A fully discrete local discontinuous Galerkin method for one-dimensional time-fractional Fisher's equation

Abstract

In this paper, we consider the local discontinuous Galerkin (LDG) finite element method for one-dimensional time-fractional Fisher's equation, which is obtained from the standard one-dimensional Fisher's equation by replacing the first-order time derivative with a fractional derivative (of order α, with 0<α<1). The proposed LDG is based on the LDG finite element method for space and finite difference method for time. We prove that the method is stable, and the numerical solution converges to the exact one with order O(h^k+1+τ^2−α), where h, τ and k are the space step size, time step size, polynomial degree, respectively. The numerical experiments reveal that the LDG is very effective.

Keywords:

• time-fractional Fisher's equation
• local discontinuous Galerkin finite element method
• Caputo fractional derivative
• fractional differential equation

2000 AMS Subject Classifications:

• 35Q99
• 65M12

Acknowledgements

We are very grateful to both referees for their carefully reading the paper and most valuable comments and suggestions. And this work is supported by the NSF of Xinjiang Uigur Autonomous Region (No. 2013211B12).