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Porous Elastic Soils with Fluid Saturation and Boundary Dissipation of Fractional Derivative Type

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Abstract

This paper deals with a one-dimensional system in the linear isothermal theory of swelling porous elastic soils subject to fractional derivative-type boundary damping. We apply the semigroup theory. We prove well-posedness by the Lumer–Phillips theorem. We show the lack of exponential stability and strong stability is proved by using general criteria due to Arendt–Batty. Polynomial stability result is obtained by applying the Borichev–Tomilov theorem.

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References

  1. Das, S.: Functional Fractional Calculus for System Identification and Control. Springer, Berlin Heidelberg, New York (2008)

    Google Scholar 

  2. Machado, J.A.T., Jesus, I.S., Barbosa, R., Silva, M., Rei, C.: Application of fractional calculus in engineering. In: Peixoto, M., Pinto, A., Rand, D. (eds.) Dynamics, Games and Science. I Springer Proceedings in Mathematics, vol. 1, pp. 619–629. Springer, Berlin, Heidelberg (2011)

    Google Scholar 

  3. Choi, J., Maccamy, R.: Fractional order Volterra equations with applications to elasticity. J. Math. Anal. Appl. 139, 448–464 (1989)

    Article  MathSciNet  Google Scholar 

  4. Eringen, A.C.: A continuum theory of swelling porous elastic soils. Int. J. Eng. Sci. 32, 1337–1349 (1994)

    Article  MathSciNet  Google Scholar 

  5. Karalis, K.: On the elastic deformation of non-saturated swelling soils. Acta Mech. 84, 19–45 (1990)

    Article  Google Scholar 

  6. Iesan, D.: On the theory of mixtures of thermoelastic solids. J. Therm. Stress. 14, 389–408 (1991)

    Article  MathSciNet  Google Scholar 

  7. Quintanilla, R.: Exponential stability for one-dimensional problem of swelling porous elastic soils with fluid saturation. J. Comput. Appl. Math. 145, 525–533 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  8. Quintanilla, R.: Exponential stability of solutions of swelling porous elastic soils. Meccanica 39, 139–145 (2004)

    Article  MathSciNet  Google Scholar 

  9. Quintanilla, R.: Existence and exponential decay in the linear theory of viscoelastic mixtures. Eur. J. Mech. A Solids 24, 311–324 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  10. Ammari, K., Hassine, F., Robbiano, L.: Stabilization for Some Fractional-Evolution Systems. Springer, Cham, Switzerland (2022)

    Book  Google Scholar 

  11. Obaya, I., El-Saka, H., Ahmed, E., Elmahdy, A.I.: On multi-strain fractional order mers-cov model. J. Fract. Calc. Appl. 9, 196–201 (2018)

    MathSciNet  Google Scholar 

  12. Sun, H.G., Zhang, Y., Baleanu, D., Chen, W., Chene, Y.Q.: A new collection of real world applications of fractional calculus in science and engineering. Commun. Nonlinear Sci. Numer. Simul. 64, 213–231 (2018)

    Article  ADS  Google Scholar 

  13. Torvik, P.J., Bagley, R.L.: On the appearance of the fractional derivative in the behavior of real materials. J. Appl. Mech. 51, 294–298 (1984)

    Article  ADS  Google Scholar 

  14. Bagley, R.L., Torvik, P.J.: A theoretical basis for the application of fractional calculus to viscoelasticity. J. Rheol. 27, 201–210 (1983)

    Article  ADS  CAS  Google Scholar 

  15. Zarraga, O., Sarría, I., García-Barruetabeña, J., Cortés, F.: An analysis of the dynamical behaviour of systems with fractional damping for mechanical engineering applications. Symmetry 11, 1499 (2019)

    Article  ADS  Google Scholar 

  16. Podlubny, I.: Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. In: Mathematics in Science and Engineering. Academic Press, Cambridge (1999)

    Google Scholar 

  17. Akil, M., Wehbe, A.: Stabilization of multidimensional wave equation with locally boundary fractional dissipation law under geometric conditions. Math. Control Relat. Fields 9, 97–116 (2019)

    Article  MathSciNet  Google Scholar 

  18. Villagran, O.P.V., Nonato, C.A., Raposo, C.A., Ramos, A.J.A.: Stability for a weakly coupled wave equations with a boundary dissipation of fractional derivative type. Rend. Circ. Mat. Palermo Ser. 2(72), 803–831 (2023)

    Article  MathSciNet  Google Scholar 

  19. Villagran, O.P.V., Raposo, C.A., Nonato, C.A., Ramos, A.J.A.: Stability of solution for Rao–Nakra sandwich beam with boundary dissipation of fractional derivative type. J. Fract. Calc. Appl. 13, 116–143 (2022)

    MathSciNet  Google Scholar 

  20. Mbodje, B.: Wave energy decay under fractional derivative controls. IMA J. Math. Control Inf. 23, 237–257 (2006)

    Article  MathSciNet  Google Scholar 

  21. Achouri, Z., Amroun, N., Benaissa, A.: The Euler–Bernoulli beam equation with boundary dissipation of fractional derivative type. Math. Methods Appl. Sci. 40, 3837–3854 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  22. Wang, J.M., Guo, B.Z.: On the stability of swelling porous elastic soils with fluid saturation by one internal damping. IMA J. Appl. Math. 71, 565–582 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  23. Apalara, T.A.: General stability result of swelling porous elastic soils with a viscoelastic damping. Z. Angew. Math. Phys. 71, 200 (2020)

    Article  MathSciNet  Google Scholar 

  24. Ramos, A.J.A., Almeida Júnior, D.S., Freitas, M.M., Noé, A.S., Dos Santos, M.J.: Stabilization of swelling porous elastic soils with fluid saturation and delay time terms. J. Math. Phys. 62, 021507 (2021)

    Article  ADS  MathSciNet  Google Scholar 

  25. Nonato, C.A.S., Ramos, A.J.A., Raposo, C.A., Dos Santos, M.J., Freitas, M.M.: Stabilization of swelling porous elastic soils with fluid saturation, time varying-delay and time-varying weights. Z. Angew. Math. Phys. 73, 20 (2022)

    Article  MathSciNet  Google Scholar 

  26. Choucha, A., Boulaaras, S.M., Ouchenane, D., Cherif, B.B., Abdalla, M.: Exponential stability of swelling porous elastic with a viscoelastic damping and distributed delay term. J. Funct. Spaces 2021, 5581634 (2021)

    MathSciNet  Google Scholar 

  27. Al-Mahdi, A.M., Al-Gharabli, M.M., Alahyane, M.: Theoretical and numerical stability results for a viscoelastic swelling porous-elastic system with past history. AIMS Math. 6, 11921–11949 (2021)

    Article  MathSciNet  Google Scholar 

  28. Baibeche, S., Bouzettouta, L., Guesmia, A., Abdelli, M.: Well-posedness and exponential stability of swelling porous elastic soils with a second sound and distributed delay term. J. Math. Comput. Sci. 12, 82 (2022)

    Google Scholar 

  29. Arendt, W., Batty, C.J.K.: Tauberian theorems and stability of one-parameter semigroups. Trans. Am. Math. Soc. 306, 837–852 (1988)

    Article  MathSciNet  Google Scholar 

  30. Borichev, A., Tomilov, Y.: Optimal polynomial decay of functions and operator semigroups. Math. Ann. 347, 455–478 (2010)

    Article  MathSciNet  Google Scholar 

  31. Mbodje, B., Montseny, G.: Boundary fractional derivative control of the wave equation. IEEE Trans. Autom. Control 40, 368–382 (1995)

    Article  MathSciNet  Google Scholar 

  32. Huang, F.L.: Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces. Ann. Diff. Eqns. 1, 43–56 (1985)

    MathSciNet  Google Scholar 

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Acknowledgements

The authors thank the anonymous referees for their suggestions, which improved this manuscript.

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Author notes

  1. Abbes Benaissa, Anderson Ramos, Carlos Raposo, Mirelson Freitas, have contributed equally to this work.

Authors and Affiliations

  1. Department of Mathematics, Ceará State University, Dr. Silas Munguba Avenue, Fortaleza, Ceará, 60714-903, Brazil

    Carlos Nonato

  2. Laboratory of Analysis and Control of PDEs, University Djillali liabes of Sidi Bel Abbes, BP 89, 22000, Sidi Bel Abbés, Algeria

    Abbes Benaissa

  3. Department of Mathematics, Federal University of Pará, Raimundo Santana Street, Salinópolis, Pará, 68721-000, Brazil

    Anderson Ramos & Mirelson Freitas

  4. Department of Mathematics, Federal University of Bahia, Milton Santos Avenue, Salvador, Bahia, 40170-110, Brazil

    Carlos Raposo

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  1. Carlos Nonato
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Correspondence to Carlos Nonato.

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Nonato, C., Benaissa, A., Ramos, A. et al. Porous Elastic Soils with Fluid Saturation and Boundary Dissipation of Fractional Derivative Type. Qual. Theory Dyn. Syst. 23, 79 (2024). https://doi.org/10.1007/s12346-023-00937-2

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  • DOI: https://doi.org/10.1007/s12346-023-00937-2

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