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Approximation of the initial value for damped nonlinear hyperbolic equations with random Gaussian white noise on the measurements
Approximation of the initial value for damped nonlinear hyperbolic equations with random Gaussian white noise on the measurements
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Date
2022
Authors
Phuong Nguyen Duc
Erkan Nane
Omid Nikan
Nguyen Anh Tuan
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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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Keywords
"wave equations,
hyperbolic equations,
Gaussian white noise,
random noise,
regularized solution,
ill-posed"