Finite Difference–Collocation Method for the Generalized Fractional Diffusion Equation

Abstract

In this paper, an approximate method combining the finite difference and collocation methods is studied to solve the generalized fractional diffusion equation (GFDE). The convergence and stability analysis of the presented method are also established in detail. To ensure the effectiveness and the accuracy of the proposed method, test examples with different scale and weight functions are considered, and the obtained numerical results are compared with the existing methods in the literature. It is observed that the proposed approach works very well with the generalized fractional derivatives (GFDs), as the presence of scale and weight functions in a generalized fractional derivative (GFD) cause difficulty for its discretization and further analysis.

Keywords

• generalized Caputo derivate
• fractional diffusion equation
• finite difference method
• collocation method
• error
• stability and convergence analysis