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.

Fourth-order fractional Sturm-Liouville problems are studied in this work. The numerical simulation uses the pseudospectral method, utilizing Chebyshev cardinal polynomials. The presented algorithm is implemented after converting the desired equation into an associated integral equation and gives us a linear system of algebraic equations. Then, we can find the eigenvalues by calculating the roots of the corresponding characteristic polynomial. What is most striking is that the proposed scheme accurately solves this type of equation. Numerical experiments confirm this claim.