High Order Numerical Scheme for Generalized Fractional Diffusion Equations

Abstract: In this paper, a higher order finite difference scheme is proposed for Generalized Fractional Diffusion Equations (GFDEs). The fractional diffusion equation is considered in terms of the generalized fractional derivatives (GFDs) which uses the scale and weight functions in the definition. The GFD reduces to the Riemann-Liouville, Caputo derivatives and other fractional derivatives in a particular case. Due to importance of the scale and the weight functions in describing behaviour of real-life physical systems, we present the solutions of the GFDEs by considering various scale and weight functions. The convergence and stability analysis are also discussed for finite difference scheme (FDS) to validate the proposed method. We consider test examples for numerical simulation of FDS to justify the proposed numerical method.