Abstract
In this paper, we study for the first time the inverse initial problem for the one-dimensional strongly damped wave with Gaussian white noise data. Under some a priori assumptions on the true solution, we propose the Fourier truncation method for stabilizing the ill-posed problem. Error estimates are given in both the L2– and Hp–norms.
Highlights
Let T be a positive number and D = (0, π)
We are interested in the problem of recovering the initial state u(x, 0), x ∈ D, for the following strongly damped wave equation utt − αuxxt − uxx = 0, (x, t) ∈ D × (0, T ), (1.1)
We cannot measure g exactly, but we observe with the presence of a Gaussian white noise process ξ gobs(x) = g(x) + ξ(x), (1.4)
Summary
Where > 0 is the amplitude of the noise It can only be observed in discretized form: gobs, φ j = g, φ j + ξ, φ j , j = 1, N,. Damped wave equation (SDWE) occurs in a wide range of applications such as modeling motion of viscoelastic materials [2, 7, 8]. From both the theoretical and numerical points of view, the initial value problem has been extensively studied (see e.g., [3, 5, 9]). The major object of this paper is to propose a stable regularized solution for the problem (1.1)-(1.3) using the Fourier truncation method
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
