An accurate numerical technique for solving fractional advection–diffusion equation with generalized Caputo derivative

In this article, we propose a numerical technique for obtaining an approximate solution of the fractional advection–diffusion equation involving a generalized Caputo derivative. This technique combines the compact finite difference method for the temporal discretization derivative with the expression of the required numerical solution in terms of the shifted ultraspherical polynomials. Stability and convergence analyses of the proposed technique are proved. Furthermore, we compare the obtained numerical solutions with exact solutions and those published in the literature to demonstrate the applicability and accuracy of the suggested technique.