Approximation of the initial value for damped nonlinear hyperbolic equations with random Gaussian white noise on the measurements

Phuong Nguyen Duc; Erkan Nane; Omid Nikan; Nguyen Anh Tuan; Phuong Nguyen Duc; Erkan Nane; Omid Nikan; Nguyen Anh Tuan

doi:10.3934/math.2022698

AIMS Mathematics

2022, Volume 7, Issue 7: 12620-12634. doi: 10.3934/math.2022698

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Approximation of the initial value for damped nonlinear hyperbolic equations with random Gaussian white noise on the measurements

1.
Faculty of Fundamental Science, Industrial University of Ho Chi Minh City, Ho Chi Minh City, Vietnam
2.
Department of Mathematics and Statistics, Auburn University, USA
3.
School of Mathematics, Iran University of Science and Technology, Narmak, Tehran
4.
Division of Applied Mathematics, Science and Technology Advanced Institute, Van Lang University, Ho Chi Minh City, Vietnam
5.
Faculty of Technology, Van Lang University, Ho Chi Minh City, Vietnam

Received: 08 March 2022 Revised: 10 April 2022 Accepted: 17 April 2022 Published: 28 April 2022
MSC : 35L15, 35R60

The main goal of this work is to study a regularization method to reconstruct the solution of the backward non-linear hyperbolic equation $ u_{tt} + \alpha\Delta^2u_t +\beta \Delta ^2u = \mathcal{F}(x, t, u) $ come with the input data are blurred by random Gaussian white noise. We first prove that the considered problem is ill-posed (in the sense of Hadamard), i.e., the solution does not depend continuously on the data. Then we propose the Fourier truncation method for stabilizing the ill-posed problem. Base on some priori assumptions for the true solution we derive the error and a convergence rate between a mild solution and its regularized solutions. Also, a numerical example is provided to confirm the efficiency of theoretical results.
- wave equations,
- hyperbolic equations,
- Gaussian white noise,
- random noise,
- regularized solution,
- ill-posed
Citation: Phuong Nguyen Duc, Erkan Nane, Omid Nikan, Nguyen Anh Tuan. Approximation of the initial value for damped nonlinear hyperbolic equations with random Gaussian white noise on the measurements[J]. AIMS Mathematics, 2022, 7(7): 12620-12634. doi: 10.3934/math.2022698

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Abstract

The main goal of this work is to study a regularization method to reconstruct the solution of the backward non-linear hyperbolic equation $ u_{tt} + \alpha\Delta^2u_t +\beta \Delta ^2u = \mathcal{F}(x, t, u) $ come with the input data are blurred by random Gaussian white noise. We first prove that the considered problem is ill-posed (in the sense of Hadamard), i.e., the solution does not depend continuously on the data. Then we propose the Fourier truncation method for stabilizing the ill-posed problem. Base on some priori assumptions for the true solution we derive the error and a convergence rate between a mild solution and its regularized solutions. Also, a numerical example is provided to confirm the efficiency of theoretical results.

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