Approximation of Caputo-Prabhakar derivative with application in solving time fractional advection-diffusion equation

Abstract

This work aims to numerically approximate the Caputo-Prabhakar derivative and use this approximation for solving the time-fractional advection-diffusion equation defined in Caputo-Prabhakar sense which is widely used in fluid dynamics. In this approach, we approximate the time-fractional derivative of the mentioned equation by two schemes, namely (Formula presented.) and (Formula presented.), using linear and quadratic interpolation functions, respectively. The convergence order of the two schemes is (Formula presented.), (Formula presented.), respectively, for (Formula presented.). The analytical error bounds for the two schemes are also discussed. Then, these schemes are applied to solve the time-fractional advection-diffusion equation defined in the Caputo-Prabhakar sense numerically. We will prove the solvability and stability of the proposed methods. Numerical examples validate the analytical results. With the reference of an example, we have shown that the schemes work well for the fractional diffusion equation also. © 2022 John Wiley & Sons Ltd.