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

AIMS Mathematics

article

Phuong Nguyen Duc¹  Erkan Nane²  Omid Nikan³  Nguyen Anh Tuan⁴ 

[1] Faculty of Fundamental Science, Industrial University of Ho Chi Minh City;Department of Mathematics and Statistics, Auburn University;School of Mathematics, Iran University of Science and Technology;Division of Applied Mathematics, Science and Technology Advanced Institute, Van Lang University;Faculty of Technology, Van Lang University

关键词: wave equations;    hyperbolic equations;    Gaussian white noise;    random noise;    regularized solution;    ill-posed;   

DOI  :  10.3934/math.2022698

学科分类：地球科学(综合)

来源: AIMS Press

【 摘 要 】

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.

【 授权许可】

CC BY