Improved efficient difference method for the modified anomalous sub-diffusion equation with a nonlinear source term

ABSTRACT

In this paper, we propose an improved efficient difference method to solve the modified anomalous sub-diffusion equation with a nonlinear source term. The weighted and shifted Grünwald difference operator and compact difference are used to approximate the Riemann–Liouville fractional derivative and the space derivative, respectively. Since the semi-discrete system obtained by discretizing in space has some stiff properties when the space mesh is dense, the second-order backward differentiation formula (BDF) is then employed to solve the semi-discrete system and a second-order interpolation formula is used to deal with the nonlinear source term. The stability and convergence of the method are discussed and the convergence order is shown to be $O({\tau }^2+h^4)$, where τ and h denote the time and space step size. The new method has higher accuracy compared with the existed methods and good stability in the numerical performance. Numerical examples are given to illustrate the theoretical results.

KEYWORDS:

• Modified anomalous sub-diffusion equation
• backward differentiation formula
• stability
• convergence
• numerical experiments

2010 AMS SUBJECT CLASSIFICATIONS:

• 26A33
• 33F05
• 65L12
• 65L20

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported by projects Hunan Province Innovation Foundation for Postgraduate [CX2012B242], NSF of China [No. 11501447], and Scientific Research Program Funded by Shaanxi Provincial Education Department [No. 2015JQ1029].