Geodesic
Neumann problem for an ordinary second-order differential equation with a distributed differentiation operator
News of the Kabardin-Balkar scientific center of RAS, no. 5 (2019), pp. 30-37

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In this paper, we study a linear ordinary second-order differential equation with a continuously distributed differentiation operator and study a two-point boundary-value problem by the Green function method. Fractional differentiation is presented in the sense of the Riemann–Liouville. Green function of the Neumann problem is constructed in term of a special function. The main properties for Green functions are proved. The explicit form of the solution for two-point boundary value problem to the equation under consideration is defined, when the solvability condition is satisfied. Requirements for the kernel of a continuously distributed differentiation operator that guarantee the fulfillment of the solvability condition for the Neumann problem are indicated.
Keywords: Neumann problem, Green's function, operator of continuously distributed differentiation, operator of fractional Riemann–Liouville differentiation. 
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     author = {B. I. Efendiev},
     title = {Neumann problem for an ordinary second-order differential equation with a distributed differentiation operator},
     journal = {News of the Kabardin-Balkar scientific center of RAS},
     pages = {30--37},
     publisher = {mathdoc},
     number = {5},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IZKAB_2019_5_a3/}
}
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B. I. Efendiev. Neumann problem for an ordinary second-order differential equation with a distributed differentiation operator. News of the Kabardin-Balkar scientific center of RAS, no. 5 (2019), pp. 30-37. http://geodesic.mathdoc.fr/item/IZKAB_2019_5_a3/