Numerical schemes with convergence for generalized fractional integro-differential equations

Abstract

In this paper, we present two numerical schemes namely linear scheme and quadratic scheme for a generalized fractional integro-differential equation (GFIDE). GFIDEs are defined in terms of the generalized fractional derivative (GFD) introduced recently. First, the linear and quadratic approximations for the GFD are presented, and its error estimates are obtained. Further, these approximations of the GFD are used to formulate numerical schemes for GFIDEs. As GFDs are defined in terms of the weight and the scale functions, thus the numerical examples are validated for different scale, and weight functions. The convergence order (CO) of the presented schemes for GFD and GFIDE is investigated as h2−α, and h3−α respectively. For numerical validation, we consider three numerical examples and perform numerical simulations varying the scale and the weight functions in the GFD. Numerical results suggest that the presented schemes work well, and validate the theoretical error estimates.