On the effective method for the space-fractional advection-diffusion equation by the Galerkin method

The present work is dedicated to a study that focuses on solving space-fractional advection-diffusion equations (SFADEs) using the Galerkin method. Through our analysis, we demonstrate the effectiveness of this approach in solving the considered equations. After introducing the Chebyshev cardinal functions (CCFs), the Caputo fractional derivative (CFD) was represented based on these bases as an operational matrix. Applying the Galerkin method reduces the desired equation to a system of algebraic equations. We have proved that the method converges analytically. By solving some numerical examples, we have demonstrated that the proposed method is effective and yields superior outcomes compared to existing methods for addressing this problem.