A second-order BDF compact difference scheme for fractional-order Volterra equation

Abstract

In this paper, a second-order backward differentiation formula compact difference scheme with the truncation error of order $1+\alpha (0<\alpha <1)$ for time and 4 for space to fractional-order Volterra equation is considered. The integral term is treated by means of the second-order convolution quadrature suggested by Lubich and fourth-order accuracy compact approximation is applied for the second-order space derivative. The stability and convergence of the compact difference scheme in a new norm are proved by the energy method. Numerical experiments that are in total agreement with our analysis are reported.

Keywords:

• fractional-order Volterra equation
• second-order BDF
• compact difference scheme
• truncation error
• stability
• convergence
• numerical experiments

2010 AMS Subject Classifications:

• 65M60
• 45K05
• 65D32

Acknowledgments

I am grateful to the referees for many helpful suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The project supported by the National Natural Science Foundation of China [no: 11171352, 10971062] and the Scientific Research Foundation of Central South University of Forestry and Technology (QJ2011001A).