Convergence and Stability Analysis of Finite Difference Methods, Caputo Derivatives, and Collocation Methods Applied to Space Fractional Diffusion Equations

The exploration of fractional diffusion equations has experienced substantial growth in recent years, driven by the investigation of various solving approaches. Applications of fractional diffusion equations span across multiple disciplines, such as physics, engineering, and petrochemical analysis. This study focuses on the numerical solution of the space fractional diffusion equation, employing shifted Gegenbauer polynomials as a basis function thereby collocating at some optimized points. The estimation of the spatial fractional derivative is accomplished through the utilization of Caputo derivatives and the finite difference method. The proposed method’s convergence and stability are supported by several theorems. To assess its effectiveness and validity, selected cases from the literature are considered. Results indicate that the proposed technique is effective and outperforms existing outcomes. All computations are conducted using the Matlab package.