Semigroup property of fractional differential operators and its applications

Nguyen Dinh Cong

doi:10.3934/dcdsb.2022064

Article Contents

2023, Volume 28, Issue 1: 1-19. Doi: 10.3934/dcdsb.2022064

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Semigroup property of fractional differential operators and its applications

Nguyen Dinh Cong

Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet Roard, 10072 Hanoi, Vietnam

Received: July 2021

Revised: March 2022

Early access: March 2022

Published: January 2023

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Abstract

We establish partial semigroup property of families of Riemann-Liouville and Caputo fractional differential operators. Using this result we prove theorems on reduction of multi-term fractional differential systems to single-term and multi-order systems. As an application we obtain existence and uniqueness of solution to multi-term Caputo fractional differential systems.

Keywords:

Riemann-Liouville fractional operators,
Caputo fractional operators,
semigroup property,
multi-term Caputo fractional differential equations,
Lyapunov stability.

Mathematics Subject Classification: Primary: 34A08, 34A12; Secondary: 34L99.

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References

[1]	P. Badri and M. Sojoodi, Stability and stabilization of fractional-order systems with different derivative orders, Asian J. Control, 21 (2019), 2270-2279. doi: 10.1002/asjc.1847.
[2]	M. Caputo, Linear models of dissipation whose Q is almost frequency independent-Ⅱ, Fract. Calc. Appl. Anal., 11 (2008), 4-14.
[3]	W. Deng, C. Li and Q. Guo, Analysis of fractional differential equations with multi-orders, Fractals, 15 (2007), 173-182. doi: 10.1142/S0218348X07003472.
[4]	K. Diethelm, The Analysis of Fractional Differential Equations. An Application-Oriented Exposition Using Differential Operators of Caputo Type, Lecture Notes in Mathematics, 2004. Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-14574-2.
[5]	K. Diethelm, S. Siegmund and H. T. Tuan, Asymptotic behavior of solutions of linear multi-order fractional differential systems, Fract. Calc. Appl. Anal., 20 (2017), 1165-1195. doi: 10.1515/fca-2017-0062.
[6]	R. Gorenflo, A. A. Kilbas, F. Mainardi and S. V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications, Springer Monographs in Mathematics. Springer, Heidelberg, 2014. doi: 10.1007/978-3-662-43930-2.
[7]	A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.
[8]	K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.
[9]	I. Podlubny, Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999.
[10]	J. Rebenda, Application of differential transform to multi-term fractional differential equations with non-commensurate orders, Symmetry, 11 (2019), 1390.
[11]	G. Vainikko, Which functions are fractionally differentiable?, Z. Anal. Anwend., 35 (2016), 465-487. doi: 10.4171/ZAA/1574.