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Cybernetics and Systems Analysis

, Volume 54, Issue 2, pp 193–204 | Cite as

Numerical Simulation of Fractional-Differential Filtration-Consolidation Dynamics Within the Framework of Models with Non-Singular Kernel

  • V. M. Bulavatsky
  • V. O. Bohaienko
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Abstract

Non-classical mathematical models to describe the fractional-differential filtration-consolidation dynamics of soil media saturated with salt solutions are constructed based on the concept of the Caputo–Fabrizio fractional derivative. The corresponding boundary-value problems for the system of fractional-differential equations of filtration and salt transfer are posed, the technique for their numerical solution is developed, an approach to the parallelization of computations is presented, and the results of numerical experiments on modeling the dynamics of the process are given.

Keywords

mathematical modeling dynamics of filtration-consolidation processes fractional-differential mathematical models models with non-singular kernel boundary-value problems finite difference solutions parallel computing 
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References

  1. 1.
    Y. Zhou, R. K. Rajapakse, and J. Graham, “A coupled thermo-poroelastic model with thermo-osmosis and thermal-filtration,” Intern. J. of Solids and Structures, Vol. 35, No. 35, 4659–4683 (1998).CrossRefzbMATHGoogle Scholar
  2. 2.
    M. Kaczmarek and T. Huekel, “Chemo-mechanical consolidation of clays: Analytical solution for a linearized one-dimensional problem,” Transport in Porous Media, Vol. 32, 49–74 (1998).CrossRefGoogle Scholar
  3. 3.
    A. Chassemi and A. Diek, “Linear chemo-poroelasticity for swelling shales: Theory and application,” J. Petroleum Sci. Eng., Vol. 38, 199–211 (2003).CrossRefGoogle Scholar
  4. 4.
    A. P. Vlasyuk and P. M. Martynyuk, Mathematical Modeling of Soil Consolidation during Filtration of Salt Solutions [in Ukrainian], Vydavn. UDUVGP, Rivne (2004).Google Scholar
  5. 5.
    V. M. Bulavatsky, Yu. G. Krivonos, and V. V. Skopetsky, Nonclassical Mathematical Models of Heat and Mass Transfer Processes [in Russian], Naukova Dumka, Kyiv (2005).Google Scholar
  6. 6.
    P. L. Ivanov, Soils and Foundations of Hydraulic Engineering Structures [in Russian], Vysshaya Shkola, Moscow (1991).Google Scholar
  7. 7.
    Yu. K. Zaretskii, Theory of Soil Consolidation [in Russian], Nauka, Moscow (1967).Google Scholar
  8. 8.
    I. Podlubny, Fractional Differential Equations, Acad. Press, New York (1999).zbMATHGoogle Scholar
  9. 9.
    A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam (2006).zbMATHGoogle Scholar
  10. 10.
    V. V. Uchaikin, Method of Fractional Derivatives, Artishok, Ul’yanovsk (2008).Google Scholar
  11. 11.
    R. Gorenflo and F. Mainardi, “Fractional calculus: Integral and differential equations of fractional order,” in: A. Carpinteri and F. Mainardi (eds.), Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag, Wien (1997), pp. 223–276.CrossRefGoogle Scholar
  12. 12.
    V. M. Bulavatsky, “Mathematical model of geoinformatics for investigation of dynamics for locally nonequilibrium geofiltration processes,” J. Autom. Inform. Sci., Vol. 43, Issue 12, 12–20 (2011).CrossRefGoogle Scholar
  13. 13.
    V. M. Bulavatsky and Yu. G. Krivonos, “Mathematical modeling in the geoinformation problem of the dynamics of geomigration under space-time nonlocality,” Cybern. Syst. Analysis, Vol. 48, No. 4, 539–546 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    V. M. Bulavatsky, “Simulation of dynamics of some locally nonequilibrium geomigration process on the basis of a fractional-differential geoinformation model,” J. Autom. Inform. Sci., Vol. 45, Issue 6, 75–84 (2013).CrossRefGoogle Scholar
  15. 15.
    V. M. Bulavatsky, “Fractional differential mathematical models of the dynamics of nonequilibrium geomigration processes and problems with nonlocal boundary conditions,” Cybern. Syst. Analysis, Vol. 50, No. 1, 81–89 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    A. Atangana and B. Alkahtani, “New model of groundwater flowing within a confine aquifer: Application of Caputo–Fabrizio derivative,” Arabian J. of Geosciences, Vol. 9, No. 1, 1–6 (2015).Google Scholar
  17. 17.
    A. Atangana and D. Baleanu, “Caputo–Fabrizio derivative applied to groundwater flow within confined aquifer,” J. of Engineering Mechanics, D 4016005 (2016).Google Scholar
  18. 18.
    M. Caputo and M. Fabrizio, “A new definition of fractional derivative without singular kernel,” Progress Fractional Differentiation and Applications, Vol. 1, No. 2, 73–85 (2015).Google Scholar
  19. 19.
    A. A. Samarskii, Theory of Difference Schemes [in Russian], Nauka, Moscow (1977).zbMATHGoogle Scholar
  20. 20.
    Y. Zhang, J. Cohen, and J. D. Owens, “Fast tridiagonal solvers on the GPU,” in: Proc. 15th ACM SIGPLAN Symp. on Principles and Practice of Parallel Programming (PPoPP 2010), Bangalore, India, Jan. 9–14 (2010), pp. 127–136.Google Scholar
  21. 21.
    A. P. Vlasyuk and P. M. Martynyuk, “Mathematical modeling of soil consolidation during filtration of salt solutions under non-isothermal conditions,” Vyd. NUVGP, Rivne (2008).Google Scholar
  22. 22.
    N. Kh. Arutyunyan, Some Problems from Creep Theory [in Russian], Gostekhizdat, Moscow–Leningrad (1952).Google Scholar
  23. 23.
    V. A. Florin, Fundamentals of Soil Mechanics [in Russian], Vol. 2, Gosstroiizdat, Moscow (1961).Google Scholar

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Authors and Affiliations

  1. 1.V. M. Glushkov Institute of CyberneticsNational Academy of Sciences of UkraineKyivUkraine

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