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

Author:

Duc Phuong Nguyen1,Nane Erkan2,Nikan Omid3,Tuan Nguyen Anh45

Affiliation:

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

Abstract

<abstract><p>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.</p></abstract>

Publisher

American Institute of Mathematical Sciences (AIMS)

Subject

General Mathematics

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