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A radial basis function (RBF)-finite difference (FD) method for the backward heat conduction problem

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  • Su, LingDe

Abstract

In this paper a numerical scheme based on the idea of radial basis function finite difference (RBF-FD) technique is considered to solve the backward heat conduction problems (BHCP). In the meshless numerical method of RBF-FD, according to the finite difference technique we approximate the required derivatives for every point xi ∈ Ω in the corresponding local-support domain Ωi. Then the partial differential equation problem is transformed into the problem of a linear system of algebraic equations. This method also belongs to localized radial basis function method or the closest point method. To compare RBF-FD method with another RBF technique, radial basis function collocation method (RBFCM) and the method of approximate particular solutions (MAPS) are also considered to solve such inverse problem, and in the computation the standard Tikhonov regularization technique with L-curve method for choose optional regularized parameter is used for solving the highly ill condition system of linear equations. Several numerical examples are presented to demonstrate the ability of the present approach for solving the backward heat conduction problem.

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  • Su, LingDe, 2019. "A radial basis function (RBF)-finite difference (FD) method for the backward heat conduction problem," Applied Mathematics and Computation, Elsevier, vol. 354(C), pages 232-247.
    • Handle: RePEc:eee:apmaco:v:354:y:2019:i:c:p:232-247
    DOI: 10.1016/j.amc.2019.02.035
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    References listed on IDEAS

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    1. Biazar, Jafar & Hosami, Mohammad, 2017. "An interval for the shape parameter in radial basis function approximation," Applied Mathematics and Computation, Elsevier, vol. 315(C), pages 131-149.
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    Cited by:

    1. Zakieh Avazzadeh & Omid Nikan & José A. Tenreiro Machado, 2020. "Solitary Wave Solutions of the Generalized Rosenau-KdV-RLW Equation," Mathematics, MDPI, vol. 8(9), pages 1-20, September.
    2. Li-Dan Hong & Cheng-Yu Ku & Chih-Yu Liu, 2022. "A Novel Space-Time Marching Method for Solving Linear and Nonlinear Transient Problems," Mathematics, MDPI, vol. 10(24), pages 1-16, December.
    3. Xubiao He & Pu Gong, 2020. "A Radial Basis Function-Generated Finite Difference Method to Evaluate Real Estate Index Options," Computational Economics, Springer;Society for Computational Economics, vol. 55(3), pages 999-1019, March.
    4. Yang Yu & Xiaochuan Luo & Huaxi (Yulin) Zhang & Qingxin Zhang, 2019. "The Solution of Backward Heat Conduction Problem with Piecewise Linear Heat Transfer Coefficient," Mathematics, MDPI, vol. 7(5), pages 1-17, April.
    5. Nikan, O. & Avazzadeh, Z., 2021. "A localisation technique based on radial basis function partition of unity for solving Sobolev equation arising in fluid dynamics," Applied Mathematics and Computation, Elsevier, vol. 401(C).
    6. Oruç, Ömer, 2021. "A radial basis function finite difference (RBF-FD) method for numerical simulation of interaction of high and low frequency waves: Zakharov–Rubenchik equations," Applied Mathematics and Computation, Elsevier, vol. 394(C).
    7. Yue, Xiaohui & Shao, Xingling & Li, Jie, 2021. "Prescribed chattering reduction control for quadrotors using aperiodic signal updating," Applied Mathematics and Computation, Elsevier, vol. 405(C).

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