# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2022064
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## Semigroup property of fractional differential operators and its applications

 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

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.

Citation: Nguyen Dinh Cong. Semigroup property of fractional differential operators and its applications. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022064
##### 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.

show all references

##### 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.
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