Gauss-Lobatto-Legendre-Birkhoff pseudospectral scheme for the time fractional reaction–diffusion equation with Neumann boundary conditions

ABSTRACT

In this paper, an efficient numerical scheme for the time fractional reaction–diffusion equation with Neumann boundary conditions is proposed combining finite-difference method in time and Gauss-Lobatto-Legendre-Birkhoff (GLLB) pseudospectral method in space. GLLB quadrature formula involves first-order derivative values at endpoints, which allows a natural and exact imposition of Neumann boundary conditions. It is proved that the scheme is unconditionally stable and convergent with order $O({\tau }^2+N^{-m})$, where τ, N and m stand for the time step, polynomial degree and spatial regularity of the exact solution. Numerical experiment is carried out to support theoretical analysis.

KEYWORDS:

• Fractional diffusion equation
• Neumann boundary conditions
• GLLB pseudospectral method
• stability
• convergence

2010 AMS SUBJECT CLASSIFICATIONS:

• 65M12
• 65M06
• 65M70
• 35R11

Acknowledgements

The authors would like to thank the anonymous referees for their valuable suggestions and comments that helped us improve our paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Haiyu Liu http://orcid.org/0000-0002-7340-575X

Additional information

Funding

This work was supported by the National Natural Science Foundation of China under Grant [number 11672011, 11272024].