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 Vestnik KRAUNC. Fiz.-Mat. Nauki, 2019, Volume 29, Number 4, Pages 48–57 (Mi vkam369)

MATHEMATICS

The Dirichlet problem for an ordinary differential equation of the second order with the operator of distributed differentiation

B. I. Efendiev

Institute of Applied Mathematics and Automation, 360000, Nalchik, Shortanova st., 89 A, Russia

Abstract: In this paper, we study a linear ordinary differential equation of the second order with operator of continuously distributed differentiation, and for him we study the two-point boundary value problem by the Greens function method. A special function is introduced, in terms of which the Green function of the Direchle problem is constructed and the main properties are proved. Sufficient conditions on the kernel of the operator of continuously distributed differentiation are determined that guarantee the fulfillment of the solvability condition for the Dirichlet problem. In the case when the homogeneous Dirichlet problem for the homogeneous equation under consideration has a nontrivial solution, an analog of the Lyapunov inequality is obtained for the kernel of a continuously distributed ifferentiation operator.

Keywords: fractional integro-differentiation operator, operator of continuously distributed differentiation, fundamental solution, Cauchy problem, Dirichlet problem, analog of Lyapunov's inequality.

DOI: https://doi.org/10.26117/2079-6641-2019-29-4-48-57

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UDC: 517.925.4
MSC: 34L99

Citation: B. I. Efendiev, “The Dirichlet problem for an ordinary differential equation of the second order with the operator of distributed differentiation”, Vestnik KRAUNC. Fiz.-Mat. Nauki, 29:4 (2019), 48–57

Citation in format AMSBIB
\Bibitem{Efe19} \by B.~I.~Efendiev \paper The Dirichlet problem for an ordinary differential equation of the second order with the operator of distributed~differentiation \jour Vestnik KRAUNC. Fiz.-Mat. Nauki \yr 2019 \vol 29 \issue 4 \pages 48--57 \mathnet{http://mi.mathnet.ru/vkam369} \crossref{https://doi.org/10.26117/2079-6641-2019-29-4-48-57}