ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik

Volume 105, Issue 2 e202400123

ORIGINAL PAPER

Theoretical study and numerical test for a delayed thermoelastic porous system with microtemperatures

Mouataz Billah Mesmouli,

Mouataz Billah Mesmouli

orcid.org/0000-0002-3963-6503

Department of Mathematics, College of Science, University of Ha'il, Ha'il, 2440 Saudi Arabia

Search for more papers by this author

Houssem Eddine Khochemane,

Corresponding Author

Houssem Eddine Khochemane

[email protected]

Department of Mathematics, Ecole Normale Supérieure d'Enseignement Technologique de Skikda, Skikda, Algeria

Correspondence

Houssem Eddine Khochemane, Ecole Normale Supérieure d'Enseignement Technologique de Skikda, Skikda, Algeria.

Email: [email protected]

Search for more papers by this author

Mouataz Billah Mesmouli,

Mouataz Billah Mesmouli

orcid.org/0000-0002-3963-6503

Department of Mathematics, College of Science, University of Ha'il, Ha'il, 2440 Saudi Arabia

Search for more papers by this author

Houssem Eddine Khochemane,

Corresponding Author

Houssem Eddine Khochemane

[email protected]

Department of Mathematics, Ecole Normale Supérieure d'Enseignement Technologique de Skikda, Skikda, Algeria

Correspondence

Houssem Eddine Khochemane, Ecole Normale Supérieure d'Enseignement Technologique de Skikda, Skikda, Algeria.

Email: [email protected]

Search for more papers by this author

First published: 31 December 2024

https://doi.org/10.1002/zamm.202400123

Citations: 1

Share a link

Email
Wechat
Bluesky

Abstract

This research paper aims to investigate a one-dimensional thermoelastic porous system with microtemperatures effects in the presence of discrete time delay in internal feedback without frictional damping to control the delay term. First, we establish the well-posedness of the system using semigroup theory. Then, based on the energy method we show an exponential stability result of the solution, under a smallness condition on the delay, irrespective of the wave speeds of the system or any other condition on the coefficients. Finally, we give numerical experiments to validate our theoretical result by carrying out an Euler scheme for time discretization and the classical finite difference method for the spatial discretization. The existing findings have been enhanced and present novel advancements compared to the established results in the literature.

CONFLICT OF INTEREST STATEMENT

The authors declare no conflicts of interest.

REFERENCES

1Datko, R.: Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks. SIAM J. Control Optim. 26(3), 697–713 (1988)
10.1137/0326040
Web of Science® Google Scholar
2Datko, R., Lagnese, J., Polis, M.P.: An example on the effect of time delays in boundary feedback stabilization of wave equations. SIAM J. Control Optim. 24(1), 152–156 (1986)
10.1137/0324007
Web of Science® Google Scholar
3Guesmia, A.: Well-posedness and exponential stability of an abstract evolution equation with infinity memory and time delay. IMA J. Math. Control Inform. 30, 507–526 (2013)
10.1093/imamci/dns039
Web of Science® Google Scholar
4Khochemane, H.E., Bouzettouta, L., Zitouni, S.: General decay of a nonlinear damping porous-elastic system with past history. Ann. Univ. Ferrara 65, 249–275 (2019)
10.1007/s11565-019-00321-6
Google Scholar
5Loucif, S., Guefaifia, R., Zitouni, S., Khochemane, H.E.: Global well-posedness and exponential decay of fully dynamic and electrostatic or quasi-static piezoelectric beams subject to a neutral delay. Z. Angew. Math. Phys. 74(3), 83 (2023). https://doi.org/10.1007/s00033-023-01972-4
10.1007/s00033-023-01972-4
Web of Science® Google Scholar
6Nicaise, S., Pignotti, C.: Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks. SIAM J. Control Optim. 45(5), 1561–1585 (2006)
10.1137/060648891
Web of Science® Google Scholar
7Nicaise, S., Pignotti, C.: Stabilization of the wave equation with boundary or internal distributed delay. Differ. Integral Equ. 21(9–10), 935–958 (2008)
10.57262/die/1356038593
Web of Science® Google Scholar
8Xu, G.Q., Yung, S.P., Li, L.K.: Stabilization of wave systems with input delay in the boundary control. ESAIM Control Optim. Calc. Var. 12(4), 770–785 (2006)
10.1051/cocv:2006021
Web of Science® Google Scholar
9Racke, R.: Instability of coupled systems with delay. Comm. Pure, Appl. Anal. 11(5), 1753–1773 (2012)
10.3934/cpaa.2012.11.1753
Web of Science® Google Scholar
10Abdallah, C., Dorato, P., Benitez-Read, J., Byrne, R.: Delayed positive feedback can stabilize oscillatory system. In: Proceedings of the ACC, pp. 3106–3107. San Francisco (1993)
Google Scholar
11Apalar, T.A., Messaoud, S.A., Mustaf, M.I.: Energy decay in thermoelastic type III with viscoelastic damping and delay. Electron. J. Differ. Equ. 2012(128), 1–15 (2012)
Google Scholar
12Foughali, F., Zitouni, S., Khochemane, H.E., Djebabla, A.: Well-posedness and exponential decay for a porous-thermoelastic system with second sound and a distributed delay term. Math. Eng. Sci. Aerospace 11(4), 1003–1020 (2020)
Google Scholar
13Mesmouli, M.B., Ardjouni, A., Djoudi, A.: Exponential stability of the heat equation with boundary time-varying delays. Int. J. Anal. Appl. 11(1), 43–53 (2016)
Google Scholar
14Zitouni, S., Ardjouni, A., Mesmouli, M.B., Amiar, R.: well posedness and stability of nonlinear wave equations with two boundary time-varying delays. Math. Eng. Sci. Aerosp. 8(2), 147–170 (2017)
Google Scholar
15Goodma, M.A., Cowi, S.C.: A continuum theory for granular materials. Arch. Ration. Mech. Anal. 44(4), 249–266 (1972)
10.1007/BF00284326
Google Scholar
16Nunziato, W., Cowin, S.C.: A nonlinear theory of elastic materials with voids. Arch. Rational Mech. Anal. 72(1979), 175–201.
Google Scholar
17Grot, R.: Thermodynamics of a continuum with microstructure. Int. J. Eng. Sci. 7, 801–814 (1969)
10.1016/0020-7225(69)90062-7
Web of Science® Google Scholar
18Dridi, H., Djebabla, A.: On the stabilization of linear porous elastic materials by microtemperature effect and porous damping. Ann. Univ. Ferrara 66, 13–25 (2020). https://doi.org/10.1007/s11565-019-00333-2
10.1007/s11565-019-00333-2
Google Scholar
19Ieşan, D.: A theory of thermoelastic materials with voids. Acta. Mech. 60(1–2), 67–89 (1986)
10.1007/BF01302942
Web of Science® Google Scholar
20Ieşan, D.: On a theory of micromorphic elastic solids with microtemperatures. J. Thermal Stresses 24(8), 737–752 (2001)
10.1080/014957301300324882
Web of Science® Google Scholar
21Ieşan, D., Quintanilla, R.: On a theory of thermoelasticity with microtemperatures. J. Thermal Stresses 23(3), 199–215 (2000)
10.1080/014957300280407
Web of Science® Google Scholar
22Quintanilla, R.: Slow decay for one-dimensional porous dissipation elasticity. Appl. Math. Lett. 16(4), 487–491 (2003)
10.1016/S0893-9659(03)00025-9
Web of Science® Google Scholar
23Apalar, T.A.: Exponential decay in one-dimensional porous dissipation elasticity. Q. J. Mech. Appl. Math. 70(4), 363–372 (2017)
10.1093/qjmam/hbx012
Google Scholar
24Apalar, T.A.: Corrigendum: Exponential decay in one-dimensional porous dissipation elasticity. Q. J. Mech. Appl. Math. 70(4), 553–555 (2017)
10.1093/qjmam/hbx027
Google Scholar
25Casa, P.S., Quintanilla, R.: Exponential stability in thermoelasticity with microtemperatures. Int. J. Eng. Sci. 43(1–2), 33–47 (2005)
10.1016/j.ijengsci.2004.09.004
Google Scholar
26Casas, P.S., Quintanilla, R.: Exponential decay in one-dimensional porous-thermo-elasticity. Mech. Res. Commun. 32(6), 652–658 (2005)
10.1016/j.mechrescom.2005.02.015
Web of Science® Google Scholar
27Magańa, A., Quintanilla, R.: On the time decay of solutions in one-dimensional theories of porous materials. Int. J. Solids Struct. 43(11–12), 3414–3427 (2006)
10.1016/j.ijsolstr.2005.06.077
Web of Science® Google Scholar
28Apalar, T.A.: General decay of solutions in one-dimensional porous-elastic system with memory. J. Math. Anal. Appl. 469(2), 457–471 (2019)
10.1016/j.jmaa.2017.08.007
Google Scholar
29Apalar, T.A.: On the stabilization of a memory-type porous thermoelastic system. Bull. Malays. Math. Sci. Soc. 43(2), 1433–1448 (2020)
10.1007/s40840-019-00748-2
Google Scholar
30Apalar, T.A.: On the stability of porous-elastic system with microtemperatures. J. Therm. Stresses 19, 222 (2018). https://doi.org/10.1080/01495739.2018.1486688
10.1080/01495739.2018.1486688
Google Scholar
31Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York (1983).
10.1007/978-1-4612-5561-1
Google Scholar
32Khochemane, H.E.: Exponential stability for a thermoelastic porous system with microtemperatures effects. Acta. Appl. Math. 173(8), (2021). https://doi.org/10.1007/s10440-021-00418-1
10.1007/s10440?021?00418?1
Google Scholar

Citing Literature

Volume105, Issue2

February 2025

e202400123

Theoretical study and numerical test for a delayed thermoelastic porous system with microtemperatures

Abstract

CONFLICT OF INTEREST STATEMENT

REFERENCES

Citing Literature

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

Theoretical study and numerical test for a delayed thermoelastic porous system with microtemperatures

Abstract

CONFLICT OF INTEREST STATEMENT

REFERENCES

Citing Literature

References

Related

Information