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June  2020, 9(2): 341-358. doi: 10.3934/eect.2020008

Regularized solution for a biharmonic equation with discrete data

1. 

Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

2. 

Institute of Research and Development, Duy Tan University, Da Nang 550000, Vietnam

3. 

School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland

Corresponding author: thnguyen2683@gmail.com(Nguyen Huy Tuan)

Received  January 2019 Revised  April 2019 Published  June 2020 Early access  August 2019

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.

Citation: Tran Ngoc Thach, Nguyen Huy Tuan, Donal O'Regan. Regularized solution for a biharmonic equation with discrete data. Evolution Equations and Control Theory, 2020, 9 (2) : 341-358. doi: 10.3934/eect.2020008
References:
[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.

show all references

References:
[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.

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
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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