Cauchy Problem for a Linear System of Ordinary Differential Equations of the Fractional Order

The paper investigates the initial problem for a linear system of ordinary differential equations with the fractional differentiation operator Dzhrbashyan -- Nersesyan with constant coefficients. The existence and uniqueness theorems of the solution of the boundary value problem under study are proved. The solution is constructed explicitly in terms of the Mittag-Leffler function of the matrix argument. The Dzhrbashyan -- Nersesyan operator is a generalization of the Riemann -- Liouville, Caputo and Miller-Ross fractional differentiation operators. The obtained results as special cases contain results related to the study of initial problems for systems of ordinary differential equations with Riemann -- Liouville, Caputo and Miller -- Ross derivativess, and the investigated initial problem generalizes them


Introduction
Consider the system of ordinary differential equations where D where D ν st is the Riemann -Liouville fractional integro-differentiation operator of order ν. The operator D ν st for ν < 0 is defined as follows [2, p. 9]: where Γ(z) is the Euler gamma function. For ν ≥ 0 the operator D ν st can be determined using the recursive relation In 1954, J.H. Barrett [3] investigated the initial problem for the equation In 1968, M.M. Dzhrbashyan and A.B. Nersesyan [1] introduced the fractional differentiation operator D {γ 0 ,γ 1 ,...,γm} 0t and investigated the Cauchy problem for the equation (1) for n = 1. Systems of linear ordinary equations of fractional order were first investigated in the works of V.K. Veber [4] - [8] and M.I. Imanaliev, V.K. Veber [5]. In 1976, V.K. Veber [4] in terms of the Mittag-Leffler function of the matrix argument, the solution of the Cauchy problem for the system of equations with constant matrix A was written. The asymptotic behavior as x → ∞ of various solutions of this system (including the fundamental matrix) was studied in [6] and [5]. Later V.K. Veber considered the Cauchy problem for an inhomogeneous system with continuous matrix function A(x) for x ≥ 0 [7], and in [8] he constructed a fundamental solution of this system with a constant matrix A in terms of the Mittag-Leffler function of a matrix argument. Also in the paper [8] are examples of some applications lead to systems of equations with fractional derivatives. A.A. Chikriy and I.I. Matychyn in [9] and [10], using the Laplace transform, obtained solutions of Cauchy problems for systems of equations of the form (1), with Riemann -Liouville, Caputo and Miller -Ross derivatives.
In the works [11] and [12] I. Matychyn and V. Onyshchenko, using the Mittag-Leffler matrix function, analytically and numerically investigated the solutions of the initial problems for systems of equations with fractional Riemann -Liouville and Caputo derivatives.
Note that [13] and [14] investigated boundary value problems for multidimensional systems of partial differential equations of fractional order. In [15], attention was drawn to the fact that in the one-dimensional case these results coincide with the results of [4].
In this paper, we investigate the initial problem for the system (1). We prove some properties of the Mittag-Leffler matrix function, obtain a general representation of the solutions to the system (1), and prove the theorem on the unique solvability of the Cauchy problem for this system.
Then there exists a unique regular in the interval (0, l) solution of Problem 1. Solution can be represented as where

Preliminaries
The formula is known as a formula of fractional integration by parts [2, p. 9].
The following formula of fractional integro-differentiation of power function which is now known as the Mittag-Leffler function.
In 1905, A. Wiman [18] generalized this function with the two-parameter Mittag-Leffler function E α,β (z) (also sometimes called the Wiman's function) The following properties of this function are valid: In 1971, an even more general function E γ α,β (z) was introduced by T.R. Prabhakar [19] is the Pohhammer symbol. This function is known as the Prabhakar function.
Here we give a definition of the Mittag-Leffler function of a matrix argument, and then examine some of its properties.
Let A be a square matrix and H a matrix reducing the matrix A to Jordan matrix i.e. where It's obvious that e 0 ≡ e α,β 0 (λ, z) = E α,β (λz).
We denote µ j = j i=0 γ i . Then, by virtue of the formula (16), for β − µ j ≥ 0 we get the following formula In particular, for β − µ j = 0 we have By using the formula (15) from (18) we obtain From the last equality, for j = m, and γ i = α i (i = 0, m), we get and for j = m, and γ i = α m−i (i = 0, m), we get 5. By virtue of formula (13) . Continuing similarly, we obtain Thus, the formula holds. Let the function V (x, t) is a solution of the equation and satisfies following conditions From the formulas (21), (22) and it follows that the function is the solution of problem (23), (24). From (1) and (23) follows that After integrating the equality (26), we obtain Using (4) and (24) from equality (27) we get Differentiating (28), taking into account the equality V (x, x) = 0, we obtain where Applying to (29) the formula that follows from (25), we get the equality (3). Lemma 1 is proved.

Proof of Theorem 1
Let us prove that the function (3) is the solution of Problem 1. We denote the last term in right hand side of (3) as u f (x) and u C (x) = u(x) − u f (x).
By virtue of (18) we get where Using the following equalities From (31), (22) and the inequality α 0 + α m > 1 it follows that Formula (32) gives the relations For s = m from (21) and (31) we get Thus, it remains for us to show that the last term in (3) is a solution to Problem 1 with homogeneous conditions. By virtue of the condition f (x) = D αm−1 0x f 0 (x), f 0 (x) ∈ L(0, l) and formulas (4) and (16), we can write The last equality yield Using formula (18) we get By virtue of (5), (16) and (22) we have Continuing in a similar way, we obtain the following equalities The relations (33), (34), (38) and (39) mean that the function (3) is the solution to Problem 1. The uniqueness of the solution to Problem 1 follows from Lemma 1.
Theorem 1 is proved.

Conclusions
The article investigates the initial problem for a linear system of ordinary differential equations with the Djrbashyan -Nersesyan fractional differentiation operator with constant coefficients. To solve the problem under study, the Green's function method is implemented, an integral representation of the solution is obtained, the properties of the matrix Mittag-Leffler function (autotransformation formula, fractional integro-differentiation formulas, etc.) are investigated. The results obtained can be used to study local and nonlocal boundary value problems for system (1).
Conflicts of Interest: The author declares no conflict of interest in this paper.