scholarly journals On inverse problem for a class of fourth order strongly damped wave equations

2018 ◽  
Vol 20 ◽  
pp. 02006
Author(s):  
Nam Danh Hua Quoc ◽  
Can Nguyen Huu ◽  
Au Vo Van ◽  
Binh Tran Thanh
Keyword(s):  
Inverse Problem ◽  
Fourth Order ◽  
Wave Equations ◽  
A Priori ◽  
Linear Wave ◽  
Ill Posed ◽  
Convergence Estimates ◽  
Linear Wave Equations ◽  
In The Beginning ◽  
Strongly Damped

In this paper, we study the initial inverse problem for a class of fourth order strongly damped linear wave equations. In the beginning, we show that the problem is ill-posed in the sense of Hadamard. Next, we propose the method called: the Fourier truncation method for stabilizing the problem. Convergence estimates are established under a priori regularity assumptions on the problem data.

scholarly journals Construction and analysis of fourth order, energy consistent, family of explicit time discretizations for dissipative linear wave equations

2020 ◽  
Vol 54 (3) ◽  
pp. 845-878
Author(s):  
Juliette Chabassier ◽  
Julien Diaz ◽  
Sébastien Imperiale
Keyword(s):  
Convergence Analysis ◽  
Fourth Order ◽  
Wave Equations ◽  
Energy Analysis ◽  
Linear Wave ◽  
Convergence Properties ◽  
Order Energy ◽  
Linear Wave Equations ◽  
The Stability ◽  
Modified Equation

This paper deals with the construction of a family of fourth order, energy consistent, explicit time discretizations for dissipative linear wave equations. The schemes are obtained by replacing the inversion of a matrix, that comes naturally after using the technique of the Modified Equation on the second order Leap Frog scheme applied to dissipative linear wave equations, by explicit approximations of its inverse. The stability of the schemes are studied using an energy analysis and a convergence analysis is carried out. Numerical results in 1D illustrate the space/time convergence properties of the schemes and their efficiency is compared to more classical time discretizations.

LOCAL EXISTENCE FOR SEMILINEAR WAVE EQUATIONS AND APPLICATIONS TO YANG–MILLS EQUATIONS

2005 ◽  
Vol 02 (01) ◽  
pp. 61-76
Author(s):  
YUNG-FU FANG
Keyword(s):  
Initial Data ◽  
Wave Equations ◽  
Nonlinear Term ◽  
A Priori ◽  
Local Existence ◽  
A Priori Estimate ◽  
Linear Wave ◽  
Yang Mills ◽  
Linear Wave Equations ◽  
Local Result

In this work we are concerned with a local existence of certain semi-linear wave equations for which the initial data has minimal regularity. Assuming the initial data are in H1+∊ and H∊ for any ∊ > 0, we prove a local result by using a fixed point argument, the main ingredient being an a priori estimate for the quadratic nonlinear term uDu. The technique applies to the Yang–Mills equations in the Lorentz gauge.

On the inverse problem for nonlinear strongly damped wave equations with discrete random noise

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Nguyen Anh Triet ◽  
Nguyen Huy Tuan ◽  
Nguyen Duc Phuong ◽  
Donal O’ Regan
Keyword(s):  
Wave Equations ◽  
Random Noise ◽  
A Priori ◽  
Expansion Method ◽  
Unstable Solution ◽  
Existence Of A Solution ◽  
Parametric Regression ◽  
Backward Problem ◽  
Ill Posed ◽  
Strongly Damped

Abstract In this paper, we consider the existence of a solution u(x, t) for the inverse backward problem for the nonlinear strongly damped wave equation with statistics discrete data. The problem is severely ill-posed in the sense of Hadamard, i.e., the solution does not depend continuously on the data. In order to regularize the unstable solution, we use the trigonometric method in non-parametric regression associated with the truncated expansion method. We investigate the convergence rate under some a priori assumptions on an exact solution in both L 2 and H q (q > 0) norms. Moreover, a numerical example is given to illustrate our results.

Sign in / Sign up

Export Citation Format

Share Document