Abstract
On the basis of fractional diffusion models of distributed order, statements are made and closed-form 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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
K. S. Basniev, I. N. Kochina, and V. M. Maximov, Underground Hydromechanics [in Russian], Nedra, Moscow (1993).
V. G. Pryazhinskaya, D. M. Yaroshevskii, and L. K. Levit-Gurevich, Computer Simulation in Water Resources Management [in Russian], Fizmatgiz, Moscow (2002).
P. Ya. Polubarinova-Kochina, V. G. Pryazhinskaya, and V. N. Emikh, Mathematical Methods in Irrigation [in Russian], Nauka, Moscow (1969).
A. Ya. Oleinik, Drainage Geohydrodynamics [in Russian], Naukova Dumka, Kyiv (1981).
A. Allwright and A. Atangana, “Fractal advection-dispersion equation for groundwater transport in fractured aquifers with self-similarities,” The European Physical J. Plus, Vol. 133 (2), 1–14 (2018).
N. Su, “Mass-time and space-time fractional partial differential equations of water movement in soils: Òheoretical framework and application to infiltration,” J. of Hydrology, Vol. 519, 1792–1803 (2014).
N. Su, P. N. Nelson, and S. Connor, “The distributer-order fractional-wave equation of groundwater flow: Theory and application to pumping and slug test,” J. of Hydrology, Vol. 529, 1263–1273 (2015).
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).
V. A. Bogaenko, V. M. Bulavatsky, and Iu. G. Kryvonos, “On mathematical modeling of fractional–differential dynamics of flushing process for saline soils with parallel algoritms usage,” J. Autom. Inform. Sci., Vol. 48, No. 1, 1–12 (2014).
V. Bogaenko and V. Bulavatsky, “Fractional-fractal modeling of filtration-consolidation processes in saline saturated soils,” Fractal and Fractional, Vol. 4, Iss. 4, 2–12 (2020). https://doi.org/10.3390/fractalfract4040059.
V. M. Bulavatsky, “Mathematical modeling of fractional differential filtration dynamics based on models with Hilfer–Prabhakar derivative,” Cybern. Syst. Analysis, Vol. 53, No. 2, 204–216 (2017). https://doi.org/10.1007/s10559-017-9920-z.
V. M. Bulavatsky and V. O. Bohaienko, “Some boundary-value problems of fractional–differential mobil/eimmobile migration dynamics in a profile filtration flow,” Cybern. Syst. Analysis, Vol. 56, No. 3, 410–425 (2020). https://doi.org/10.1007/s10559-020-00257-2.
T. Sandev and Z. Tomovsky, Fractional Equations and Models, Theory and Applications, Springer Nature Switzerland AG, Cham, Switzerland (2019).
W. Ding, S. Patnaik, S. Sidhardh, and F. Semperlotti, “Applications of distributed-order fractional operators: A review,” Entropy, Vol. 23, Iss. 1 (2021). https://doi.org/10.3390/e23010110.
A. V. Chechkin, R. Gorenflo, I. M. Sokolov, and V. Y. Gonchar, “Distributed order time fractional diffusion equation,” Fract. Calc. Appl. Anal., No. 6, 259–257 (2003).
M. Naber, “Distributed order fractional sub-diffusion,” Fractals, No. 12, 23–32 (2004).
F. Mainardi and G. Pagnini, “The role of the Fox–Wright functions in fractional sub-diffusion of distributed order,” J. Comput. Appl. Math., Vol. 207, 245–257 (2007).
A. N. Kochubei, “Distributed order calculus and equations of ultraslow diffusion,” J. Math. Anal. Appl., Vol. 340, 252–281 (2008).
E. Bazhlekova and I. Bazhlekov, “Application of Dimovski’s conolutional calculus to distributed-order time-fractional diffusion equation on a bounded domain,” J. Inequalities and Special Functions, Vol. 8, Iss. 1, 68–83 (2017).
A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam (2006).
G. I. Barenblatt, V. M. Entov, and V. M. Ryzhik, Motion of Liquids and Gases in Natural Strata [in Russian], Nedra, Moscow (1984).
I. Sneddon, The Use of Integral Transform, Mc. Graw-Hill Book Comp., New York (1973).
W. Rundell and Z. Zhang, “Fractional diffusion: Recovering the distributed fractional derivative from overposed data,” Inverse Problems, Vol. 33, No. 3, 1–24 (2017).
V. S. Vladimirov, Equations of Mathematical Physics [in Russian], Nauka, Moscow (1967).
R. Gorenflo, A. A. Kilbas, F. Mainardi, and S. V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications, Springer Verlag, Berlin (2014).
N. I. Ionkin, “Solution of one boundary-value heat conduction problem with a nonclassical boundary condition,” Diff. Uravneniya, Vol. 13, No. 2, 294–304 (1977).
N. I. Ionkin and E. I. Moiseyev, “A problem for the heat conduction equation with two-point boundary conditions,” Diff. Uravneniya, Vol. 15, No. 7, 1284–1295 (1979).
E. I. Moiseyev, “Solution of one non-local boundary-value problem by the spectral method,” Diff. Uravneniya, Vol. 35, No. 8, 1094–1100 (1999).
X. Cheng, C. Yuan, K. Liang, “Inverse source problem for a distributed-order time fractional diffusion equation,” J. Inverse Ill-Posed Probl., Vol. 28(1), 17–32 (2020).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Kibernetyka ta Systemnyi Analiz, No. 1, January–February, 2022, pp. 77–89.
Rights and permissions
About this article
Cite this article
Bulavatsky, V.M. Some Boundary-Value Problems of Filtration Dynamics Corresponding to Models of Fractional Diffusion of Distributed Order. Cybern Syst Anal 58, 65–76 (2022). https://doi.org/10.1007/s10559-022-00436-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10559-022-00436-3