# Vestnik КRAUNC. Fiz.-Mat. Nauki. 2023. vol. 42. no. 1. P. 98-107. ISSN 2079-6641

MATHEMATICS https://doi.org/10.26117/2079-6641-2023-42-1-98-107
Research Article
Full text in Russian
MSC 34A12, 34K09

Contents of this issue

The Cauchy Problem for the Delay Differential Equation with Dzhrbashyan – Nersesyan Fractional Derivative

M. G. Mazhgikhova^*

Institute of Applied Mathematics and Automation of Kabardino-Balkarian Scientific Center of RAS, 360000, Nalchik, Shortanova st., 89A, Russia

Abstract. In recent, the number of works devoted to the study of problems for fractional order differential equations is growing noticeably. The interest of researchers is due to the fact that the number of areas of science in which equations containing fractional derivatives are used varies from biology and medicine to control theory, engineering, finance, as well as optics, physics, and so on. The inclusion of delay in the fractional order equation significantly affects the course of the process described by this equation, since the unknown function is given for different values of the argument, which includes a history effect into the equation. Therefore, mathematical models containing a fractional operator and a delay argument are more accurate than models containing integer derivatives. In this paper, we study the Cauchy problem for a linear ordinary delay differential equation with the Dzhrbashyan – Nersesyan fractional differentiation operator, which is generalizing the Riemann – Liouville and Gerasimov – Caputo fractional operators. The results of the work are obtained using the methods of the theory of integer and fractional calculus, methods of the theory of delay differential equations, the method of special functions. In this paper proves a theorem on the validity of an analogue of the Lagrange formula. It is also proved that the special function W_{\gamma_m}\left(t\right), which is defined in terms of the generalized Mittag-Leffler function (or the Prabhakar function), satisfies the equation and conditions associated with the one under study, and is the fundamental solution of the considered equation. The main result is that the existence and uniqueness theorem to the initial value problem is proved. The solution to the problem is written out in terms of the special function W_\nu\left(t\right).

Key words: Dzhrbashyan – Nersesyan derivative, fractional differential equation, delay differential equation, Lagrange formula, fundamental solution, generalized Mittag – Leffler function.

Received: 20.12.2022; Revised: 21.03.2023; Accepted: 24.03.2023; First online: 16.04.2023

For citation. Mazhgikhova M. G. The Cauchy problem for the delay differential equation with Dzhrbashyan – Nersesyan fractional derivative. Vestnik KRAUNC. Fiz.-mat. nauki. 2023, 42: 1, 98-107. EDN: DMUMKE. https://doi.org/10.26117/2079-6641-2023-42-1-98-107.

Funding. The study was carried out without financial support from foundations.

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

Contribution and Responsibility. The author participated in the writing of the article and is fully responsible for
submitting the final version of the article to print.

The content is published under the terms of the Creative Commons Attribution 4.0 International License

© Institute of Cosmophysical Research and Radio Wave Propagation, 2023 (original layout, design, compilation)

References

1. Dzhrbashyan M. M., Nersesyan A. B. Fractional derivatives and the Cauchy problem for differential equations of fractional order, Izv. AN Arm. SSR., 1968, 36, 1, 3–29. (In Russian)
2. Pskhu A. V. The fundamental solution of a diffusion-wave equation of fractional order, Izv. Math., 2009, 73, 2, 351–392. DOI: 10.4213/im2429
3. Nakhushev A.M. Drobnoe ischislenie i ego primenenie [Fractional calculus and its application]. Fizmatlit, Moskva, 2003, 272 (In Russian)
4. Kilbas A. A., Srivastava H. M., Trujillo J. J. Theory and Applications of Factional Differential Equations. Elsevier, Amsterdam, 2006, 523
5. Pskhu A. V. Uravneniya v chastnykh proizvodnykh drobnogo poryadka [Fractional order partial differential equations]. Nauka, Moskva, 2005, 199. (In Russian)
6. Oldham K. B., Spanier J. The fractional calculus. N.-Y. L., Acad. press., 1974, 234.
7. Barrett J. H. Differential equation of non-integer order, Can. J. Math., 1954, 6, 4, 529–541.
8. Pskhu A. V. Initial-value problem for a linear ordinary differential equation of noninteger order, Sb. Math., 2011, 202, 4, 571–582.
9. Fedorov V. E., Plekhanova M. V., Izhberdeeva E. M. Initial value problems of linear equations with the Dzhrbashyan – Nersesyan derivative in Banach spaces, Symmetry, 2021, 13, 6, 1058.
10. Volkova A. R., IzhberdeevaE. M., Fedorov V. E. Initial value problems for equations with a composition of fractional derivatives, Chel. fiz.-mat. zh., 2021, 6, 3, 269–277. (In Russian)
11. Bogatyreva F. T. Initial value problem for fractional order equation with constant coefficients, Vest. KRAUNTs. Fiz.-mat. nauki., 2016, 16, 4–1, 21–6. (In Russian)
12. Bellman R. E., Cooke K. L. Differential-Difference Equations, N-Y., Acad. Press, 1963, 462.
13. Hale J. K, Lunel S. M. V. Introduction to Functional Differential Equations, N-Y, Springer, 1993. 449.
14. El’sgol’ts L. E. Introduction to the theory of differential equations with deviating argument. Nauka, Moskva, 1971, 296. (In Russian)
15. Myshkis A. D. Lineynye differentsial’nye uravneniya s zapazdyvayushchim argumentom [Linear delay differential equations]. Nauka, Moskva, 1972. (In Russian)
16. Norkin S. B. On solutions of a linear homogeneous delay differential equation, UMN, 1959, 14, 1:85, 199–206. (In Russian)
17. Mazhgikhova M. G. Cauchy problem for ordinary differential equation with Riemann-Liouville operator with delay, Izvestiya KBNTs RAN, 2017, 75, 1, 24–28. (In Russian)
18. Mazhgikhova M. G.Initial and boundary value problems for ordinary differential equation of fractional order with delay, Chel. Phys. and Math. Journal, 2018, 3, 1, 27–37. (In Russian)
19. Mazhgikhova M. G. Dirichlet problem for a fractional-order ordinary differential equation with retarded argument, Differential equations, 2018, 54, 2, 187–194. (In Russian)
20. Prabhakar T. R. A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math. J., 1971, 19, 7–15.
21. Naymark M. A. Lineynye differentsial’nye operatory [Linear differential operators]. Nauka,
Moskva, 1969, 528. (In Russian)