A formally second-order BDF finite difference scheme for the integro-differential equations with the multi-term kernels

Abstract

In this article, a formally second-order backward differentiation formula (BDF) finite difference scheme is presented for the integro-differential equations with the multi-term kernels. In the time direction, the time derivative is approximated by a second-order BDF scheme and the Riemann-Liouville (R-L) fractional integral terms are discretized by the second-order convolution quadrature rule. We construct a fully discrete difference scheme with the space discretization by the standard central difference formula. The $L^2$ and $H^1$-norms stability, and convergence in $L^2$-norm are derived by the discrete energy method. In the numerical experiments, the results are consistent with the theoretical analysis.

Keywords:

• Integro-differential equations
• second-order BDF scheme
• multi-term kernels
• second-order convolution quadrature rule
• stability and convergence

2010 AMS Subject Classifications:

• 45K05
• 65M06
• 65M12
• 65M15

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Hongbin Chen http://orcid.org/0000-0002-8989-3743

Additional information

Funding

The project is supported by the National Natural Science Foundation of China (No. 11671131), the Natural Science Foundation of Hunan Province (No. 2018JJ2669), the Scientific Research Foundation of Hunan Provincial Education Department (No. 17B277) and the Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering (No. 2017TP1017), and this paper is supported by Construct Program of the Key Discipline in Hunan Province, Performance Computing and Stochastic Information Processing (Ministry of Education of China).