A unified difference-spectral method for time–space fractional diffusion equations

ABSTRACT

In this paper we develop a unified difference-spectral method for stably solving time–space fractional sub-diffusion and super-diffusion equations. Based on the equivalence between Volterra integral equations and fractional ordinary differential equations with initial conditions, this proposed method is constructed by combining the spectral Galerkin method in space and the fractional trapezoid formula in time. Numerical experiments are carried out to verify the effectiveness of the method, and demonstrate that the unified method can achieve spectral accuracy in space and second-order accuracy in time for solving two kinds of time–space fractional diffusion equations.

KEYWORDS:

• Fractional diffusion equation
• spectral method
• Legendre polynomials
• fractional trapezoid formula
• unified numerical method

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 45D05
• 65M06
• 65M12
• 65M70

Acknowledgments

We are grateful for the insightful comments and suggestions by the reviewers.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research is supported by National Natural Science Foundation of China [Grant No. 11426141], and by Jiangsu Overseas Research or Training Program for University Prominent Young Faculty and Presidents.