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A high-order L2-compact difference method for Caputo-type time-fractional sub-diffusion equations with variable coefficients

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  • Wang, Yuan-Ming
  • Ren, Lei

Abstract

A high-order compact finite difference method is proposed for solving a class of time-fractional sub-diffusion equations. The diffusion coefficient of the equation may be spatially variable and the time-fractional derivative is in the Caputo sense with the order α ∈ (0, 1). The Caputo time-fractional derivative is discretized by a (3−α) th-order numerical formula (called the L2 formula here) which is constructed by piecewise quadratic interpolating polynomials but does not require any sub-stepping scheme for the approximation at the first-time level. The variable coefficient spatial differential operator is approximated by a fourth-order compact finite difference operator. By developing a technique of discrete energy analysis, a full theoretical analysis of the stability and convergence of the method is carried out for the general case of variable coefficient and for all α ∈ (0, 1). The optimal error estimate is obtained in the L2 norm and shows that the proposed method has the temporal (3−α) th-order accuracy and the spatial fourth-order accuracy. Further approximations are also considered for enlarging the applicability of the method while preserving its high-order accuracy. Applications are given to three model problems, and numerical results are presented to demonstrate the theoretical analysis results.

Suggested Citation

  • Wang, Yuan-Ming & Ren, Lei, 2019. "A high-order L2-compact difference method for Caputo-type time-fractional sub-diffusion equations with variable coefficients," Applied Mathematics and Computation, Elsevier, vol. 342(C), pages 71-93.
    • Handle: RePEc:eee:apmaco:v:342:y:2019:i:c:p:71-93
    DOI: 10.1016/j.amc.2018.09.007
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    Citations

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    Cited by:

    1. 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).
    2. Adán J. Serna-Reyes & Jorge E. Macías-Díaz & Nuria Reguera, 2021. "A Convergent Three-Step Numerical Method to Solve a Double-Fractional Two-Component Bose–Einstein Condensate," Mathematics, MDPI, vol. 9(12), pages 1-22, June.
    3. Kolade M. Owolabi & Sonal Jain & Edson Pindza, 2024. "Investigating the Dynamic Behavior of Integer and Noninteger Order System of Predation with Holling’s Response," Mathematics, MDPI, vol. 12(10), pages 1-27, May.
    4. Ahmed S. Hendy & Jorge E. Macías-Díaz, 2020. "A Discrete Grönwall Inequality and Energy Estimates in the Analysis of a Discrete Model for a Nonlinear Time-Fractional Heat Equation," Mathematics, MDPI, vol. 8(9), pages 1-15, September.

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