Analysis of finite difference schemes for Volterra integro-differential equations involving arbitrary order derivatives

In this work, we consider a Volterra integro-differential equation involving Caputo fractional derivative of order alpha is an element of (0, 1). To approximate the solution, we propose two finite difference schemes that use L1 and L1-2 discretization to approximate the differential part and a composite trapezoidal rule to approximate an integral part. The error estimates for both schemes are established. It is shown that the approximate solution obtained by using the L1-2 scheme converges to the exact solution more rapidly than the L1 scheme. Finally, some numerical experiments are carried out to show the validity and accuracy of the proposed schemes.