A Galerkin Finite Element Method to Solve Fractional Diffusion and Fractional Diffusion-Wave Equations

Abstract

In the present study, numerical solutions of the fractional diffusion and fractional diffusion-wave equations where fractional derivatives are considered in the Caputo sense have been obtained by a Galerkin finite element method using quadratic B-spline base functions. For the fractional diffusion equation, the L1 discretizaton formula is applied, whereas the L2 discretizaton formula is applied for the fractional diffusion-wave equation. The error norms L ₂ and L _∞ are computed to test the accuracy of the proposed method. It is shown that the present scheme is unconditionally stable by applying a stability analysis to the approximation obtained by the proposed scheme.

Keywords:

• finite element method
• Galerkin method
• fractional diffusion equation
• fractional diffusion-wave equation
• quadratic B-spline

AMS Subject Classification:

• 97N40
• 65D07
• 74S05
• 26A33
• 34A08
• 65L60