Figure 1.
The exact and regularized solutions at $ y = 0 $ (a) and its errors (b)
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On the three dimensional Kelvin-Voigt fluids: global solvability, exponential stability and exact controllability of Galerkin approximations
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 |
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.
Keywords:
Ill-posed problem,
regularized method,
estimate,
biharmonic equation,
Cauchy problem,
discrete data.
Mathematics Subject Classification:
35K05, 35K99, 47J06, 47H10.
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
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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 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)
Table 1.
The errors between the exact solution and the regularized solution at $ y \in \{0.1, \, 0.2, \, 0.3\} $
| Errors | |||
| 0.016631443540398 | 0.017785494891671 | 0.016885191385811 | |
| 0.018572286169882 | 0.016934217726270 | 0.017559567667446 | |
| 0.012505879576731 | 0.012581413283707 | 0.015940532272324 |
| Errors | |||
| 0.016631443540398 | 0.017785494891671 | 0.016885191385811 | |
| 0.018572286169882 | 0.016934217726270 | 0.017559567667446 | |
| 0.012505879576731 | 0.012581413283707 | 0.015940532272324 |
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