A Galerkin finite element scheme for time–space fractional diffusion equation

Abstract

In this paper, a Galerkin finite element scheme to approximate the time–space fractional diffusion equation is studied. Firstly, the fractional diffusion equation is transformed into a fractional Volterra integro-differential equation. And a second-order fractional trapezoidal formula is used to approach the time fractional integral. Then a Galerkin finite element method is introduced in space direction, where the semi-discretization scheme and fully discrete scheme are given separately. The stability analysis of semi-discretization scheme is discussed in detail. Furthermore, convergence analysis of semi-discretization scheme and fully discrete scheme are given in details. Finally, two numerical examples are displayed to demonstrate the effectiveness of the proposed method.

Keywords:

• Riemann–Liouville derivative
• Caputo derivative
• time–space fractional diffusion equation
• fractional Volterra integro-differential equation
• fractional trapezoidal formula
• Galerkin finite element method

2010 AMS Subject Classifications:

• 35R11
• 78M10

Acknowledgments

The authors wish to thank the referees for their constructive comments and suggestions which improved the present paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China under [grant number 11301333]; Funding Scheme for Training Young Teachers in Shanghai Colleges under [grant number zzhg12001]; Innovation Program of Shanghai Municipal Education Commission under [grant number 14YZ165]; Natural Science Foundation of Anhui province under [grant number 1408085MA14]; Funding Scheme for Training Young Teachers in Shanghai colleges under [grant number 14AZ17].