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Regularized solution for a biharmonic equation with discrete data

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  • In this work, we focus on the Cauchy problem for the biharmonic equation associated with random data. In general, the problem is severely ill-posed in the sense of Hadamard, i.e, the solution does not depend continuously on the data. To regularize the instable solution of the problem, we apply a nonparametric regression associated with the Fourier truncation method. Also we will present a convergence result.

    Mathematics Subject Classification: 35K05, 35K99, 47J06, 47H10.


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  • Figure 1.  The exact and regularized solutions at $ y = 0 $ (a) and its errors (b)

    Figure 2.  The exact and regularized solutions at $ y = 0.1 $ (a) and its errors (b)

    Figure 3.  The exact and regularized solutions at $ y = 0.2 $ (a) and its errors (b)

    Figure 4.  The exact and regularized solutions at $ y = 0.3 $ (a) and its errors (b)

    Figure 5.  The exact solution $ u $ (a) and the regularized solution $ \widehat v $ (b)

    Table 1.  The errors between the exact solution and the regularized solution at $ y \in \{0.1, \, 0.2, \, 0.3\} $

    Errors $ n = 20 $ $ n=50 $ $ n=100 $
    $ \mathrm{Err}(0.1) $ 0.016631443540398 0.017785494891671 0.016885191385811
    $ \mathrm{Err}(0.2) $ 0.018572286169882 0.016934217726270 0.017559567667446
    $ \mathrm{Err}(0.3) $ 0.012505879576731 0.012581413283707 0.015940532272324
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    [2] A. Benrabah and N. Bousetila, Modified nonlocal boundary value problem method for an ill-posed problem for the biharmonic equation, Inverse Problem in Science and Engnineering, 27 (2019), 340–368, https://doi.org/10.1080/17415977.2018.1461859. doi: 10.1080/17415977.2018.1461859.
    [3] R. L. EubankNonparametric Regression and Spline Smoothing, CRC Press, 1999. 
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    [5] J. C. Li, Application of radial basis meshless methods to direct and inverse biharmonic boundary value problems, Commun. Numer. Methods Eng., 21 (2005), 169-182.  doi: 10.1002/cnm.736.
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    [7] L. Marin and D. Lesnic, The method of fundamental solutions for inverse boundary value problems associated with the two-dimensional biharmonic equation, Math. Comput. Modelling, 42 (2005), 261-278.  doi: 10.1016/j.mcm.2005.04.004.
    [8] V. V. Meleshko, Selected topics in the history of the two-dimensional biharmonic problem, Appl. Mech. Rev., 56 (2003), 33-85.  doi: 10.1115/1.1521166.
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    [11] A. B. Tsybakov, Introduction to Nonparametric Estimation, Springer, New York, 2009. doi: 10.1007/b13794.
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