Article Contents
Article Contents

# Regularized solution for a biharmonic equation with discrete data

• 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.

 Citation:

• 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
•  [1] D. Adams, $L^p$ potential theory techniques and nonlinear PDE. Potential theory, Potential Theory (Nagoya, 1990), Berlin: de Gruyter, (1992), 1–15. [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. Eubank,  Nonparametric Regression and Spline Smoothing, CRC Press, 1999. [4] D. Lesnic and A. Zeb, The method of fundamental solutions for an inverse internal boundary value problem for the biharmonic equation, Int. J. Comput. Methods, 6 (2009), 557-567.  doi: 10.1142/S0219876209001991. [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. [6] T. N. Luan, T. T. Khieu and T. Q. Khanh, Regularized solution of the cauchy problem for the biharmonic equation, Bulletin of the Malaysian Mathematical Sciences Society, to appear, (2018), 1–26. doi: 10.1007/s40840-018-00711-7. [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. [9] S. G. Mikhlin, Integral Equations and Their Applications to Certain Problems in Mechanics, Mathematical Physics and Technology, ed. 2, The Macmillan Co., New York, 1964. [10] P. W. Schaefer, On existence in the Cauchy problem for the biharmonic equation, Compositio Math., 28 (1974), 203-207. [11] A. B. Tsybakov, Introduction to Nonparametric Estimation, Springer, New York, 2009. doi: 10.1007/b13794. [12] A. Zeb, L. Elliott, D. B. Ingham and D. Lesnic, Cauchy problem for the biharmonic equation solved using the regularization method, Boundary Element Research in Europe (Southampton, 1998), Comput. Mech., Southampton, (1998), 285–294.

Figures(5)

Tables(1)