Effective numerical treatment of sub-diffusion equation with non-smooth solution

ABSTRACT

In this paper we investigate a sub-diffusion equation for simulating the anomalous diffusion phenomenon in real physical environment. Based on an equivalent transformation of the original sub-diffusion equation followed by the use of a smooth operator, we devise a high-order numerical scheme by combining the Nyström method in temporal direction with the compact finite difference method and the spectral method in spatial direction. The distinct advantage of this approach in comparison with most current methods is its high convergence rate even though the solution of the anomalous sub-diffusion equation usually has lower regularity on the starting point. The effectiveness and efficiency of our proposed method are verified by several numerical experiments.

KEYWORDS:

• Fractional derivative
• anomalous sub-diffusion
• weakly singular
• Volterra integral equation
• spectral method

2010 AMS SUBJECT CLASSIFICATIONS:

• 35R11
• 65M70
• 45D05

Acknowledgments

The authors would like to thank the editor and the anonymous referees for their valuable comments and helpful suggestions that improve the quality of our paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Jungang Wang  http://orcid.org/0000-0002-3702-1548

Additional information

Funding

This research was supported by the National Natural Science Foundation of China (grant nos.11601432,11471262) and the Strategic Research Grant of the City University of Hong Kong (grant no. 7004446).