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

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

doi:10.3934/math.2021692

AIMS Mathematics

2021, Volume 6, Issue 11: 11921-11949. doi: 10.3934/math.2021692

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Research article

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

1.
The Preparatory Year Program, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
2.
The Interdisciplinary Research Center in Construction and Building Materials, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
3.
Department of Mathematics, RISE, University of Sharjah, P.O. Box 27272, Sharjah, UAE

Received: 08 April 2021 Accepted: 08 August 2021 Published: 16 August 2021
MSC : 35B40, 45K05, 74S05, 93D15, 93D20

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.
- swelling porous problem,
- viscoelastic,
- general decay,
- convex functions,
- finite element and Crank-Nicolson methods
Citation: Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Mohamed Alahyane. Theoretical and numerical stability results for a viscoelastic swelling porous-elastic system with past history[J]. AIMS Mathematics, 2021, 6(11): 11921-11949. doi: 10.3934/math.2021692

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Abstract

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.

References

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