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The Existence and Uniqueness of Solution for Fractional Order Matrix Differential Equation

Xiaoxu Chen

Zibo Vocational Institute, 255314, China.

*Corresponding author: Xiaoxu Chen

Published: 16 November 2022 How to cite this paper


Due to the introduction of fractional calculus which is a generalization of ordi-nary (integer order) differentiation and integration to its fractional (non-integer) order counter- part, almost every problem in calculus can be revisited at a whole new level, where one does not necessarily restrict oneself to an integer order derivative or integral, which allows much more flexibility in solving real-life problems. We are motivated by the article of (Al-Zhour, 2005), in this article the biggest bright spot is that method of Kronecker products of matrices and the method of vector operators. This method of proof not mentioned in the previous knowledge, but it is not restricted by fractional order, and reduce the difficulty of solving equation which is known. It makes the unknown knowledge through variable substitution into the known knowledge, which is very easy to promote. Since previous studies did not give any method of solving matrix fractional dif-ferential equations and the formula, the solving process is too complicated and difficult. Formulas are concluded in this paper, and we also provide strict prove.

KEYWORDS: Kronecker products, Vector-operator, Mittag-Leffler matrix, Matrix fractional differential systems


A.A.Kilbas, H.M.Srivastava, J.J.Trujillo. Theory and Applications of Fractional differential equations, Elsevier, Amsterdam, 2006.

A.A.Kilbas, M.Saigo. Fractional integrals and derivatives of functions of Mittag Leffler type. Doki. Akad. Nauk Belarusi, 39(4), 91995, 22-26.

A.AChilkrii, II, Matichin. Presentation of solutions of linear systems with linear systems with fractional derivatives of the sense of Riemann-Liouville, Caputo and Miller Ross, J, Automat,Sci., 40(2008), 1-11.

A.Graham, Kronecker. Products and Matrix Calculus with Applications, first ed., Ellis Horwood Ltd, UK, 1981.

A.Kilicmane, Z. Al-Zhour. The general common exact solutions of coupled linear matrix and matrix differential equations. J.Anal.Comput., 1(1)(2005), 15-30.

D, Matignon. Stability results of fractional differential system with applications to control in: Proceeding of IMACS, IEEE-SMC, 1996, pp.963-968, Lille, France.

D. Delbosco, L. Rodinn. Existence and uniqueness for a nonlinear fractional differential equations. J. Math Appl. Anal., 204(1996), 609-625.

Ding Tongren. Application of qualitative methods for ordinary differential equations. Beijing: Peking University Press, 1987.

G.F.Van Loan. The ubiquitous Kronecker Products. J.Comput.Appl.Math., 123(2000), 85-100.

G.Wang. Weighted Moore-Penrose, Drazin and group inverses of the Kronecker and products  , and some applications, Linear Algebra Appl., 250(1997), 39-50.

J.R.Magnus, H.Neudecker. Matrix Differential Calculus with Applications Statistics and Econometrics John Wiley& Sons, Ltd., 1999.

L. Podlubny. The Laplace Transform Method for Linear Differential Equations of Franctional Order, Slovac Republic, Slovac Academy of Science, 1994.

M.M.Al-Hajri. Use of fractional calculus to solve ordinary differential equations (Master thesis), Kuwait University, Kuwait, 1999.

R.Gorenflo, F, Mainardi. Fractional calculus: fractional of order, in: Carpintrri, Mainardi (Eds.). Fractals and Fractional calculus, 1997. New York.

S. Barnett. Introduction to Mathematical Control Theory, Oxford University Press, Oxford, 1975.

W. Debg, C. Li, J. Lu. Stability analysis of linear fractional differential system with multiple time delays. Nonlinear Dynam, 48(2007), 409-416.

Wang Kaohsiung, Zhou Zhiming, Zhu Siming, Wang Shousong. Ordinary Differential Equations. Higher Education Press, 2006.

Z. Al-Zhour, A. Kilicman. Matrix equalities and matrix inequalities involving Khatri-rao and Tracy-Singh sums. J.Ineq. Pure Appl.Math., 7(1)(2006), 496-513. Article 34.

How to cite this paper

Xiaoxu Chen. The Existence and Uniqueness of Solution for Fractional Order Matrix Differential Equation. OA Journal of Education Research, 2022, 1(4), 297-309.

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