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Numerical algorithms to estimate relaxation parameters and Caputo fractional derivative for a fractional thermal wave model in spherical composite medium

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  • Yu, Bo
  • Jiang, Xiaoyun
  • Wang, Chu

Abstract

In this paper, we formulate a fractional thermal wave model for a bi-layered spherical tissue. Implicit finite difference method is employed to obtain the solution of the direct problem. The inverse analysis for simultaneously estimating the Caputo fractional derivative and the relaxation time parameters is implemented by means of the Levenberg–Marquardt method. Compared with the experimental data, we can obviously find out that the estimated temperature increase values are excellently consistent with the measured temperature increase values in the experiment. We have also discussed the effect of the fractional derivative, the relaxation time parameters, the initial guess as well as the sensitivity problem. All the results show that the proposed fractional thermal wave model is efficient and accurate in modeling the heat transfer in the hyperthermia experiment, and the proposed numerical method for simultaneously estimating multiple parameters for the fractional thermal wave model in a spherical composite medium is effective.

Suggested Citation

  • Yu, Bo & Jiang, Xiaoyun & Wang, Chu, 2016. "Numerical algorithms to estimate relaxation parameters and Caputo fractional derivative for a fractional thermal wave model in spherical composite medium," Applied Mathematics and Computation, Elsevier, vol. 274(C), pages 106-118.
    • Handle: RePEc:eee:apmaco:v:274:y:2016:i:c:p:106-118
    DOI: 10.1016/j.amc.2015.10.081
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    References listed on IDEAS

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    1. Tian, WenYi & Li, Can & Deng, Weihua & Wu, Yujiang, 2012. "Regularization methods for unknown source in space fractional diffusion equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 85(C), pages 45-56.
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    Cited by:

    1. Zhang, Hui & Jiang, Xiaoyun & Yang, Xiu, 2018. "A time-space spectral method for the time-space fractional Fokker–Planck equation and its inverse problem," Applied Mathematics and Computation, Elsevier, vol. 320(C), pages 302-318.
    2. M. Bishehniasar & S. Salahshour & A. Ahmadian & F. Ismail & D. Baleanu, 2017. "An Accurate Approximate-Analytical Technique for Solving Time-Fractional Partial Differential Equations," Complexity, Hindawi, vol. 2017, pages 1-12, December.
    3. Shi, Z.G. & Zhao, Y.M. & Liu, F. & Wang, F.L. & Tang, Y.F., 2018. "Nonconforming quasi-Wilson finite element method for 2D multi-term time fractional diffusion-wave equation on regular and anisotropic meshes," Applied Mathematics and Computation, Elsevier, vol. 338(C), pages 290-304.
    4. Wang, Jinfeng & Yin, Baoli & Liu, Yang & Li, Hong & Fang, Zhichao, 2021. "Mixed finite element algorithm for a nonlinear time fractional wave model," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 188(C), pages 60-76.

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