Boundary-Value Problem with Shift for a Linear Ordinary Differential Equation with the Operator of Discretely Distribute
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BOUNDARY-VALUE PROBLEM WITH SHIFT FOR A LINEAR ORDINARY DIFFERENTIAL EQUATION WITH THE OPERATOR OF DISCRETELY DISTRIBUTED DIFFERENTIATION UDC 517.91
L. Kh. Gadzova
Abstract. In this paper, we study a boundary-value problem with local shift for a linear ordinary differential equation with the operator of discretely distributed differentiation, which links values of the solution at the endpoints of the considered interval with values at interior points. Keywords and phrases: fractional differentiation operator, Caputo derivative, boundary-value problem. AMS Subject Classification: 34A08
1. Introduction and statement of the problem. In the interval 0 < x < 1, let us consider the equation m α βj ∂0xj u(x) + λu(x) = f (x), (1) j=1
γ u(x) is a regularized fractional where αj ∈ ]1, 2[, λ, βj ∈ R, β1 > 0, α1 > α2 > . . . > αm , and ∂0x derivative (the Caputo derivative, see [12, p. 11]):
γ D0x
γ γ−n (n) u(x) = D0x u (x), ∂0x
n − 1 < γ ≤ n.
(2)
Here is the Riemann–Liouville fractional integro-differentiation operator of order γ defined as follows (see [12, p. 9]): ⎧ x ⎪ 1 ⎪ ⎪ u(t)(x − t)−γ−1 dt, γ < 0, ⎪ ⎪ ⎪ Γ(−γ) ⎨ 0 γ u(x) = D0x ⎪ u(x), γ = 0, ⎪ ⎪ ⎪ n ⎪ ⎪ ⎩ d D γ−n u(x), n − 1 < γ ≤ n, n ∈ N. dxn 0x Linear fractional ordinary differential equations were studied by many authors; a detailed bibliography on this subject can be found in [12, 14, 15, 19, 20]. A significant contribution to the study of fractional differential equations was made by the authors of [2, 4, 5]. Equations with operators of fractional discretely distributed differentiation were investigated in [17, 18]. In [11], an equation with the Caputo operator of discretely distributed order was examined by numerical methods. A priori estimates of differential and finite-difference problems with discretely distributed operators were obtained in [1]. We also note the work [3], in which a boundary-value problem with displacement for an ordinary differential equation with the Dzhrbashyan–Nersesyan fractional differentiation operator was studied. For Eq. (1), the Dirichlet and Neumann problems are solved, the corresponding Green functions are constructed, and properties of real eigenvalues are examined in [6–8]. In recent years, interest in boundary-value problems for fractional differential equations has increased significantly due to applications found in physics and mathematical modeling of fractal media (see [10, 12, 13, 20]). Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 149, Proceedings of the International Conference “Actual Problems of Applied Mathematics and Physics,” Kabardino-Balkaria, Nalchik, May 17–21, 2017, 2018.
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In this paper, we consider a nonlocal boundary-value problem for an ordinary differential equation with a discretely distributed differentiation operator and its specific case, namely, the two-point boundary-value problem. A regular solution of Eq. (1) is a function u = u(x), which has absolutely continuo
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