Two new approximations for generalized Caputo fractional derivative and their application in solving generalized fractional sub-diffusion equations

Abstract

In this paper, we propose two new approximation methods on a general mesh for the generalized Caputo fractional derivative of order α∈ (0 , 1 ). The accuracy of these two methods is shown to be of order (3 - α) which improves some previous work done to date. To demonstrate the accuracy and usefulness of the proposed approximations, we carry out experiment on test examples and apply these approximations to solve generalized fractional sub-diffusion equations. The numerical results indicate that the proposed methods perform well in practice. Our contributions lie in two aspects: (i) we propose high order approximations that work on a general mesh; (ii) we establish the well-posedness of generalized fractional sub-diffusion equations and develop numerical schemes using the new high order approximations.