scholarly journals Theoretical and numerical stability results for a viscoelastic swelling porous-elastic system with past history

2021 ◽  
Vol 6 (11) ◽  
pp. 11921-11949
Author(s):  
Adel M. Al-Mahdi ◽  
◽  
Mohammad M. Al-Gharabli ◽  
Mohamed Alahyane ◽  
◽  
...  
Keyword(s):  
Elastic System ◽  
Past History ◽  
Stability Result ◽  
Kernel Functions ◽  
Memory Term ◽  
Relaxation Kernel ◽  
Wave Speeds ◽  
Swelling Porous Media ◽  
Stability Results ◽  
Theoretical Results

<abstract><p>The purpose of this paper is to establish a general stability result for a one-dimensional linear swelling porous-elastic system with past history, irrespective of the wave speeds of the system. First, we establish an explicit and general decay result under a wider class of the relaxation (kernel) functions. The kernel in our memory term is more general and of a broader class. Further, we get a better decay rate without imposing some assumptions on the boundedness of the history data considered in many earlier results in the literature. We also perform several numerical tests to illustrate our theoretical results. Our output extends and improves some of the available results on swelling porous media in the literature.</p></abstract>

General decay of a nonlinear damping porous-elastic system with past history

2019 ◽  
Vol 65 (2) ◽  
pp. 249-275
Author(s):  
Houssem Eddine Khochemane ◽  
Lamine Bouzettouta ◽  
Salah Zitouni
Keyword(s):  
Elastic System ◽  
Past History ◽  
Nonlinear Damping ◽  
General Decay

A Reduced Second-Order Approach for Linear Viscoelastic Oscillators

2010 ◽  
Vol 77 (4) ◽  
Author(s):  
Sondipon Adhikari
Keyword(s):  
Past History ◽  
Kernel Functions ◽  
Second Order ◽  
Order System ◽  
Closed Form Expression ◽  
Internal Variables ◽  
Dynamical Response ◽  
Convolution Integrals ◽  
History Of ◽  
Second Order Systems

This paper proposes a new approach for the reduction in the model-order of linear multiple-degree-of-freedom viscoelastic systems via equivalent second-order systems. The assumed viscoelastic forces depend on the past history of motion via convolution integrals over kernel functions. Current methods to solve this type of problem normally use the state-space approach involving additional internal variables. Such approaches often increase the order of the eigenvalue problem to be solved and can become computationally expensive for large systems. Here, an approximate reduced second-order approach is proposed for this type of problems. The proposed approximation utilizes the idea of generalized proportional damping and expressions of approximate eigenvalues of the system. A closed-form expression of the equivalent second-order system has been derived. The new expression is obtained by elementary operations involving the mass, stiffness, and the kernel function matrix only. This enables one to approximately calculate the dynamical response of complex viscoelastic systems using the standard tools for conventional second-order systems. Representative numerical examples are given to verify the accuracy of the derived expressions.

scholarly journals Construction of modified Godunov-type schemes accurate at any Mach number for the compressible Euler system

2016 ◽  
Vol 26 (13) ◽  
pp. 2525-2615 ◽  
Author(s):  
S. Dellacherie ◽  
J. Jung ◽  
P. Omnes ◽  
P.-A. Raviart
Keyword(s):  
Wave Equation ◽  
Mach Number ◽  
Stability Result ◽  
Linear Analysis ◽  
Euler System ◽  
Linear Wave ◽  
Godunov Scheme ◽  
Linear Wave Equation ◽  
Number Problem ◽  
Theoretical Results

This paper is composed of three self-consistent sections that can be read independently of each other. In Sec. 1, we define and analyze the low Mach number problem through a linear analysis of a perturbed linear wave equation. Then, we show how to modify Godunov-type schemes applied to the linear wave equation to make this scheme accurate at any Mach number. This allows to define an all Mach correction and to propose a linear all Mach Godunov scheme for the linear wave equation. In Sec. 2, we apply the all Mach correction proposed in Sec. 1 to the case of the nonlinear barotropic Euler system when the Godunov-type scheme is a Roe scheme. A linear stability result is proposed and a formal asymptotic analysis justifies the construction in this nonlinear case by showing how this construction is related with the linear analysis of Sec. 1. At last, we apply in Sec. 3 the all Mach correction proposed in Sec. 1 in the case of the full Euler compressible system. Numerous numerical results proposed in Secs. 1–3 justify the theoretical results and show that the obtained all Mach Godunov-type schemes are both accurate and stable for all Mach numbers. We also underline that the proposed approach can be applied to other schemes and allows to justify other existing all Mach schemes.

scholarly journals Existence and stability results of a nonlinear Timoshenko equation with damping and source terms

2021 ◽  
pp. 2-2
Author(s):  
Amar Ouaoua ◽  
Aya Khaldi ◽  
Messaoud Maouni
Keyword(s):  
Galerkin Method ◽  
Integral Inequality ◽  
Local Existence ◽  
Stability Result ◽  
Initial Energy ◽  
Positive Initial Energy ◽  
Existence And Stability ◽  
The Stability ◽  
Damping And Source Terms ◽  
Stability Results

In this paper, we consider a nonlinear Timoshenko equation. First, we prove the local existence solution by the Faedo-Galerkin method, and, under suitable assumptions with positive initial energy, we prove that the local existence is global in time. Finally, the stability result is established based on Komornik?s integral inequality.

scholarly journals Uniform stability for a type III thermoelastic laminated beam with structural damping

2022 ◽  
Author(s):  
Fayssal Djellali
Keyword(s):  
Heat Flux ◽  
Exponential Stability ◽  
Semigroup Theory ◽  
Stability Result ◽  
Uniform Stability ◽  
Well Posedness ◽  
Structural Damping ◽  
Laminated Beam ◽  
Wave Speeds ◽  
Lack Of Exponential Stability

In this work, we consider a thermoelastic laminated beam with structural damping, where the heat flux is given by Green and Naghdi theories. We establish the well-posedness of the system using semigroup theory. Moreover, under the condition of equal wave speeds, we prove an exponential stability result for the considered system. In the case of lack of exponential stability we show that the solution decays polynomially.

scholarly journals Eigenvalue Problems for Exponential-Type Kernels

2020 ◽  
Vol 20 (1) ◽  
pp. 61-78
Author(s):  
Difeng Cai ◽  
Panayot S. Vassilevski
Keyword(s):  
Exponential Type ◽  
Eigenvalue Problems ◽  
Order Approximation ◽  
Mesh Size ◽  
Kernel Functions ◽  
Nyström Methods ◽  
Functions Of Exponential Type ◽  
Largest Eigenvalues ◽  
Independent Constant ◽  
Theoretical Results

AbstractWe study approximations of eigenvalue problems for integral operators associated with kernel functions of exponential type. We show convergence rate {\lvert\lambda_{k}-\lambda_{k,h}\rvert\leq C_{k}h^{2}} in the case of lowest order approximation for both Galerkin and Nyström methods, where h is the mesh size, {\lambda_{k}} and {\lambda_{k,h}} are the exact and approximate kth largest eigenvalues, respectively. We prove that the two methods are numerically equivalent in the sense that {|\lambda^{(G)}_{k,h}-\lambda^{(N)}_{k,h}|\leq Ch^{2}}, where {\lambda^{(G)}_{k,h}} and {\lambda^{(N)}_{k,h}} denote the kth largest eigenvalues computed by Galerkin and Nyström methods, respectively, and C is a eigenvalue independent constant. The theoretical results are accompanied by a series of numerical experiments.

Stability results for fixed point sets of α*- Ψ contractive multivalued mappings

Filomat ◽  
2017 ◽  
Vol 31 (12) ◽  
pp. 3665-3670
Author(s):  
Binayak Choudhury ◽  
Chaitali Bandyopadhyay ◽  
Rajendra Pant
Keyword(s):  
Fixed Point ◽  
Contraction Mapping ◽  
Stability Result ◽  
Multivalued Mappings ◽  
Point Sets ◽  
Generalized Contraction ◽  
Class Of Functions ◽  
Stability Results ◽  
Fixed Point Sets

In this paper, we established a stability result for fixed point sets associated with a sequence of multivalued mappings which belong to class of functions obtained by a multivalued extension of certain generalized contraction mapping. Certain other aspects of these mappings are already studied in the existing literatures. We also construct an illustrative example.

A stability result of a Timoshenko system in thermoelasticity of second sound with a delay term in the internal feedback

2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Djamel Ouchenane
Keyword(s):  
Heat Conduction ◽  
Stability Result ◽  
Well Posedness ◽  
Second Sound ◽  
Timoshenko System ◽  
Wave Speeds ◽  
One Dimensional ◽  
Delay Term ◽  
Semigroup Method ◽  
Cattaneo’S Law

AbstractIn this paper, we consider a one-dimensional linear thermoelastic system of Timoshenko type with a delay term in the feedback. The heat conduction is given by Cattaneo's law. Under an appropriate assumption between the weight of the delay and the weight of the damping, we prove the well-posedness of the problem. Furthermore, an exponential stability result is shown without the usual assumption on the wave speeds. To achieve our goals, we make use of the semigroup method and the energy method.

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