proposing a general framework of multi-Galerkin and iterated multi-Galerkin methods for the reduced system of equations. Moreover, we prove that the order of convergence of the proposed methods increases theoretically and numerically with the increasing order of the fractional derivatives. Finally, numerical implementations and illustrative examples are provided to demonstrate our theoretical aspects.