Second boundary-value problem for the generalized Aller--Lykov equation

M. A. Kerefov ; S. Kh. Gekkieva

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 23 (2019) no. 4, pp. 607-621.

Résumé

The equations that describe a new type of wave motion arise in the course of mathematical modeling for continuous media with memory. This refers to differential equations of fractional order, which form the basis for most mathematical models describing a wide class of physical and chemical processes in media with fractal geometry. The paper presents a qualitatively new equation of moisture transfer, which is a generalization of the Aller–Lykov equation, by introducing the concept of the fractal rate of change in humidity clarifying the presence of flows affecting the potential of humidity. We have studied the second boundary value problem for the Aller–Lykov equation with the fractional Riemann–Liouville derivative. The existence of a solution to the problem has been proved by the Fourier method. To prove the uniqueness of the solution we have obtained an a priori estimate, in terms of a fractional Riemann–Liouville using the energy inequality method.

Détail

Keywords: second boundary-value problem, Fourier method, fractional Riemann–Liouville operator of fractional integro-differentiation, method of energy inequalities.
Mots-clés : Aller–Lykov equation

Bibliographie

[1] Chudnovsky A. F., Teplofizika pochv [Thermophysics of soils], Nauka, Moscow, 1976, 352 pp. (In Russian)

[2] Nakhushev A. M., Drobnoe ischislenie i ego primenenie [Fractional calculus and its applications], Fizmatlit, Moscow, 2003, 352 pp. (In Russian) | Zbl

[3] Nakhushev A. M., Uravneniia matematicheskoi biologii [Equations of mathematical biology], Vysshaia Shkola, Moscow, 1995, 301 pp. (In Russian) | Zbl

[4] Kulik V. Ya., “The study of the soil moisture movement from the point of view of invariance of continuous groups of transformations”, Issledovanie processov obmena jenergiej i veshhestvom v sisteme pochva-rastenie-vozduh [The Study of the Processes of Energy and Mass-Transfer in the System Soil-Plant-Air], Nauka, Leningrad, 1972 (In Russian)

[5] Arkhestova S. M., Shkhanukov–Lafishev M. Kh., “Difference schemes for the Aller–Lykov moisture transfer equation with a nonlocal condition”, Izvestiya KBSC RAS, 2012, no. 3, 7–16 (In Russian)

[6] Lafisheva M. M., Kerefov M. A., Dyshekova R. V., “Difference schemes for the Aller–Lykov moisture transfer equations with a nonlocal condition”, Vladikavkaz. Mat. Zh., 19:1 (2017), 50–58 (In Russian) | MR

[7] Gekkieva S. Kh., “The first boundary-value problem for the Aller–Lykov moisture transfer equations with a fractional time derivative”, Ustoychivoe razvitie: problemy, kontseptsii, modeli [Sustainable Development. Problems, Concepts, Models], KBSC RAS, Nal'chik, 2017, 99–102 (In Russian)

[8] Gekkieva S. Kh., Kerefov M. A., “The boundary-value problem for the generalized moisture transfer equation”, Vestnik KRAUNC. Fiz.-Mat. Nauki, 2018, no. 1(21), 21–31 (In Russian) | DOI | MR | Zbl

[9] Gekkieva S. Kh., “Nonlocal boundary-value problem for the generalized Aller–Lykov moisture transport equation”, Vestnik KRAUNTs. Fiz.-mat. nauki, 2018, no. 4(24), 19–28 (In Russian) | DOI | MR | Zbl

[10] Gekkieva S. Kh., “A boundary-value problem for the generalized transport equation with a fractional time derivative”, Dokl. Adygskoi (Cherkesskoi) Mezhdunar. Akad. Nauk, 1:1 (1994), 17–18 (In Russian)

[11] Agrawal O. P., “Solution for a fractional diffusion-wave equation defined in a bounded domain”, Nonlinear Dynamics, 29:1 (2002), 145–155 | DOI | MR | Zbl

[12] Nakhusheva V. A., Differentsial'nye uravneniia matematicheskikh modelei nelokal'nykh protsessov [Differential Equations of Mathematical Models of Nonlocal Processes], Nauka, Moscow, 2006, 173 pp. (In Russian)

[13] Turmetov B. Kh., Torebek B. T., “On solvability of some boundary value problems for a fractional analogue of the Helmholtz equation”, New York J. Math., 20 (2014), 1237–1251 http://nyjm.albany.edu/j/2014/20-57p.pdf | MR | Zbl

[14] Masaeva O. Kh., “Uniqueness of solutions to Dirichlet problems for generalized Lavrent'ev–Bitsadze equations with a fractional derivative”, Electron. J. Differ. Equ., 2017 (2017), 1–8 https://ejde.math.txstate.edu/Volumes/2017/74/masaeva.pdf | MR

[15] Shogenov V. Kh., Kumykova S. K., Shkhanukov–Lafishev M. Kh., “Generalized transport equations and fractional derivatives”, Dop. Nats. Akad. Nauk Ukr., no. 12, 47–55 (In Russian) | MR

[16] Kerefov M. A., Boundary-value problems for a modified moisture transfer equation with a fractional time derivative, Cand. Phys. Math. Sci. Diss., Nal'chik, 2000, 175 pp. (In Russian)

[17] Yangarber V. A., “The mixed problem for a modified moisture-transfer equation”, J. Appl. Mech. Tech. Phys., 8:1 (1967), 62–64 | DOI

[18] Vladimirov V. S., Uravneniia matematicheskoi fiziki [Equations of mathematical physics], Nauka, Moscow, 1981, 512 pp. (In Russian) | MR

[19] Pskhu A. V., Uravneniia v chastnykh proizvodnykh drobnogo poriadka [Partial differential equations of fractional order], Nauka, Moscow, 2005, 199 pp. (In Russian)

[20] Kilbas A. A., Srivastava H. M., Trujillo J. J., Theory and applications of fractional differential equations, North-Holland Mathematics Studies, 204, Elsevier, Amsterdam, 2006, xv+523 pp. | MR | Zbl

[21] Pskhu A. V., “Initial-value problem for a linear ordinary differential equation of noninteger order”, Sb. Math., 202:4 (2011), 571–582 | DOI | DOI | MR | Zbl

[22] Samarskiy A. A., Teoriia raznostnykh skhem [Theory of Difference Schemes], Nauka, Moscow, 1971, 552 pp. (In Russian) | MR