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Long time behavior of Robin boundary sub-diffusion equation with fractional partial derivatives of Caputo type in differential and difference settings

Author

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  • Hendy, Ahmed S.
  • Zaky, Mahmoud A.
  • Abbaszadeh, Mostafa

Abstract

A Robin boundary sub-diffusion equation is considered with fractional partial derivatives of the Caputo type. The model is an extension of various well-known equations from mathematical physics, biology, and chemistry. Initial–boundary data are imposed upon a closed and bounded spatial domain. We state and prove two main theorems in differential and difference settings to ensure the algebraic decay rate of the long-time behavior for that kind of problem. The dissipation of the continuous solution for such a problem is discussed in the first theorem based on energy inequalities and by the aid of Grönwall inequalities. It demonstrates that with an L2(Ω)-bounded absorbing set, the solution is dissipated with respect to time. The numerical dissipativity is proved in the second theorem by using discrete energy inequalities and the discrete Paley–Wiener inequality. Finally, an example is provided to illustrate the main outcomes.

Suggested Citation

  • Hendy, Ahmed S. & Zaky, Mahmoud A. & Abbaszadeh, Mostafa, 2021. "Long time behavior of Robin boundary sub-diffusion equation with fractional partial derivatives of Caputo type in differential and difference settings," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 190(C), pages 1370-1378.
    • Handle: RePEc:eee:matcom:v:190:y:2021:i:c:p:1370-1378
    DOI: 10.1016/j.matcom.2021.07.006
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    Cited by:

    1. Sarita Nandal & Mahmoud A. Zaky & Rob H. De Staelen & Ahmed S. Hendy, 2021. "Numerical Simulation for a Multidimensional Fourth-Order Nonlinear Fractional Subdiffusion Model with Time Delay," Mathematics, MDPI, vol. 9(23), pages 1-15, November.
    2. Xu, Da, 2023. "The long time error estimates for the second order backward difference approximation to sub-diffusion equations with boundary time delay and feedback gain," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 208(C), pages 186-206.

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