scholarly journals Lp-asymptotic stability of 1D damped wave equations with localized and linear damping

2021 ◽  
Author(s):  
Meryem Kafnemer ◽  
Benmiloud Mebkhout ◽  
Frédéric Jean ◽  
Yacine Chitour
Keyword(s):  
Asymptotic Stability ◽  
Wave Equations ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Semi Group ◽  
One Dimensional ◽  
Exponentially Stable ◽  
Linear Damping ◽  
The One ◽  
Well Posed

<p>In this paper, we study the L<sup>p</sup>-asymptotic stability of the one dimensional linear damped<br />wave equation with Dirichlet boundary conditions in <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math>, with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>&#8712;</mo><mo>(</mo><mn>1</mn><mo>,</mo><mo>&#8734;</mo><mo>)</mo></math>. The damping<br />term is assumed to be linear and localized&nbsp; to an arbitrary open sub-interval of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math>. We prove that the&nbsp;<br />semi-group <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mi>p</mi></msub><mo>(</mo><mi>t</mi><msub><mo>)</mo><mrow><mi>t</mi><mo>&#8805;</mo><mn>0</mn></mrow></msub></math> associated with the previous equation is well-posed and exponentially stable.<br />The proof relies on the multiplier method and depends on whether <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>&#8805;</mo><mn>2</mn></math>&nbsp;or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>&#60;</mo><mi>p</mi><mo>&#60;</mo><mn>2</mn></math>.</p>

scholarly journals On a stochastic Burgers equation with Dirichlet boundary conditions

2003 ◽  
Vol 2003 (43) ◽  
pp. 2735-2746 ◽  
Author(s):  
Ekaterina T. Kolkovska
Keyword(s):  
Weak Solution ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Burgers Equation ◽  
Martingale Problem ◽  
Weak Limit ◽  
Dirichlet Boundary Conditions ◽  
One Dimensional ◽  
Stochastic Burgers Equation ◽  
The One

We consider the one-dimensional Burgers equation perturbed by a white noise term with Dirichlet boundary conditions and a non-Lipschitz coefficient. We obtain existence of a weak solution proving tightness for a sequence of polygonal approximations for the equation and solving a martingale problem for the weak limit.

scholarly journals Time periodic solutions of one-dimensional forced Kirchhoff equations with x -dependent coefficients

2018 ◽  
Vol 474 (2213) ◽  
pp. 20170620 ◽  
Author(s):  
Mu Ma ◽  
Shuguan Ji
Keyword(s):  
Periodic Solutions ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Kirchhoff Equations ◽  
One Dimensional ◽  
Period 2 ◽  
Time Periodic Solutions ◽  
The One ◽  
Time Periodic ◽  
Iteration Technique

In this paper, we consider the one-dimensional Kirchhoff equation with x -dependent coefficients under Dirichlet boundary conditions, which models the forced vibrations of a clamped inhomogeneous string in the presence of a time periodic external forcing with period 2 π / ω and amplitude ϵ . By using the Nash–Moser iteration technique, we obtain the existence, regularity and local uniqueness of time periodic solutions with period 2 π / ω and order ϵ . Such results hold for parameters ( ω , ϵ ) in a positive measure Cantor set that has asymptotically full measure as the amplitude ϵ goes to zero.

Initial-boundary value problems for the one-dimensional linear advection–dispersion equation with decay

2020 ◽  
Vol 75 (8) ◽  
pp. 713-725 ◽  
Author(s):  
Guenbo Hwang
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
One Dimensional ◽  
Advection Dispersion ◽  
Asymptotically Periodic ◽  
The One ◽  
Initial Boundary Value

AbstractInitial-boundary value problems for the one-dimensional linear advection–dispersion equation with decay (LAD) are studied by utilizing a unified method, known as the Fokas method. The method takes advantage of the spectral analysis of both parts of Lax pair and the global algebraic relation coupling all initial and boundary values. We present the explicit analytical solution of the LAD equation posed on the half line and a finite interval with general initial and boundary conditions. In addition, for the case of periodic boundary conditions, we show that the solution of the LAD equation is asymptotically t-periodic for large t if the Dirichlet boundary datum is periodic in t. Furthermore, it can be shown that if the Dirichlet boundary value is asymptotically periodic for large t, then so is the unknown Neumann boundary value, which is uniquely characterized in terms of the given asymptotically periodic Dirichlet boundary datum. The analytical predictions for large t are compared with numerical results showing the excellent agreement.

A numerical algorithm based on modified cubic trigonometric B-spline functions for computational modelling of hyperbolic-type wave equations

2017 ◽  
Vol 34 (4) ◽  
pp. 1257-1276 ◽  
Author(s):  
Ali Saleh Alshomrani ◽  
Sapna Pandit ◽  
Abdullah K. Alzahrani ◽  
Metib Said Alghamdi ◽  
Ram Jiwari
Keyword(s):  
Numerical Algorithm ◽  
Wave Equations ◽  
Dirichlet Boundary ◽  
Computational Modelling ◽  
Hyperbolic Type ◽  
Dirichlet Boundary Conditions ◽  
Spline Functions ◽  
Content Type ◽  
B Spline ◽  
Type Wave

Purpose The main purpose of this work is the development of a numerical algorithm based on modified cubic trigonometric B-spline functions for computational modelling of hyperbolic-type wave equations. These types of equations describe a variety of physical models in the vibrations of structures, nonlinear optics, quantum field theory and solid-state physics, etc. Design/methodology/approach Dirichlet boundary conditions cannot be handled easily by cubic trigonometric B-spline functions. Then, a modification is made in cubic trigonometric B-spline functions to handle the Dirichlet boundary conditions and a numerical algorithm is developed. The proposed algorithm reduced the hyperbolic-type wave equations into a system of first-order ordinary differential equations (ODEs) in time variable. Then, stability-preserving SSP-RK54 scheme and the Thomas algorithm are used to solve the obtained system. The stability of the algorithm is also discussed. Findings A different technique based on modified cubic trigonometric B-spline functions is proposed which is quite different from the schemes developed (Abbas et al., 2014; Nazir et al., 2016) and depicts the computational modelling of hyperbolic-type wave equations. Originality/value To the best of the authors’ knowledge, this technique is novel for solving hyperbolic-type wave equations and the developed algorithm is free from quasi-linearization process and finite difference operators for time derivatives. This algorithm gives better results than the results discussed in literature (Dehghan and Shokri, 2008; Batiha et al., 2007; Mittal and Bhatia, 2013; Jiwari, 2015).

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