Geodesic
On boundary value problem for generalized Aller equation
Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ, Tome 26 (2020) no. 2, pp. 7-14 Cet article a éte moissonné depuis la source Math-Net.Ru

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The mathematical models of fluid filtration processes in porous media with a fractal structure and memory are based on differential equations of fractional order in both time and space variables. The dependence of the soil water content can significantly affect the moisture transport in capillary-porous media. The paper investigates the generalized Aller equation widely used in mathematical modeling of the processes related to water table dynamics in view of fractal structure. As a mathematical model of the Aller equation with Riemann–Liouville fractional derivatives, a loaded fractional order equation is proposed, and a solution to the Goursat problem has been written out for this model in explicit form.
Mots-clés : Aller equation, Goursat problem, Laplace convolution.
Keywords: Riemann–Liouville fractional integrodifferential operator, moisture transfer equation, generalized Newton–Leibniz formula, loaded equation, Volterra equation of the second kind 
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S.Kh. Gekkieva; M. M. Karmokov; M. A. Kerefov. On boundary value problem for generalized Aller equation. Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ, Tome 26 (2020) no. 2, pp. 7-14. http://geodesic.mathdoc.fr/item/VSGU_2020_26_2_a0/

[1] Yangarber V. A., “The mixed problem for a modified moisture-transfer equation”, Journal of Applied Mechanics and Technical Physics, 8:1 (1967), 62–64 | DOI

[2] Coleman B. D., Duffin R. J., Mizel V. J., “Instability, uniqueness and nonexistence theorems for the equation $u_t=u_{xx}-u_{xtx}$ on a strip”, Arch. Rat. Mech. Anal., 19:2 (1965), 100–116 | DOI | MR | Zbl

[3] Nakhushev A. M., Loaded equations and their applications, Nauka, M., 2012, 232 pp. (In Russ.)

[4] Vogakhova V. A., “A boundary value problem with A. M. Nakhushev's nonlocal condition for a pseudoparabolic equation of moisture transfer”, Differential Equations, 18:2 (1982), 280–285 (In Russ.) | MR | Zbl

[5] Soldatov A. P., Shhanukov M. H., “Boundary value problems with A. A. Samarskii general nonlocal condition for higher-order pseudoparabolic equations”, Doklady Mathematics, 36:3 (1988), 507–511 (In Russ.)

[6] Evdokimova N. N., Pulkina L. S. Non-local problem for a single degenerate hyperbolic equation, Vestnik of Samara State University. Natural Science Series, 1999, no. 2, 67–70 (In Russ.) | Zbl

[7] Karsanova Zh.T., Nakhusheva F. M., “On a nonlocal boundary value problem for a third-order pseudoparabolic equation”, Vladikavkaz Mathematical Journal, 4:2 (2002), 31–37 (In Russ.) | MR | Zbl

[8] Kozhanov A. I., “On a non-local boundary value problem with variable coefficients for the heat conduction and Aller equation”, Differential Equations, 40:6 (2004), 815–826 (In Russ.) | DOI | MR | Zbl

[9] Khubiev K. U., “On mathematical models of the Aller equation”, Bulletin KRASEC. Physical and Mathematical Sciences, 2016, no. 4–1, 56–65 (In Russ.) | DOI | MR | Zbl

[10] Beshtokov M.Kh., “Differential and Difference Boundary Value Problem for Loaded Third-Order Pseudo-Parabolic Differential Equations and Difference Methods for Their Numerical Solution”, Computational Mathematics and Mathematical Physics, 57:12 (2017), 1973–1993 | DOI | DOI | MR | Zbl

[11] Nakhushev A. M., “An approximate method for solving boundary value problems for differential equations and its application to the dynamics of ground moisture and ground water”, Differential Equations, 18:1 (1982), 72–81 (In Russ.) | MR | Zbl

[12] Nakhushev A. M., “A non-local problem and the Goursat problem for a loaded equation of hyperbolic type, and their applications to the prediction of ground moisture”, Dokl. Akad. Nauk SSSR, 242:5 (1978), 1008–1011 (In Russ.) | MR | Zbl

[13] Nakhushev A. M., Fractional calculation and its application, Fizmatlit, M., 2003, 272 pp. (In Russ.)

[14] Kerefov M. A., Boundary value problems for a modified moisture transfer equation with a time-fractional derivative, Candidate's of Physical and Mathematical Sciences thesis, Nalchik, 2000, 75 pp. (In Russ.)

[15] Kerefov M. A., Gekkieva S.Kh., “A nonlocal boundary value problem for the generalized equation of moisture transfer”, Proceedings of Voronezh State University. Series: Physics. Mathematics, 2017, no. 2, 106–112 (In Russ.) | Zbl

[16] Beshtokov M.Kh., “Local and non-local boundary value problems for degenerate and non-degenerate pseudo-parabolic equations with the Riemann – Liouville fractional derivative”, Differential Equations, 54:6 (2018), 763–778 (In Russ.) | DOI | MR | Zbl

[17] Beshtokov M. K., “Boundary value problems for a loaded modified fractional-order moisture transfer equation with the Bessel operator and difference methods for their solution”, Bulletin of Udmurt University. Mathematics, Mechanics, Computer Science, 30:2 (2020), 158–175 (In Russ.) | DOI | MR

[18] Pskhu A. V., Partial differential equations of fractional order, Nauka, M., 2005, 199 pp. (In Russ.)

[19] Pskhu A. V., “Solution of a two-dimensional Abel integral equation of the second kind”, News of the Kabardin-Balkar Scientific Center of the Russian Academy of Sciences, 2016, no. 6(74), 75–80 (In Russ.)

[20] Wright E. M., “On the coefficients of power series having exponential singularities”, J. London Math. Soc., 8:29 (1933), 71–79 | DOI | MR | Zbl