High Order Approximation of Generalized Caputo Fractional Derivative and its Application

Xuhao Li, Qinxu Ding, Patricia J. Y. Wong

Abstract: In this paper, we propose a high order approximation for generalized Caputo fractional derivative of order <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\alpha\in(0,1)$</tex> . The approximation order is shown to be <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(\tau^{3-\alpha})$</tex> which improves some previous work done to date. We then apply the new approximation to solve a class of generalized time fractional sub-diffusion problem. Some experiments are carried out to demonstrate the accuracy of the proposed methods. The numerical results indicate consistency with the theoretical results and good performance of the methods.