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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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/

[1] A. M. Nakhushev, “On continuous differential equations and their difference analogues”, Reports of the USSR Academy of Sciences, 300:4 (1988), 796–799 | MR | Zbl

[2] A. M. Nakhushev, “Positiveness of the operators of continual and discrete differentiation and integration, which are quite important in the fractional calculus and in the theory of mixed-type equations”, Differential Equations, 34:1 (1998), 101–109 | MR | Zbl

[3] A. M. Nakhushev, Fractional calculus and its application, Fizmatlit, Moscow, 2003, 272 pp.

[4] A. V. Pskhu, Equations in partial derivative of fractional order, Nauka, Moscow, 2005, 199 pp.

[5] A. V. Pskhu, “The Boundary Value Problem for a Fractional Differential Equation with Partial Derivatives”, News of Kabardin-Balkar Scientific Center of the Russian Academy of Sciences, 2002, no. 1, 76–78

[6] V. E. Fedorov, E. M. Streletskaya, “Initial-value problems for linear distributed-order differential equations in Banach spaces”, Electronic Journal of Differential Equations, 2018:176 (2018), 1–17 | MR | Zbl

[7] A. N. Kochubei, “Distributed order calculus and equations of ultraslow diffusion”, Journal of Mathematical Analysis and Applications, 340 (2008), 252–281 | DOI | MR | Zbl

[8] B. I. Efendiev, “Cauchy problem for a second order ordinary differential equation with a continual derivative”, Differential Equations, 47:9 (2011), 1364–1368 | MR | Zbl

[9] B. I. Efendiev, “Dirichlet problem for a second-order ordinary differential equation with a continual derivative”, Mathematical Notes, 97:4 (2015), 620–628

[10] B. I. Efendiev, “Neumann problem for a second order ordinary differential equation with a continual derivative”, Reports of the International Adyghe (Circassian) Academy of Sciences, 8:2 (2006), 87–89 | MR

[11] B. I. Efendiev, “On the fundamental solution to an ordinary differential equation with a continuous distributed differentiation operator”, News of Kabardin-Balkar Scientific Center of the Russian Academy of Sciences, 2018, no. 6 (86), 48–52