UDC 517.9: 519.6
1 V. M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine
 v_bulav@ukr.net
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SOME BOUNDARY-VALUE PROBLEMS OF FILTRATION DYNAMICS
CORRESPONDING TO FRACTIONAL DIFFUSION MODELS OF DISTRIBUTED ORDER
Abstract. On the basis of distributed-order fractional diffusion models, statements are made and closed solutions
are obtained for some boundary-value problems of anomalous geofiltration dynamics, in particular, the problem of inflow
to a gallery located between two supply lines in a three-layer geoporous medium. For a simplified version of the filtration model
of distributed order, solutions are obtained for the direct and inverse boundary-value problems of filtration dynamics,
as well as for the filtration problem with nonlocal boundary conditions.
Keywords: mathematical modeling, fractional-differential dynamics of filtration processes, geoporous media, non-classical models, model of filtration with distributed order derivative, boundary-value problem, closed-form solution.
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REFERENCES
- Basniev K.S., Kochina I.N., Maksimov V.M. Underground hydromechanics [in Russian]. Moscow: Nedra, 1993. 416 p.
- Pryazhinskaya V.G., Yaroshevsky D.M., Levit-Gurevich L.K. Computer modeling in water resources management [in Russian]. Moscow: Fizmatgiz, 2002. 496 p.
- Polubarinova-Kochina P.Ya., Pryazhinskaya V.G., Emikh V.N. Mathematical methods in irrigation matters [in Russian]. Moscow: Nauka, 1969. 414 p.
- Oleinik A.Ya. Drainage geohydrodynamics [in Russian]. Kiev: Naukova Dumka, 1981. 283 p.
- Allwright A., Atangana, A. Fractal advection-dispersion equation for groundwater transport in fractured aquifers with self-similarities. The European Physical Journal Plus. 2018. Vol. 133 (2). P. 1–14.
- Su N. Mass-time and space-time fractional partial differential equations of water movement in soils: Тheoretical framework and application to infiltration. Journal of Hydrology. 2014. Vol. 519. P. 1792–1803.
- Su N., Nelson P.N., Connor S. The distributer-order fractional-wave equation of groundwater flow: theory and application to pumping and slug test. Journal of Hydrology. 2015. Vol. 529. P. 1263–1273.
- Shumer R., Benson D.A., Meershaert M.M., Baeumer B. Fractal mobile immobile solute transport. Water Resour. Res. 2003. Vol. 39, N 10. P. 1296–1309.
- Bogaenko V.A., Bulavatsky V.M., Kryvonos Yu.G. On mathematical modeling of fractional-differential dynamics of flushing process for saline soils with parallel algoritms usage. Journ. Autom. and Informat. Sci. 2014. Vol. 48, N 1. P. 1–12.
- Bogaenko V., Bulavatsky V. Fractional-fractal modeling of filtration-consolidation processes in saline saturated soils. Fractal and fractional. 2020. Vol. 4, Iss. 4. P. 2–12. https://doi.org/10.3390/fractalfract4040059.
- Bulavatsky V.M. Mathematical modeling of fractional differential filtration dynamics based on models with Hilfer-Prabhakar derivative. Cybernetics and Systems Analysis. 2017. Vol. 53, N 2. P. 204–216.
- Bulavatsky V.M., Bohaienko V.O. Some boundary-value problems of fractional-differential mobil/eimmobile migration dynamics in a profile filtration flow. Cybernetics and Systems Analysis. 2020. Vol. 56, N 3. P. 410–425.
- Sandev T., Tomovsky Z. Fractional equations and models. Theory and applications. Cham, Switzerland: Springer Nature Switzerland AG. 2019. 344 p.
- Ding W., Patnaik S., Sidhardh S., Semperlotti F. Applications of distributed-order fractional operators: a review. Entropy. 2021. Vol. 23, Iss. 1. https://doi.org/10.3390/e23010110.
- Chechkin A.V., Gorenflo R., Sokolov I.M., Gonchar V.Y. Distributed order time fractional diffusion equation. Fract. Calc. Appl. Anal. 2003. N 6. P. 259–257.
- Naber M. Distributed order fractional sub-diffusion. Fractals. 2004. N 12. P. 23–32.
- Mainardi F., Pagnini G. The role of the Fox–Wright functions in fractional sub-diffusion of distributed order. Journ. Comput. Appl. Math. 2007. Vol. 207. P. 245–257.
- Kochubei A.N. Distributed order calculus and equations of ultraslow diffusion. Journ. Math. Anal. Appl. 2008. Vol. 340. P. 252–281.
- Bazhlekova E., Bazhlekov I. Application of Dimovski’s conolutional calculus to distributed-order time-fractional diffusion equation on a bounded domain. Journal Inequalities and Special Functions. 2017. Vol. 8, Iss. 1. P. 68–83.
- Kilbas A.A., Srivastava H.M., Trujillo J.J. Theory and applications of fractional differential equations. Amsterdam: Elsevier, 2006. 523 p.
- Barenblatt G.I., Entov V.M., Ryzhik V.M. The movement of liquids and gases in natural formations [in Russian]. Moscow: Nedra, 1984. 211 p.
- Sneddon I. The use of integral transform. New York: Mc. Graw-Hill Book Comp., 1973. 539 p.
- Rundell W., Zhang Z. Fractional diffusion: recovering the distributed fractional derivative from overposed data. Inverse problems. 2017. Vol. 33, N 3. P. 1–24.
- Vladimirov V.S. Equations of mathematical physics [in Russian]. Moscow: Nauka, 1967. 436 p.
- Gorenflo R., Kilbas A.A., Mainardi F., Rogosin S.V. Mittag-Leffler functions, related topics and applications. Berlin: Springer Verlag. 2014. 454 p.
- Ionkin N.I. Solution of a boundary value problem in the theory of heat conduction with a nonclassical boundary condition. Differentsial'nyye uravneniya. 1977. Vol. 13, N 2. P. 294–304.
- Ionkin N.I., Moiseev E.I. On the problem for the heat equation with two-point boundary conditions. Differentsial'nyye uravneniya. 1979. Vol. 15, N 7. P. 1284–1295.
- Moiseev E.I. On the solution of a nonlocal boundary value problem by the spectral method. Differentsial'nyye uravneniya. 1999. Vol. 35, N 8. P. 1094–1100.
- Cheng X., Yuan C., Liang K. Inverse source problem for a distributed-order time fractional diffusion equation. J. Inverse ill-Posed Probl. 2020. Vol. 28(1). P. 17–32.
