@article{Duc2022, 
author = {Phuong Nguyen Duc and Erkan Nane and Omid Nikan and Nguyen Anh Tuan},
title = {Approximation of the initial value for damped nonlinear hyperbolic equations with random Gaussian white noise on the measurements},
year = {2022},
journal = {AIMS Mathematics},
volume = {7},
number = {7},
pages = {12620-12634},
keywords = {wave equations, hyperbolic equations, Gaussian white noise, random noise, regularized solution, ill-posed},
url = {https://www.sciopen.com/article/10.3934/math.2022698},
doi = {10.3934/math.2022698},
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          t      t        +  α      Δ    2        u    t    +  β      Δ    2    u  =      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.}
}