A posteriori error analysis for a fractional differential equation

ABSTRACT

Numerical treatment for a fractional differential equation (FDE) is proposed and analysed. The solution of the FDE may be singular near certain domain boundaries, which leads to numerical difficulty. We apply the upwind finite difference method to the FDE. The stability properties and a posteriori error analysis for the discrete scheme are given. Then, a posteriori adapted mesh based on a posteriori error analysis is established by equidistributing arc-length monitor function. Numerical experiments illustrate that the upwind finite difference method on a posteriori adapted mesh is more accurate than the method on uniform mesh.

KEYWORDS:

• Fractional differential equation
• Caputo fractional derivative
• finite difference scheme
• a posteriori error estimate
• mesh equidistribution

AMS SUBJECT CLASSIFICATIONS:

• 65L05
• 65L12
• 65L20

Acknowledgments

We would like to thank the anonymous referee for several suggestions for the improvement of this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work was supported by the Project of Philosophy and Social Science Research in Zhejiang Province [Grant No. 15NDJC243YB], Humanities and Social Sciences Planning Fund of Ministry of education of China [Grant No. 14YJC790006], Ningbo Municipal Soft Science Foundation [Grant No. 2015A10045] and Key Research Center of Philosophy and Social Science of Zhejiang Province-Modern Port Service Industry and Creative Culture Research Center.