Regularized solution for a biharmonic equation with discrete data

Tran Ngoc Thach; Nguyen Huy Tuan; Donal O'Regan

doi:10.3934/eect.2020008

Article Contents

2020, Volume 9, Issue 2: 341-358. Doi: 10.3934/eect.2020008

This issue Previous Article On the three dimensional Kelvin-Voigt fluids: global solvability, exponential stability and exact controllability of Galerkin approximations Next Article Decay rates for second order evolution equations in Hilbert spaces with nonlinear time-dependent damping

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

Received: December 31, 2018

Revised: March 31, 2019

Early access: August 2019

Published: June 2020

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Abstract

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.

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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)

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

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

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

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

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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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