Numerical Analysis of Volterra Integro-differential Equations with Caputo Fractional Derivative

This article deals with a fully discretized numerical scheme for solving fractional order Volterra integro-differential equations involving Caputo fractional derivative. Such problem exhibits a mild singularity at the initial time t = 0. To approximate the solution, the classical L1 scheme is introduced on a uniform mesh. For the integral part, the composite trapezoidal approximation is used. It is shown that the approximate solution converges to the exact solution. The error analysis is carried out. Due to presence of weak singularity at the initial time, we obtain the rate of convergence is of order O(tau) on any subdomain away from the origin whereas it is of order O(tau(alpha)) over the entire domain. Finally, we present a couple of examples to show the efficiency and the accuracy of the numerical scheme.