Numerical methods for fractional partial differential equations

ABSTRACT

In this review paper, we are mainly concerned with the finite difference methods, the Galerkin finite element methods, and the spectral methods for fractional partial differential equations (FPDEs), which are divided into the time-fractional, space-fractional, and space-time-fractional partial differential equations (PDEs). Besides, fast algorithms for the FPDEs are included in order to stimulate more efficient algorithms for high-dimensional FPDEs.

KEYWORDS:

• Finite difference methods
• Galerkin finite element methods
• spectral methods
• fast algorithms
• fractional partial differential equations

2010 AMS SUBJECT CLASSIFICATIONS:

• 26A33
• 35R11
• 65M06
• 65M60

Acknowledgments

The authors wish to thank Professor Abdul Q. M. Khaliq, Editor-in-Chief of International Journal Computer Mathematics for his cordial invitation. The authors are also grateful to Professor Khaliq and the referees for the valuable comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work was partially supported by the National Natural Science Foundation of China under grant nos. [11372170, 11671251, and 11632008].