Finite difference method for time–space linear and nonlinear fractional diffusion equations

ABSTRACT

In this paper a finite difference method is presented to solve time–space linear and nonlinear fractional diffusion equations. Specifically, the centred difference scheme is used to approximate the Riesz fractional derivative in space. A trapezoidal formula is used to solve a system of Volterra integral equations transformed from spatial discretization. Stability and convergence of the proposed scheme is discussed which shows second-order accuracy both in temporal and spatial directions. Finally, examples are presented to show the accuracy and effectiveness of the schemes.

KEYWORDS:

• Fractional diffusion equation
• finite difference method
• stability and convergence analysis
• trapezoidal formula
• Riesz derivative
• Caputo derivative

2010 MSC SUBJECT CLASSIFICATIONS:

• 35R11
• 65Z05
• 65N06
• 65N12

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research is supported by the State Key Program of National Natural Science Foundation of China (Grant No. 11631013), the National Natural Science Foundation of China (Grant Nos 11371357, 11472247, 11601460), the ITER-China Program (Grant No. 2014GB124005) and the Research Foundation of Education Commission of Hunan Province of China (Grant No. 16C1540).