Boundary value problem for the Aller–Lykov moisture transport generalized equation with concentrated heat capacity

M. A. Kerefov ; F. M. Nakhusheva ; S. Kh. Gekkieva

Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ, Tome 24 (2018) no. 3, pp. 23-29

Résumé

The article considers the Aller–Lykov equation with a Riemann–Liouville fractional time derivative, boundary conditions of the third kind and with the concentrated specific heat capacity on the boundary of the domain. Similar conditions arise in the case with a material of a higher thermal conductivity when solving a temperature problem for restricted environment with a heater as a concentrated heat capacity. Analogous conditions also arise in practices for regulating the water-salt regime of soils, when desalination of the upper layer is achieved by draining of a surface of the flooded for a while area. Using energy inequality methods, we obtained an a priori estimate in terms of the Riemann–Liouville fractional derivative, which revealed the uniqueness of the solution to the problem under consideration.

Détail

Mots-clés : Aller's–Lykov equation
Keywords: fractional derivative, nonlocal problem, moisture transfer generalized equation, concentrated heat capacity, inequalities method, a priori estimate, boundary value problem.

Bibliographie

[1] Barenblat G.I., Zheltov Yu.P., Kochina I.N., “About the main submissions of the theory of filtration of uniform liquids in jointed breeds”, Journal of Applied Mathematics and Mechanics, 25:5 (1960), 852–864 (in Russian)

[2] Dzektser E.S., “Equations of the movement of underground waters with a free surface in multilayered environments”, Doklady Earth Sciences, 220:3 (1975), 540–543 (in Russian) | Zbl

[3] Rubinshtein L.I., “To the question of the process of distribution of heat in heterogeneous environments”, Izvestiya RAN (Akad. Nauk SSSR). Seriya Geograficheskaya, 12:1 (1948), 27–45 (in Russian)

[4] Ting T., Cooling A., “Process according to two temperature theory of heat conduction”, J. Math. Anal. Appl., 45:9 (1974), 23 pp. (in English) | DOI | MR

[5] Hallaire M., “L'eau et la production vegetable”, Institut National de la Recherche Agronomique, 1964, no. 9 (in French)

[6] Chudnovsky A.F., Thermophysics of soils, Nauka, M., 1976, 352 pp. (in Russian)

[7] Nerpin S.V., Chudnovsky A.F., Power- and a mass exchange in system plant-soil-air, Gidrometeoizdat, L., 1975, 358 pp. (in Russian)

[8] Nakhusheva F.M., Vodakhova V.A., Kudaeva F.Kh., Abaeva Z.V., “Locally one-dimensional difference schemes for the fractional order diffusion equation with a concentrated heat capacity”, Modern problems of science and education, 2015, no. 2, 763 (in Russian)

[9] Nakhusheva F.M., Kudaeva F.Kh., Kaygermazov A.A., Karmokov M.M., “Difference schemes for the fractional order diffusion equation with a concentrated heat capacity”, Modern problems of science and education, 2015, no. 2, 839 (in Russian)

[10] Shkhanukov-Lafishev M.Kh., Lafisheva M.M., Nakhusheva F.M., Mambetova A.B., “Local and one-dimensional scheme for the heat conductivity equation with the concentrated thermal capacity”, Vladikavkaz Mathematical Journal, 15:4 (2013), 58–64 (in Russian) | DOI | MR | Zbl

[11] Nakhushev A.M., Fractional calculation and its application, Fizmatlit, M., 2003, 272 pp. (in Russian)

[12] Kulik V.Ya., “Research of the movement of soil moisture from the point of view of invariancy of rather continuous groups of transformations”, Collection “Research of the Processes of Exchange of Energy and Substance in the System Soil-Plant-Air”, Nauka, L., 1972 (in Russian) | Zbl

[13] Lafisheva M.M., Kerefov M.A., Dyshekova R.V., “Differential schemes for the equation of moisture transfer of Aller–Lykov with not local condition”, Vladikavkaz Mathematical Journal, 19:1 (2017), 50–58 (in Russian) | DOI | MR

[14] Gekkieva S.Kh., “First boundary-value problem for Aller–Lykov moisture transfer equation with time fractional derivative”, Materials of the All-Russian conference with the international participation “Sustainable development: problems, concepts, models” (Nalchik, 2017), 99–102 (in Russian)

[15] Arkhestova S.M., Shkhanukov-Lafishev M.Kh., “Differential schemes for the equation of moisture transfer of Aller–Lykov with not local condition”, News of Kabardino-Balkarian scientific center of the Russian Academy of Sciences, 2012, no. 3(47), 7–16 (in Russian)

[16] Gekkieva S.Kh., Kerefov M.A., “The boundary value problem for the generalized moisture transfer equation”, Bulletin KRASEC. Physical Mathematical Sciences, 2018, no. 1(21), 21–32 (in Russian) | DOI | MR

[17] Pskhu A.V., Equations in private derivatives of a fractional order, Nauka, M., 2005, 199 pp. (in Russian)

[18] Ladyzhenskaya A. A., Boundary value problems of mathematical physics, Nauka, M., 1973, 407 pp. (in Russian)