A backward Euler difference scheme for the integro-differential equations with the multi-term kernels

ABSTRACT

A backward Euler difference scheme is formulated and analyzed for the integro-differential equations with the multi-term kernels. The convolution quadrature is used to deal with the Riemann-Liouville fractional integral terms. A fully discrete difference scheme is constructed with time derivative by backward Euler scheme and space discretization by standard central difference approximation. We prove the stability and convergence based on the nonnegative character of the real quadratic form. The $L^2$ and $H^1$-norms stability and convergence are given. Finally, numerical experiments validate the theoretical results.

KEYWORDS:

• Integro-differential equations
• multi-term kernels
• finite difference scheme
• stability
• convergence

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 45K05
• 65M06
• 65M12
• 65M15

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

S. F. Hu  http://orcid.org/0000-0003-3044-6014

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

Additional information

Funding

The project supported by the Natural Science Foundation of Hunan Province (No: 2018JJ2669, 14JJ3114), the Scientific Reasearch Foundation of Hunan Provincial Eduction Department (No: 17B277), the Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering (No: 2017TP1017) and the Startup Scientific Research Foundation of CSUFT (No: 2017YJ026), Bridge non-destructive testing and engineering calculation key laboratory project of universities in sichuan province (2015QYJ05).