Abstract
The analytical solutions of boundary-value problems with nonlocal boundary conditions are presented for two fractional differential mathematical models of the dynamics of a geomigration process non-equilibrium in time. The models based on the equations with the Caputo and Hilfer derivatives of fractional order are considered.
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A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics [in Russian], Nauka, Moscow (1972).
L. I. Sedov, Continuum Mechanics, Vol. 2 [in Russian], Nauka, Moscow (1973).
M. M. Khasanov and G. T. Bulgakova, Nonlinear and Nonequilibrium Effects in Rheologically Complex Media [in Russian], Inst. Komp. Issled., Moscow–Izhevsk (2003).
S. L. Sobolev, “Locally nonequilibrium models of transition processes,” Usp. Fiz. Nauk, 167, No. 10, 1095–1106 (1997).
A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam (2006).
I. Podlubny, Fractional Differential Equations, Acad. Press, New York (1999).
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.
S. G. Samko, A. A. Kilbas, and O. I. Marichev, Integrals and Derivatives of Fractional Order and some of their Applications [in Russian], Nauka i Tekhnika, Minsk (1987).
V. V. Uchaikin, The Method of Fractional Derivatives [in Russian], Artishok, Ul’yanovsk (2008).
R. Gorenflo, F. Mainardi, D. Moretti, and P. Paradisi, “Time fractional diffusion: A discrete random walk approach,” Nonlinear Dynamics, 29, No. 1–4, 129–143 (2002).
P. Paradisi, R. Cesari, F. Mainardi, and F. Tampieri, “The fractional Fick’s law for nonlocal transport processes,” Physica A, 293, No. 1–2, 130–142 (2001).
H. G. Sun, W. Chen, C. Li, and Y. Q. Chen, “Fractional differential models for anomalous diffusion” Physica A, 389, 2719–2724 (2010).
V. M. Bulavatskiy, “Some mathematical models of geoinformatics for describing mass transfer processes under time-nonlocal conditions,” J. Autom. Inform. Sci., 43, Issue 6, 49–59 (2011).
V. M. Bulavatskiy, “Mathematical model of geoinformatics for investigation of dynamics for locally nonequilibrium geofiltration processes,” J. Autom. Inform. Sci., 43, Issue 12, 12–20 (2011).
V. M. Bulavatskiy and Yu. G. Krivonos, “On one geoinformation fractional differential model of the variable order,” J. Autom. Inform. Sci., 44, Issue 6, 1–7 (2012).
V. M. Bulavatsky, “Nonclassical mathematical model in geoinformatics to solve dynamic problems for nonequilibrium nonisothermal seepage fields,” Cybern. Syst. Analysis, 47, No. 6, 898–906 (2011).
V. M. Bulavatsky and Yu. G. Krivonos, “Mathematical modeling in the geoinformational problem of the dynamics of geomigration under space–time nonlocality,” Cybern. Syst. Analysis, 48, No. 4, 539–546 (2012).
I. I. Lyashko, L. I. Demchenko, and G. E. Mistetskii, Numerical Solution of Heat- and Mass Transfer Problems in Porous Media [in Russian], Naukova Dumka, Kyiv (1991).
M. Kaczmarek and T. Huekel, “Chemo-mechanical consolidation of clays: analytical solution for a linearized one-dimensional problem,” Transport in Porous Media, 32, 49–74 (1998).
N. I. Ionkin, “Solving one boundary-value problem of the thermal conduction theory with the nonclassical boundary condition,” Diff. Uravn., 13, No. 2, 294–304 (1977).
N. I. Ionkin, “On the stability of one problem of the thermal conduction theory with the nonclassical boundary condition,” Diff. Uravn., 15, No. 7, 1280–1283 (1979).
V. S. Vladimirov, The Equations of Mathematical Physics [in Russian], Nauka, Moscow (1981).
R. Hilfer, Fractional Time Evolution, in: R. Hilfer (ed.), Applications of Fractional Calculus in Physics, World Scientific, Singapore (2000), pp. 87–130.
R. Hilfer, Y. Luchko, and Z. Tomovski, “Operational method for the solution of fractional differential equations with generalized Riemann–Liouville fractional derivatives,” Fraction. Calculus and Appl. Analysis, 12, No. 3, 299–318 (2009).
Z. Tomovski, R. Hilfer, and H. M. Srivastava, “Fractional and operational calculus with generalized fractional derivative operators and Mittag–Leffler type functions,” Integral Transforms and Special Functions, No. 11, 797–814 (2010).
T. Sandev, R. Metzler, and Z. Tomovski, “Fractional diffusion equation with a generalized Riemann–Liouville time fractional derivative,” J. Physics A, 44, 5–52 (2011).
N. I. Ionkin and E. I. Moiseev, “On the problem for the heat conduction equation with two-point boundary conditions,” Diff. Uravn., 15, No. 7, 1284–1295 (1979).
A. Yu. Mokin, “On a family of initial-boundary value problems for the heat equation,” 45, No. 1, 126–141 (2009).
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Translated from Kibernetika i Sistemnyi Analiz, No. 1, January–February, 2014, pp. 93–101.
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Bulavatsky, V.M. Fractional Differential Mathematical Models of the Dynamics of Nonequilibrium Geomigration Processes and Problems with Nonlocal Boundary Conditions. Cybern Syst Anal 50, 81–89 (2014). https://doi.org/10.1007/s10559-014-9594-8
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DOI: https://doi.org/10.1007/s10559-014-9594-8