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Initial-boundary value problems for multi-term time-fractional diffusion equations with positive constant coefficients

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  • Li, Zhiyuan
  • Liu, Yikan
  • Yamamoto, Masahiro

Abstract

In this paper, we investigate the well-posedness and the long-time asymptotic behavior for initial-boundary value problems for multi-term time-fractional diffusion equations. The governing equation under consideration includes a linear combination of Caputo derivatives in time with decreasing orders in (0,1) and positive constant coefficients. By exploiting several important properties of multinomial Mittag–Leffler functions, various estimates follow from the explicit solutions in form of these special functions. Then we prove the uniqueness and continuous dependency on initial values and source terms, from which we further verify the Lipschitz continuous dependency of solutions with respect to coefficients and orders of fractional derivatives. Finally, by a Laplace transform argument, it turns out that the decay rate of the solution as t→∞ is given by the minimum order of the time-fractional derivatives.

Suggested Citation

  • Li, Zhiyuan & Liu, Yikan & Yamamoto, Masahiro, 2015. "Initial-boundary value problems for multi-term time-fractional diffusion equations with positive constant coefficients," Applied Mathematics and Computation, Elsevier, vol. 257(C), pages 381-397.
    • Handle: RePEc:eee:apmaco:v:257:y:2015:i:c:p:381-397
    DOI: 10.1016/j.amc.2014.11.073
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    References listed on IDEAS

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    1. Giona, Massimiliano & Cerbelli, Stefano & Roman, H.Eduardo, 1992. "Fractional diffusion equation and relaxation in complex viscoelastic materials," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 191(1), pages 449-453.
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    Cited by:

    1. Masahiro Yamamoto, 2022. "Fractional Calculus and Time-Fractional Differential Equations: Revisit and Construction of a Theory," Mathematics, MDPI, vol. 10(5), pages 1-55, February.
    2. Lele Yuan & Kewei Liang & Huidi Wang, 2023. "Solving Inverse Problem of Distributed-Order Time-Fractional Diffusion Equations Using Boundary Observations and L 2 Regularization," Mathematics, MDPI, vol. 11(14), pages 1-20, July.
    3. Xiaozhong Yang & Lifei Wu, 2020. "A New Kind of Parallel Natural Difference Method for Multi-Term Time Fractional Diffusion Model," Mathematics, MDPI, vol. 8(4), pages 1-19, April.
    4. She, Mianfu & Li, Dongfang & Sun, Hai-wei, 2022. "A transformed L1 method for solving the multi-term time-fractional diffusion problem," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 193(C), pages 584-606.
    5. Morales-Delgado, V.F. & Taneco-Hernández, M.A. & Vargas-De-León, Cruz & Gómez-Aguilar, J.F., 2023. "Exact solutions to fractional pharmacokinetic models using multivariate Mittag-Leffler functions," Chaos, Solitons & Fractals, Elsevier, vol. 168(C).
    6. Yuriy Povstenko, 2021. "Some Applications of the Wright Function in Continuum Physics: A Survey," Mathematics, MDPI, vol. 9(2), pages 1-14, January.
    7. Wang, Yuan-Ming & Wen, Xin, 2020. "A compact exponential difference method for multi-term time-fractional convection-reaction-diffusion problems with non-smooth solutions," Applied Mathematics and Computation, Elsevier, vol. 381(C).

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