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

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 NS1 and NS2, using linear and quadratic interpolation functions, respectively. The convergence order of the two schemes is 2-alpha, 3-alpha, respectively, for 0<1. 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.

Automated summary

This work aims to numerically approximate the Caputo-Prabhakar derivative and use it to solve the time-fractional advection-diffusion equation defined in the Caputo-Prabhakar sense. Two schemes are proposed to approximate the time-fractional derivative using different interpolation functions, and their convergence order and analytical error bounds are discussed. Numerical examples validate the feasibility and stability of these schemes.