AIMS Mathematics | |
Approximation of the initial value for damped nonlinear hyperbolic equations with random Gaussian white noise on the measurements | |
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
【 预 览 】
Files | Size | Format | View |
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RO202302200001926ZK.pdf | 729KB | download |