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Solving Inverse Conductivity Problems in Doubly Connected Domains by the Homogenization Functions of Two Parameters

Author

Listed:
  • Jun Lu

    (Nanjing Hydraulic Research Institute, Nanjing 210029, China)

  • Lianpeng Shi

    (College of Mechanics and Materials, Hohai University, Nanjing 210098, China)

  • Chein-Shan Liu

    (Center of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung 20224, Taiwan)

  • C. S. Chen

    (Department of Mathematics, University of Southern Mississippi, Hattiesburg, MS 39406, USA)

Abstract

In the paper, we make the first attempt to derive a family of two-parameter homogenization functions in the doubly connected domain, which is then applied as the bases of trial solutions for the inverse conductivity problems. The expansion coefficients are obtained by imposing an extra boundary condition on the inner boundary, which results in a linear system for the interpolation of the solution in a weighted Sobolev space. Then, we retrieve the spatial- or temperature-dependent conductivity function by solving a linear system, which is obtained from the collocation method applied to the nonlinear elliptic equation after inserting the solution. Although the required data are quite economical, very accurate solutions of the space-dependent and temperature-dependent conductivity functions, the Robin coefficient function and also the source function are available. It is significant that the nonlinear inverse problems can be solved directly without iterations and solving nonlinear equations. The proposed method can achieve accurate results with high efficiency even for large noise being imposed on the input data.

Suggested Citation

  • Jun Lu & Lianpeng Shi & Chein-Shan Liu & C. S. Chen, 2022. "Solving Inverse Conductivity Problems in Doubly Connected Domains by the Homogenization Functions of Two Parameters," Mathematics, MDPI, vol. 10(13), pages 1-17, June.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:13:p:2256-:d:849221
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    References listed on IDEAS

    as
    1. Xiao-Xiao Li & Heng Zhen Guo & Shi Min Wan & Fan Yang, 2012. "Inverse Source Identification by the Modified Regularization Method on Poisson Equation," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-13, January.
    2. Ma, Yong-Ki & Prakash, P. & Deiveegan, A., 2018. "Generalized Tikhonov methods for an inverse source problem of the time-fractional diffusion equation," Chaos, Solitons & Fractals, Elsevier, vol. 108(C), pages 39-48.
    3. Djennadi, Smina & Shawagfeh, Nabil & Abu Arqub, Omar, 2021. "A fractional Tikhonov regularization method for an inverse backward and source problems in the time-space fractional diffusion equations," Chaos, Solitons & Fractals, Elsevier, vol. 150(C).
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