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A Discrete Grönwall Inequality and Energy Estimates in the Analysis of a Discrete Model for a Nonlinear Time-Fractional Heat Equation

Author

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  • Ahmed S. Hendy

    (Department of Computational Mathematics and Computer Science, Institute of Natural Sciences and Mathematics, Ural Federal University, 19 Mira St., 620002 Yekaterinburg, Russia
    Department of Mathematics, Faculty of Science, Benha University, Benha 13511, Egypt)

  • Jorge E. Macías-Díaz

    (Department of Mathematics, School of Digital Technologies, Tallinn University, Narva Rd. 25, 10120 Tallinn, Estonia
    Departamento de Matemáticas y Física, Centro de Ciencias Básicas, Universidad Autónoma de Aguascalientes, Avenida Universidad 940, Ciudad Universitaria, Aguascalientes 20121, Ags., Mexico)

Abstract

In the present work, we investigate the efficiency of a numerical scheme to solve a nonlinear time-fractional heat equation with sufficiently smooth solutions, which was previously reported in the literature [Fract. Calc. Appl. Anal. 16 : 892–910 (2013)]. In that article, the authors established the stability and consistency of the discrete model using arguments from Fourier analysis. As opposed to that work, in the present work, we use the method of energy inequalities to show that the scheme is stable and converges to the exact solution with order O ( τ 2 − α + h 4 ) , in the case that 0 < α < 1 satisfies 3 α ≥ 3 2 , which means that 0.369 ⪅ α ≤ 1 . The novelty of the present work lies in the derivation of suitable energy estimates, and a discrete fractional Grönwall inequality, which is consistent with the discrete approximation of the Caputo fractional derivative of order 0 < α < 1 used for that scheme at t k + 1 / 2 .

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:9:p:1539-:d:410864
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    References listed on IDEAS

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    1. Li, Lili & Zhou, Boya & Chen, Xiaoli & Wang, Zhiyong, 2018. "Convergence and stability of compact finite difference method for nonlinear time fractional reaction–diffusion equations with delay," Applied Mathematics and Computation, Elsevier, vol. 337(C), pages 144-152.
    2. 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.
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