Adyghe Int. Sci. J. Vol. 24, No. 1. P. 11-22. 

Read article                                                                                                                     Contents of this issue

DOI: https://doi.org/10.47928/1726-9946-2024-24-1-11-22
EDN: INICMW

MATHEMATICS

MSC 35E99

Original Article

On the question of the existence of a solution to the first boundary value problem for the Aller – Lykov moisture transfer equation with the operator of fractional discretely distributed differentiation

Sakinat Khasanovna Gekkieva
Candidate of Physical and Mathematical Sciences, Senior Researcher at the Department of Computational Methods, Institute of Applied Mathematics and Automation of Kabardin-Balkar Scientific Centre of RAS (89A, Shortanova str., Nal’chik, 360000, Russian Federation), https://orcid.org/0000-0002-2135-2115, gekkieva_s@mail.ru
Marat Aslanbievich Kerefov
Candidate of Physical and Mathematical Sciences, Associate Professor of the Department of Applied Mathematics and Computer Science, Kabardino-Balkarian State University named after H.M. Berbekov (173, Chernyshevsogo str., Nal’chik, 360004, Russian Federation), https://orcid.org/0000-0002-7442-5402, kerefov@mail.ru

Abstract. The paper investigates the first boundary value problem for the Aller – Lykov moisture transfer equation with the operator of fractional discretely distributed differentiation. Fractional derivatives included in the equation are understood in the Riemann – Liouville sense. The equation in question is a generalization of the classical Aller – Lykov equation. It takes into account the colloidal capillary-porous structure of the soil, including the presence of flows against the moisture potential. The existence of a solution to the first boundary value problem is proved by the Fourier method.

Keywords: fractional order derivative, Cauchy problem, fractional order differential equation, Aller – Lykov moisture transfer equation.

Funding. The work was not carried out within the framework of funds.

Competing interests. There are no conflicts of interest regarding authorship and publication.

Contribution and Responsibility. All authors contributed to this article. Authors are solely responsible for providing the final version of the article in print. The final version of the manuscript was approved by all authors.

For citation. Gekkieva S. Kh., Kerefov M. A. On the question of the existence of a solution to~the~first boundary value problem for the Aller – Lykov moisture transfer equation with the operator of fractional discretely distributed differentiation. Adyghe Int. Sci. J. 2024. Vol. 24, No. 1. Pp. 11–22. DOI: https://doi.org/10.47928/1726-9946-2024-24-1-11-22; EDN: INICMW.

Submitted 23.01.2024; approved after reviewing 07.03.2024; aeeepted for publication 15.03.2024.

© Gekkieva S. Kh., Kerefov M. A., 2024

REFERENCES

1. Chudnovskiy A. F. Teplofizika pochv (Thermophysics of soils). Moscow: Nauka, 1976. 353 p.
2. Nakhushev A. M. Drobnoe ischislenie i ego primenenie (Fractional calculus and its applications). Moscow: Fizmatlit, 2003. 272 p.
3. Kulik V. Ja. Investigation of soil moisture movement from the point of view of invariance with respect to continuous transformation groups, Issledovanie processov obmena jenergiej I veshhestvom v sisteme pochva-rastenie-vozduh (Investigation of energy and substance exchange processes in the soil-plant-air system). Leningrad: Nauka, 1972. 315 p. (in Russian).
4. Arkhestova S. M., Shkhanukov-Lafishev M. Kh. Difference schemes for moisture transport equation of Aller-Lykov with non-local condition. Izvestiya KBNTs RAN. 2012. No. 3 (47). Pp. 7–16. https://elibrary.ru/item.asp?id=17822798 (in Russian)
5. Lafisheva M. M., Kerefov M. A., Dyshekova R. V. Difference schemes for the Aller – Lykov moisture transfer equations with a nonlocal condition. Vladikavkazskiy matematicheskiy zhurnal. 2017. Vol. 19, issue 1. Pp. 50–58. http://mi.mathnet.ru/vmj607 (in Russian)
6. Gekkieva S. Kh. Nonlocal boundary-value problem for the generalized Aller – Lykov moisture transport equation. Vestnik KRAUNC. Fiz.-mat. nauki. 2018. No. 4 (24). Pp. 19–28. https://doi.org/10.18454/2079-6641-2018-24-4-19-28. (in Russian)
7. Kerefov M. A., Nakhusheva F. M., Gekkieva S. Kh. Boundary value problem for the Aller – Lykov moisture transport generalized equation with concentrated heat capacity, Vestnik Samarskogo universiteta. Estestvennonauchnaya seriya. 2018. Vol. 24, No. 3. Pp. 23–29. https://elibrary.ru/item.asp?id=36731739. (in Russian)
8. Gekkieva S. Kh., Kerefov M. A. Boundary-value problem for the Aller–Lykov nonlocal moisture transfer equation. Proceedings of the IV International Scientific Conference “Actual Problems of Applied Mathematics”. Kabardino-Balkar Republic, Nalchik, Elbrus Region, May 22–26. 2018. Part III, Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., 167, VINITI, Moscow, 2019. Pp. 27–33. https://doi.org/10.36535/0233-6723-2019-167-27-33
9. Gekkieva S. Kh., Kerefov M. A. First boundary value problem for the Aller-Lykov moisture transfer equation with a time-fractional derivative. Ufimskiy matematicheskiy zhurnal. 2019. Vol. 11, No.~2. Pp. 72–82. http://mi.mathnet.ru/ufa472. (in Russian)
10. Kerefov M. A., Gekkieva S. Kh. Second boundary-value problem for the generalized Aller – Lykov equation, Vestnik Samarskogo gosudarstvennogo tekhnicheskogo universiteta. Seriya Fiziko-matematicheskiye nauki. 2019. Vol. 23, No. 4. Pp. 607–621. https://doi.org/10.14498/vsgtu1686. (in Russian)
11. Kerefov M. A., Gekkieva S. Kh. Numerical-analytical method for solving boundary value problem for the generalized moisture transport equation. Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki. 2021. Vol. 23, No. 1. Pp. 19–34. https://doi.org/10.35634/vm210102. (in Russian)
12. Gekkieva S. Kh., Kerefov M. A., Nakhusheva F. M. Local and nonlocal boundary value problems for generalized Aller–Lykov equation. Ufimsk. Mat. Zh. 2023. Vol. 15, No. 1. Pp. 22–34. https://doi.org/10.13108/2023-15-1-21
13. Kilbas A. A., Srivastava H. M., Trujillo J. J. Theory and applications of fractional differential equations. North-Holland Math. Stud. Vol. 204. Elsevier, Amsterdam. 2006. 499 p.
14. Pskhu A. V. Initial-value problem for a linear ordinary differential equation of noninteger order. Mat. Sb. 2011. Vol. 202, No. 4. Pp. 111–122. https://doi.org/10.1070/SM2011v202n04ABEH004156
15. Pskhu A. V. Uravneniya v chastnykh proizvodnykh drobnogo poryadka (Partial differential equations of fractional order). Moscow: Nauka, 2005. 199 p.
16. Smirnov V. I. Kurs vysshey matematiki (Higher Mathematics Course). Vol. 2. SPb.: BKhV-Peterburg, 2008. 848 p.
17. Dzhrbashyan M. M. Integral’nye preobrazovaniya i predstavleniya funktsiy v kompleksnoy oblasti (Integral transformations and representations of functions in the complex domain). Moscow: Nauka, 1966. 672 p.

This work is licensed under a Creative Commons Attribution 4.0 License.