Theoretical and Computational Results of a Memory-Type Swelling Porous-Elastic System
Abstract
:1. Introduction
- We produce some numerical experiments to illustrate the energy decay results, for this purpose, we develop a second-order numerical scheme to solve the problem (4) based on finite element discretization and the Crank–Nicolson method in time that has the property to be unconditionally stable.
- The result is significant to engineers and architects as it might help to attenuate the harmful effects of swelling soils swiftly.
2. Assumptions
3. Technical Lemmas
4. The Main Result
- (1)
- Let , , are constants, and a is chosen so that is satisfied; thenThus, under the assumptions of Theorem 1, we conclude that the solution of (4) satisfies, for two constants , the energy estimate
- (2)
- For , for , , and a is chosen so that condition is satisfied, thenThus, under the assumptions of Theorem 1, we conclude that the solution of (4) satisfies, for some constant , the energy estimate
- (3)
- Consider the following relaxation function,
5. Numerical Tests
- Test 1: For the first numerical test, we choose the following entries:Thus, under the assumptions of Theorem 1, the solution of (4) satisfies the energy estimate
- Test 2: In the second numerical test, we consider the following entries so that condition is satisfiedThen, under the assumptions of Theorem 1, the solution of (4) satisfies the energy estimate
- Test 3: For last test, we consider the third case of Example 1 with the same entries of Test 1 and with an polynomial relaxation functionUnder the assumptions of Theorem 1, the solution of (4) satisfies the energy estimate
6. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
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Al-Mahdi, A.M.; Al-Gharabli, M.M.; Alahyane, M. Theoretical and Computational Results of a Memory-Type Swelling Porous-Elastic System. Math. Comput. Appl. 2022, 27, 27. https://doi.org/10.3390/mca27020027
Al-Mahdi AM, Al-Gharabli MM, Alahyane M. Theoretical and Computational Results of a Memory-Type Swelling Porous-Elastic System. Mathematical and Computational Applications. 2022; 27(2):27. https://doi.org/10.3390/mca27020027
Chicago/Turabian StyleAl-Mahdi, Adel M., Mohammad M. Al-Gharabli, and Mohamed Alahyane. 2022. "Theoretical and Computational Results of a Memory-Type Swelling Porous-Elastic System" Mathematical and Computational Applications 27, no. 2: 27. https://doi.org/10.3390/mca27020027