Compact scheme for fractional diffusion-wave equation with spatial variable coefficient and delays

ABSTRACT

In this paper, we study the spatial variable coefficient fractional convection–diffusion wave equation with the singe delay and multi-delay numerically when the exact solution satisfies a certain regularity. First, via the well-known exponential transformation, the delay problem can be greatly simplified, which allows us to use the variable coefficient four-order compact operator. Next, the numerical scheme is derived based on the compact operator and the reduction order method, followed by a linearized technique. Convergence of the full discrete numerical scheme is obtained with convergence order $O({\tau }^{3-\alpha }+h^4)$ under the maximum norm by the energy argument. We prove the almost unconditional stability of the scheme under very mild conditions. Extending the numerical method to the multi-delay case is available. Extensive computational results are presented including single delay and double delay problems, which demonstrate the effectiveness and correctness of the developed schemes.

KEYWORDS:

• Fractional wave equation with delays
• spatial variable coefficient
• exponential transformation
• maximum error estimate
• numerical stability

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 65M06
• 65M12

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work was supported in part by Natural Science Foundation of China [grant number 11971010, 11501514], in part by Natural Sciences Foundation of Zhejiang Province [grant number LY19A010026], in part by China Postdoctoral Science Foundation [grant number 2018M642131], in part by Zhejiang Province ‘Yucai’ Project (2019) and in part by Fundamental Research Funds of Zhejiang Sci-Tech University [grant number 2019Q072]. The authors would like to thank the referees for their valuable comments and suggestions which improve the present paper.