    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  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 & Control Theory, doi: 10.3934/eect.2020008
##### References:
  D. Adams, $L^p$ potential theory techniques and nonlinear PDE. Potential theory, Potential Theory (Nagoya, 1990), Berlin: de Gruyter, (1992), 1–15. Google Scholar  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.  Google Scholar  R. L. Eubank, Nonparametric Regression and Spline Smoothing, CRC Press, 1999. Google Scholar  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.  Google Scholar  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.  Google Scholar  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. Google Scholar  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.  Google Scholar  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. Google Scholar  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. Google Scholar  P. W. Schaefer, On existence in the Cauchy problem for the biharmonic equation, Compositio Math., 28 (1974), 203-207. Google Scholar  A. B. Tsybakov, Introduction to Nonparametric Estimation, Springer, New York, 2009. doi: 10.1007/b13794.  Google Scholar  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. Google Scholar

show all references

##### References:
  D. Adams, $L^p$ potential theory techniques and nonlinear PDE. Potential theory, Potential Theory (Nagoya, 1990), Berlin: de Gruyter, (1992), 1–15. Google Scholar  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.  Google Scholar  R. L. Eubank, Nonparametric Regression and Spline Smoothing, CRC Press, 1999. Google Scholar  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.  Google Scholar  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.  Google Scholar  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. Google Scholar  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.  Google Scholar  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. Google Scholar  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. Google Scholar  P. W. Schaefer, On existence in the Cauchy problem for the biharmonic equation, Compositio Math., 28 (1974), 203-207. Google Scholar  A. B. Tsybakov, Introduction to Nonparametric Estimation, Springer, New York, 2009. doi: 10.1007/b13794.  Google Scholar  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. Google Scholar The exact and regularized solutions at $y = 0$ (a) and its errors (b) The exact and regularized solutions at $y = 0.1$ (a) and its errors (b) The exact and regularized solutions at $y = 0.2$ (a) and its errors (b) The exact and regularized solutions at $y = 0.3$ (a) and its errors (b) The exact solution $u$ (a) and the regularized solution $\widehat v$ (b)
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
  Misha Perepelitsa. An ill-posed problem for the Navier-Stokes equations for compressible flows. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 609-623. doi: 10.3934/dcds.2010.26.609  Stefan Kindermann. Convergence of the gradient method for ill-posed problems. Inverse Problems & Imaging, 2017, 11 (4) : 703-720. doi: 10.3934/ipi.2017033  Peter I. Kogut, Olha P. Kupenko. On optimal control problem for an ill-posed strongly nonlinear elliptic equation with $p$-Laplace operator and $L^1$-type of nonlinearity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1273-1295. doi: 10.3934/dcdsb.2019016  Alfredo Lorenzi, Luca Lorenzi. A strongly ill-posed integrodifferential singular parabolic problem in the unit cube of $\mathbb{R}^n$. Evolution Equations & Control Theory, 2014, 3 (3) : 499-524. doi: 10.3934/eect.2014.3.499  Paola Favati, Grazia Lotti, Ornella Menchi, Francesco Romani. An inner-outer regularizing method for ill-posed problems. Inverse Problems & Imaging, 2014, 8 (2) : 409-420. doi: 10.3934/ipi.2014.8.409  Li Liang. Increasing stability for the inverse problem of the Schrödinger equation with the partial Cauchy data. Inverse Problems & Imaging, 2015, 9 (2) : 469-478. doi: 10.3934/ipi.2015.9.469  Hiroyuki Hirayama, Mamoru Okamoto. Random data Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6943-6974. doi: 10.3934/dcds.2016102  V. Varlamov, Yue Liu. Cauchy problem for the Ostrovsky equation. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 731-753. doi: 10.3934/dcds.2004.10.731  Adrien Dekkers, Anna Rozanova-Pierrat. Cauchy problem for the Kuznetsov equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 277-307. doi: 10.3934/dcds.2019012  Markus Haltmeier, Richard Kowar, Antonio Leitão, Otmar Scherzer. Kaczmarz methods for regularizing nonlinear ill-posed equations II: Applications. Inverse Problems & Imaging, 2007, 1 (3) : 507-523. doi: 10.3934/ipi.2007.1.507  Matthew A. Fury. Estimates for solutions of nonautonomous semilinear ill-posed problems. Conference Publications, 2015, 2015 (special) : 479-488. doi: 10.3934/proc.2015.0479  Matthew A. Fury. Regularization for ill-posed inhomogeneous evolution problems in a Hilbert space. Conference Publications, 2013, 2013 (special) : 259-272. doi: 10.3934/proc.2013.2013.259  Rudong Zheng, Zhaoyang Yin. The Cauchy problem for a generalized Novikov equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3503-3519. doi: 10.3934/dcds.2017149  Felix Lucka, Katharina Proksch, Christoph Brune, Nicolai Bissantz, Martin Burger, Holger Dette, Frank Wübbeling. Risk estimators for choosing regularization parameters in ill-posed problems - properties and limitations. Inverse Problems & Imaging, 2018, 12 (5) : 1121-1155. doi: 10.3934/ipi.2018047  Guozhi Dong, Bert Jüttler, Otmar Scherzer, Thomas Takacs. Convergence of Tikhonov regularization for solving ill-posed operator equations with solutions defined on surfaces. Inverse Problems & Imaging, 2017, 11 (2) : 221-246. doi: 10.3934/ipi.2017011  Youri V. Egorov, Evariste Sanchez-Palencia. Remarks on certain singular perturbations with ill-posed limit in shell theory and elasticity. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1293-1305. doi: 10.3934/dcds.2011.31.1293  Johann Baumeister, Barbara Kaltenbacher, Antonio Leitão. On Levenberg-Marquardt-Kaczmarz iterative methods for solving systems of nonlinear ill-posed equations. Inverse Problems & Imaging, 2010, 4 (3) : 335-350. doi: 10.3934/ipi.2010.4.335  Olha P. Kupenko, Rosanna Manzo. On optimal controls in coefficients for ill-posed non-Linear elliptic Dirichlet boundary value problems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1363-1393. doi: 10.3934/dcdsb.2018155  Sergiy Zhuk. Inverse problems for linear ill-posed differential-algebraic equations with uncertain parameters. Conference Publications, 2011, 2011 (Special) : 1467-1476. doi: 10.3934/proc.2011.2011.1467  Faker Ben Belgacem. Uniqueness for an ill-posed reaction-dispersion model. Application to organic pollution in stream-waters. Inverse Problems & Imaging, 2012, 6 (2) : 163-181. doi: 10.3934/ipi.2012.6.163

2018 Impact Factor: 1.048