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Some Boundary-Value Problems of Fractional-Differential Mobile–Immobile Migration Dynamics in a Profile Filtration Flow

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Cybernetics and Systems Analysis Aims and scope

Abstract

Within the framework of fractional-differential mathematical model, the formulation of boundary-value problems of convective diffusion of soluble substances with regard to immobilization under the conditions of stationary profile filtration of groundwater from reservoir to drainage is performed. In case of averaging the filtration rate over the complex potential region, closed solutions of boundary-value problems, corresponding to classical and nonlocal boundary conditions are obtained. In the general case of the variable filtration velocity, a technique is developed for the numerical solution to the boundary-value problem of convective diffusion in a fractional-differential formulation, the problems of parallelizing computations are covered, and the results of the computer experiments are presented.

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References

  1. V. I. Lavrik, V. P. Filchakova, and A. A. Yashyn, Conformal Mappings of Physical Topological Models [in Russian], Naukova Dumka, Kyiv ((1990).

  2. I. I. Lyashko, L. I. Demchenko, and G. E. Mistetskii, Numerical Solution of Problems of Heat and Mass Transfer in Porous Media [in Russian], Naukova Dumka, Kyiv (1991).

    Google Scholar 

  3. G. E. Mistetskii, Hydro Construction. Automated Calculation of Mass Transfer in Soils [in Russian], Budivelnyk, Kyiv (1985).

  4. P. Ya. Polubarinova-Kochina, Theory of Groundwater Motion [in Russian], Nauka, Moscow (1977).

    Google Scholar 

  5. V. M. Bulavatsky, “Mathematical modeling of dynamics of the process of filtration convective diffusion under the condition of time nonlocality,” J. Autom. Inform. Sci., Vol. 44, No. 4. P. 13–22 (2012).

    Article  MathSciNet  Google Scholar 

  6. V. M. Bulavatsky, “Numerical modeling of the dynamics of a convection diffusion process locally non-equilibrium in time,” Cybern. Syst. Analysis, Vol. 48, No. 6, 861–869 (2012).

    Article  MathSciNet  Google Scholar 

  7. V. M. Bulavatsky and V. A. Bogaenko, “Mathematical modeling of the dynamics of nonequilibrium in time convection–diffusion processes in domains with free boundaries,” Cybern. Syst. Analysis, Vol. 52, No. 3, 427–440 (2016).

    Article  MathSciNet  Google Scholar 

  8. H. A. Deans, and U. Rice, “A mathematical model for dispersion in the direction of flow in porous media,” Soc. Petrol. Eng. Journal, Vol. 3, Iss. 01, 49–52 (1963).

    Article  Google Scholar 

  9. M. Th. van Genuchten and P. J. Wierenga, “Mass transfer studies in sorbing porous media, 1: Analytical solutions,” Soil Science Society of America Journal, Vol. 40, 473–480 (1976).

    Article  Google Scholar 

  10. R. Shumer, D. A. Benson, M. M. Meershaert, and B. Baeumer, “Fractal mobile/immobile solute transport,” Water Resour. Res., Vol. 39, No. 10, 1296–1309 (2003).

    Google Scholar 

  11. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam (2006).

    MATH  Google Scholar 

  12. N. Su, “Distributed-order infiltration, absorption and water exchange in mobile and immobile zones of swelling soils,” J. of Hydrology, Vol. 468–469, 1–10 (2012).

    Article  Google Scholar 

  13. I. N. Sneddon, The Use of Integral Transform, Mc. Graw-Hill Book Comp., New York (1973).

    Google Scholar 

  14. Yu. Luchko and R. Gorenflo, “An operational method for solving fractional differential equations with Caputo derivatives,” Acta Mathematica Vietnamica, Vol. 24, No. 2, 207–233 (1999).

    MathSciNet  MATH  Google Scholar 

  15. M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York (1965).

    Google Scholar 

  16. N. I. Ionkin, “ Solution of one boundary-value problem of the theory of thermal conductivity with the nonclassical boundary condition,” Diff. Uravneniya, Vol. 13, No. 2, 294–304 (1977).

    Google Scholar 

  17. E. I. Moiseev, “The solution of a nonlocal boundary-value problem by the spectral method,” Diff. Uravneniya, Vol. 35, No. 8, 1094–1100 (1999).

    MathSciNet  Google Scholar 

  18. A. Yu. Mokin, “On a family of initial-boundary value problems for the heat equation,” Diff. Uravneniya, Vol. 45, No. 1, 123–137 (2009).

    MathSciNet  Google Scholar 

  19. A. Yu. Mokin, “Application of nonclassical separation of variables to a nonlocal heat problem,” Diff. Equat., Vol. 49, No. 1, 59–67 (2013).

    Article  MathSciNet  Google Scholar 

  20. A. A. Samarskii, The Theory Of Difference Schemes [in Russian], Nauka, Moscow (1977).

    Google Scholar 

  21. A. A. Samarskii and P. N. Vabishchevich, Computational Heat Transfer, Vol. 2, Wiley, New York (1995).

    Google Scholar 

  22. K. Diethelm, “An efficient parallel algorithm for the numerical solution of fractional differential equations,” Fract. Calc. Appl. Anal., Vol. 14, No. 3, 475–490 (2011).

    Article  MathSciNet  Google Scholar 

  23. C. Gong, W. Bao, and G. Tang, “A parallel algorithm for the Riesz fractional reaction-diffusion equation with explicit finite difference method,” Fract. Calc. Appl. Anal., Vol. 16, No. 3, 654–669 (2013).

    Article  MathSciNet  Google Scholar 

  24. T. A. Biala and A. Q. M. Khaliq, “Parallel algorithms for nonlinear time–space fractional parabolic PDEs,” J. of Computational Physics, Vol. 375, 135–154 (2018).

    Article  MathSciNet  Google Scholar 

  25. I. Podlubny, Fractional Differential Equations, Academic Press, New York (1999).

    MATH  Google Scholar 

  26. C. Gong, W. Bao, and J. Liu, “A piecewise memory principle for fractional derivatives,” Fract. Calc. Appl. Anal. Vol. 20, No. 4, 1010–1022 (2017).

    Article  MathSciNet  Google Scholar 

  27. N. J. Ford and A. C. Simpson, “The numerical solution of fractional differential equations: Speed versus accuracy,” Numerical Algorithms, Vol. 26, No. 4, 333–346 (2001).

    Article  MathSciNet  Google Scholar 

  28. V. O. Bohaienko, “A fast finite-difference algorithm for solving space-fractional filtration equation with a generalized Caputo derivative,” Computational and Applied Mathematics, Vol. 38, No. 3, Article 105 (2019). https://doi.org/10.1007/s40314-019-0878-5.

  29. V. O. Bohaienko, “Numerical schemes for modelling time-fractional dynamics of non-isothermal diffusion in soils,” Mathematics and Computers in Simulation, Vol. 157, 100–114 (2019).

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. V. M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine

    V. M. Bulavatsky & V. O. Bohaienko

Authors
  1. V. M. Bulavatsky
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  2. V. O. Bohaienko
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Correspondence to V. M. Bulavatsky or V. O. Bohaienko.

Additional information

Translated from Kibernetika i Sistemnyi Analiz, No. 3, May–June, 2020, pp. 80–96.

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Bulavatsky, V.M., Bohaienko, V.O. Some Boundary-Value Problems of Fractional-Differential Mobile–Immobile Migration Dynamics in a Profile Filtration Flow. Cybern Syst Anal 56, 410–425 (2020). https://doi.org/10.1007/s10559-020-00257-2

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  • DOI: https://doi.org/10.1007/s10559-020-00257-2

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