Vestnik КRAUNC. Fiz.-Mat. nauki. 2023. vol. 44. no. 3. P. 58-66. ISSN 2079-6641


Research Article

Full text in Russian

MSC 26A33, 34B05

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Nonlocal Boundary Value Problem for an Equation with Fractional Derivatives with Different Origins

L. M. Eneeva^\ast

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

Abstract. We consider a linear ordinary differential equation of fractional order with a composition of left and right-sided fractional derivative operators in the principal part. Equations containing a composition of fractional order differentiation operators with different origins appear when modeling various physical and geophysical phenomena. Their appearance is caused by the use of the concept of the effective rate of change in the parameters of the simulated processes. In particular, equations of the type considered in this work arise when describing dissipative oscillatory systems. Fractional differentiation is understood in the sense of Riemann-Liouville and Gerasimov-Caputo. For the equation under study, a nonlocal boundary value problem is investigated. The nonlocal boundary condition is specified in the form of an integral operator of the desired solution. Under a certain condition on the kernel of the operator appearing in the nonlocal condition, the problem under consideration is equivalently reduced to the Fredholm integral equation of the second kind. Sufficient conditions for the unique solvability of the problem under study are found, including an integral constraint on the variable potential. As a corollary, the Lyapunov inequality for solutions to the nonlocal problem under consideration is obtained. It is shown that the condition on the kernel of the integral operator from the nonlocal condition that arises in the solution of the problem is necessary in the sense that if this condition is violated, the uniqueness of the solution to the problem is lost.

Key words: fractional differential equation with different origins, nonlocal boundary value problem, Riemann–Liouville derivative, Gerasimov–Caputo derivative, Lyapunov inequality.

Received: 20.10.2023; Revised: 26.10.2023; Accepted: 28.10.2023; First online: 02.11.2023

For citation. Eneeva L. M. Nonlocal boundary value problem for an equation with fractional derivatives with different origins. Vestnik KRAUNC. Fiz.-mat. nauki. 2023, 44: 3, 58-66. EDN: YOUDLG.

Funding. The study was carried out without 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 forsubmitting the final version of the article to the press.

^\astCorrespondence: E-mail:

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

© Eneeva L. M., 2023

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


  1. Nakhushev A. M. Drobnoye ischisleniye i yego primeneniye. [Fractional calculus and its application]. Moscow. Fizmatlit, 2003. 272 p.(In Russian).
  2. Rekhviashvili S. Sh. Lagrange formalism with fractional derivative in problems of mechanics, Technical Physics Letters, 2004, vol. 30, no. 2, pp. 33–37.
  3. Rekhviashvili S. Sh. Fractional derivative physical interpretation, Nonlinear world, 2007, vol. 5, no. 4, pp. 194–197.(In Russian).
  4. Eneeva L. M. Boundary value problem for differential equation with fractional order derivatives with different origins, Vestnik KRAUNC. Fiz.-Mat. Nauki, 2015, vol. 3, no. 2(11), pp. 39–44.(In Russian).
  5. Stankovi´c B. An equation with left and right fractional derivatives, Publications de l’institut math´ematique. Nouvelle s´erie, 2006. Т. 80(94), С. 259–272.
  6. Atanackovic T. M., Stankovic B. On a differential equation with left and right fractional derivatives, Fractional Calculus and Applied Analysis, 2007. Т. 10, №2, С. 139–150.
  7. Torres C. Existence of a solution for the fractional forced pendulum, Journal of Applied Mathematics and Computational Mechanics, 2014. Т. 13, №1, С. 125–142.
  8. Tokmagambetov N., Torebek B. T. Fractional Analogue of Sturm-Liouville Operator Documenta Mathematica, 2016. Т. 21, С. 1503–1514.
  9. Eneeva L., Pskhu A., Rekhviashvili S. Ordinary Differential Equation with Left and Right Fractional Derivatives and Modeling of Oscillatory Systems. Mathematics, 2020. Т. 8(12), С. 2122.
  10. Eneeva L. M. Mixed boundary value problem for an ordinary differential equation with fractional derivatives with different origins, Vestnik KRAUNC. Fiz.-Mat. Nauki, 2021, vol. 36, no. 3, pp. 65–71.(In Russian).
  11. Eneeva L. M. Solution of a mixed boundary value problem for an equation with fractional derivatives with different origins, Vestnik KRAUNC. Fiz.-Mat. Nauki, 2022, no. 3(40), pp. 64–71.(In Russian).

Information about author

Eneeva Liana Magometovna – Ph. D. (Phys. & Math.), Senior Researcher at the Institute of Applied Mathematics and Automation of the Kabardino-Balkarian Scientific Center of the Russian Academy of Sciences, Nalchik, Russia, ORCID 0000-0003-2530-5022.