Theoretical and computational decay results for a Bresse system with one infinite memory in the longitudinal displacement

Mohamed Alahyane; Mohammad M. Al-Gharabli; Adel M. Al-Mahdi

doi:10.3934/eect.2022027

Article Contents

2023, Volume 12, Issue 1: 191-212. Doi: 10.3934/eect.2022027

This issue Previous Article A general decay result for the Cauchy problem of plate equations with memory Next Article Controllability results for Sobolev type

$ \psi - $

Hilfer fractional backward perturbed integro-differential equations in Hilbert space

Theoretical and computational decay results for a Bresse system with one infinite memory in the longitudinal displacement

1.
Arts et Metiers Institute of Technology, Centrale Lille, Junia, ULR2697 - L2EP, University of Lille, France
2.
The Preparatory Year Program, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
3.
IRC for construction and building materials, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

^* Corresponding author: Adel M. Al-Mahdi
^* Corresponding author: Adel M. Al-Mahdi

Received: June 30, 2021

Revised: February 28, 2022

Early access: May 2022

Published: February 2023

The second and third authors are supported by KFUPM under Project No. SB201012

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Abstract

In this paper, we consider a one-dimensional linear Bresse system with only one infinite memory term acting in the third equation (longitudinal displacements). Under a general condition on the memory kernel (relaxation function), we establish a decay estimate of the energy of the system. Our decay result extends and improves some decay rates obtained in the literature such as the one in [27], [4], [33], [58] and [34]. The proof is based on the energy method together with convexity arguments. Numerical simulations are given to illustrate the theoretical decay result.

Keywords:

Bresse system,
relaxation functions,
convexity,
stability,
finite element and finite difference methods.

Mathematics Subject Classification: Primary: 35B40, 93D20, 35L45, 74H40.

Citation:

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Figure 1. Test 1: The approximate solution $ (\varphi, \psi, w) $ in the $ x $-$ t $ plane

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Figure 2. Test 1: Energy decay

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Figure 3. Test 2: The approximate solution $ (\varphi, \psi, w) $ in the $ x $-$ t $ plane

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Figure 4. Test 2: Energy decay

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Figure 5. Test 3: The approximate solution $ (\varphi, \psi, w) $ in the $ x $-$ t $ plane

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Figure 6. Test 3: Energy decay

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