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Robust finite difference scheme for the non-linear generalized time-fractional diffusion equation with non-smooth solution

Author

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  • Kedia, Nikki
  • Alikhanov, Anatoly A.
  • Singh, Vineet Kumar

Abstract

The present paper aims to develop a stable multistep numerical scheme for the non-linear generalized time-fractional diffusion equations (GTFDEs) with non-smooth solutions. Mesh grading technique is used to discretize the temporal direction, which results in 2−α order of convergence (0<α<1). The spatial direction is discretized using a second order difference operator and the non-linear term is approximated using Taylor’s series. Theoretical stability and convergence analysis is established in the L2-norm. Moreover, some random noise perturbations are added to investigate the numerical stability of the developed scheme. Finally, numerical simulations are performed on three test examples to verify the robustness and efficiency of the scheme.

Suggested Citation

  • Kedia, Nikki & Alikhanov, Anatoly A. & Singh, Vineet Kumar, 2024. "Robust finite difference scheme for the non-linear generalized time-fractional diffusion equation with non-smooth solution," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 219(C), pages 337-354.
    • Handle: RePEc:eee:matcom:v:219:y:2024:i:c:p:337-354
    DOI: 10.1016/j.matcom.2023.12.034
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    References listed on IDEAS

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    1. Robert J. Elliott & John Van Der Hoek, 2003. "A General Fractional White Noise Theory And Applications To Finance," Mathematical Finance, Wiley Blackwell, vol. 13(2), pages 301-330, April.
    2. Peter Carr & Hélyette Geman & Dilip B. Madan & Marc Yor, 2003. "Stochastic Volatility for Lévy Processes," Mathematical Finance, Wiley Blackwell, vol. 13(3), pages 345-382, July.
    3. D. Baleanu & A. H. Bhrawy & T. M. Taha, 2013. "Two Efficient Generalized Laguerre Spectral Algorithms for Fractional Initial Value Problems," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-10, June.
    4. Alikhanov, Anatoly A. & Huang, Chengming, 2021. "A high-order L2 type difference scheme for the time-fractional diffusion equation," Applied Mathematics and Computation, Elsevier, vol. 411(C).
    5. repec:dau:papers:123456789/1392 is not listed on IDEAS
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    More about this item

    Keywords

    Fractional derivative with generalized memory kernel; Non-smooth solution; Weight function; Non-linear; Generalized L1 scheme; Convergence and stability;
    All these keywords.

    JEL classification:

    • L1 - Industrial Organization - - Market Structure, Firm Strategy, and Market Performance

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