Vestnik KRAUNC. Fiz.-Mat. Nauki. 2022. vol. 40. no. 3. pp. 42–52. ISSN 2079-6641
Non-local boundary value problem for a system of ordinary differential equations with Riemann–Liouville derivatives
M. O. Mamchuev¹, T. I. Zhabelova²
¹Institute of Applied Mathematics and Automation of KBSC of RAS, 360017, Nalchik, Shortanov str. 89-A, Russia,
²Scientific and educational center of KBSC of RAS, 360010, Nalchik, Balkarov str., 4, Russia
We study a nonlocal boundary value problem for a linear system of ordinary differential equations of fractional order with constant coefficients on the interval \left[0, l\right] . The fractional derivative of order \alpha\in \left(0,1\right] is understood in the Riemann–Liouville sense. The boundary conditions connect the trace of the fractional integral of the desired vector function at the left end of the segment – at the point x = 0 , with the trace of the vector function itself at the right end of the segment at the point x = l . The purpose of this work is to construct an explicit representation of the solution of this problem in terms of the Green’s function. The structure of the solution to the boundary value problem is investigated, the corresponding Green’s function is defined and constructed, and the representation of the solution is obtained. A theorem on the unique solvability of the boundary value problem under study is proved.
Key words: system of ordinary differential equations, fractional derivatives, non-local boundary value problem, Green’s function.
Original article submitted: 02.10.2022
Revision submitted: 17.10.2022
For citation. Mamchuev M. O., Zhabelova T. I. Non-local boundary value problem for a system of ordinary differential equations with Riemann–Liouville derivatives. Vestnik KRAUNC. Fiz.-mat. nauki. 2022, 40: 3, 42-52. DOI: 10.26117/2079-6641-2022-40-3-42-52
Competing interests. The authors declare that 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.
The content is published under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0/deed.ru)
© Mamchuev M.O., Zhabelova T. I., 2022
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Mamchuev Murat Osmanovich – D. Sci. (Phys. & Math.), Head of the Department of Fractional Calculus, Institute of Applied Mathematics and Automation, KBSC RAS, Nalchik, Russia, ORCID 0000-0002-7986-456X.
Zhabelova Tanzilya Ibragimovna – Post-graduate student of the Scientific and Educational Center of the KBSC RAS, Nalchik, Russia, ORCID 0000-0001-8447-071X.