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Asymptotic Stability of Compact and Linear \(\theta \)-Methods for Space Fractional Delay Generalized Diffusion Equation

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Abstract

It is essential to explore the asymptotic stability of the numerical methods for fractional differential equations with delay since their exact solutions are unavailable or difficult to obtain. In this paper, we firstly derive a sufficient condition of the exact solution to be delay-independently asymptotically stable for the space fractional generalized diffusion equation with delay. Next, by applying the linear \(\theta \)-method to temporal dimension, we have established two classes of finite difference schemes independent of the delay for that, which have second-order and fourth-order accuracy in spatial dimension respectively. The unique solvability is proved and the local truncation error is derived. Furthermore, when \(\theta \in [0,1/2)\), we have obtained the necessary and sufficient condition of the asymptotic stability of both schemes under the specific conditions for the temporal step size \(\Delta t\) and the spatial step size h. When \(\theta \in [1/2,1]\), both schemes are unconditionally asymptotically stable. Moreover, the convergence results in the maximum norm are obtained according to the consistency analysis and the Lax theorem. The numerical examples are implemented to verify the theoretical results, which demonstrate that the asymptotic stabilities of both schemes are depending on the parameter \(\theta \) and the fractional derivative \(\alpha \), however, independent of the delay \(\tau \).

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Acknowledgements

The authors are grateful to the editor and the anonymous reviewers for their careful reading and many patient checking of the whole manuscript. Their valuable suggestions and comments significantly improve the quality of the manuscript.

Funding

This work was supported by Natural Science Foundation of China (Grant No. 11501514), Natural Science Foundation of Zhejiang Province (Grant No. LY19A010026), Zhejiang Province “Yucai” Project (Grant No. 2019YCGC012), Visiting Scholar Program of China Scholarship Council (Grant No. 201908330528), and Fundamental Research Funds of Zhejiang Sci-Tech University (Grant No. 2019Q072).

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  1. Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou, 310018, China

    Qifeng Zhang & Tingyue Li

  2. Department of Mathematics, Southeast University, Nanjing, 210096, China

    Qifeng Zhang

  3. School of Mathematics, Shanghai University of Finance and Economics, Shanghai, 200433, China

    Tingyue Li

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Zhang, Q., Li, T. Asymptotic Stability of Compact and Linear \(\theta \)-Methods for Space Fractional Delay Generalized Diffusion Equation. J Sci Comput 81, 2413–2446 (2019). https://doi.org/10.1007/s10915-019-01091-1

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