Analysis of the generalized fractional differential system

Jianhua Tang; Chuntao Yin; Jianhua Tang; Chuntao Yin

doi:10.3934/math.2022484

AIMS Mathematics

2022, Volume 7, Issue 5: 8654-8684. doi: 10.3934/math.2022484

Previous Article Next Article

Research article Special Issues

Analysis of the generalized fractional differential system

Jianhua Tang ¹,
Chuntao Yin ^{1,2
,
,}

1.
Department of Mathematics, Shanghai University, Shanghai 200444, China
2.
Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China

Received: 03 December 2021 Revised: 17 January 2022 Accepted: 16 February 2022 Published: 02 March 2022
MSC : 26A33, 34A08, 34A12, 34D20, 34D30

In this paper, we study the existence, uniqueness, and stability of the solution of the fractional differential system with the generalized fractional derivative. First, the solution of the generalized fractional differential system is obtained by the transformation method. Based on the fixed point theorems, we establish the existing and unique theories of the solution. Furthermore, the sufficient criteria of local stabilities of one-dimensional, two-dimensional, and $ n $ -dimensional linear generalized fractional differential systems are dealt with. In addition, the linearization and stability theorems of the nonlinear generalized fractional differential systems are discussed. Finally, we take the generalized fractional Chen system as an example to illustrate the correctness of the theoretical analysis.
- Erdélyi-Kober fractional derivative,
- generalized fractional derivative,
- existence,
- uniqueness,
- stability
Citation: Jianhua Tang, Chuntao Yin. Analysis of the generalized fractional differential system[J]. AIMS Mathematics, 2022, 7(5): 8654-8684. doi: 10.3934/math.2022484

Related Papers:

Abstract

In this paper, we study the existence, uniqueness, and stability of the solution of the fractional differential system with the generalized fractional derivative. First, the solution of the generalized fractional differential system is obtained by the transformation method. Based on the fixed point theorems, we establish the existing and unique theories of the solution. Furthermore, the sufficient criteria of local stabilities of one-dimensional, two-dimensional, and $ n $ -dimensional linear generalized fractional differential systems are dealt with. In addition, the linearization and stability theorems of the nonlinear generalized fractional differential systems are discussed. Finally, we take the generalized fractional Chen system as an example to illustrate the correctness of the theoretical analysis.

References

[1]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Amsterdam: Elsevier, 2006.
[2]	C. P. Li, M. Cai, Theory and numerical approximations of fractional integrals and derivatives, Philadelphia: SIAM, 2019. https://doi.org/10.1137/1.9781611975888
[3]	C. P. Li, F. H. Zeng, Numerical methods for fractional calculus, USA: Chapman and Hall/CRC, 2015. https://doi.org/10.1201/b18503
[4]	I. Podlubny, Fractional differential equations, San Diego: Academic Press, 1998.
[5]	S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives: Theory and applications, Amsterdam: Gordon and Breach Science, 1993.
[6]	E. Y. Fan, C. P. Li, Z. Q. Li, Numerical approaches to Caputo-Hadamard fractional derivatives with applications to long-term integration of fractional differential systems, Commun. Nonlinear Sci. Numer. Simul., 106 (2022), 106096. https://doi.org/10.1016/j.cnsns.2021.106096 doi: 10.1016/j.cnsns.2021.106096
[7]	G. Pagnini, Erdélyi-Kober fractional diffusion, Fract. Calc. Appl. Anal., 15 (2012), 117–127. https://doi.org/10.2478/s13540-012-0008-1 doi: 10.2478/s13540-012-0008-1
[8]	V. S. Kiryakova, Generalized fractional calculus and applications, CRC Press, 1993.
[9]	Z. Odibat, D. Baleanu, On a new modification of the Erdélyi-Kober fractional derivative, Fractal Fract., 5 (2021), 121. https://doi.org/10.3390/fractalfract5030121 doi: 10.3390/fractalfract5030121
[10]	A. Erdélyi, On fractional integration and its application to the theory of Hankel transforms, Quart. J. Math., 11 (1940), 293–303. https://doi.org/10.1093/qmath/os-11.1.293 doi: 10.1093/qmath/os-11.1.293
[11]	H. Kober, On a fractional integral and derivative, Quart. J. Math., 11 (1940), 193–211. https://doi.org/10.1093/qmath/os-11.1.193 doi: 10.1093/qmath/os-11.1.193
[12]	I. N. Sneddon, The use in mathematical physics of Erdélyi-Kober operators and of some of their generalizations, In: Fractional calculus and its applications, Germany: Springer, Berlin/Heidelberg, 1975. https://doi.org/10.1007/BFb0067097
[13]	I. N. Sneddon, The use of operators of fractional integration in applied mathematics, Warszawa-Poznan, 1979.
[14]	Y. Luchko, Operational rules for a mixed operator of the Erdélyi-Kober type, Fract. Calc. Appl. Anal., 7 (2004), 339–364.
[15]	M. Saigo, On the Hölder continuity of the generalized fractional integrals and derivative, Math. Rep., Kyushu Univ., 12 (1980), 55–62. https://doi.org/10.15017/1449020 doi: 10.15017/1449020
[16]	S. B. Yakubovich, Y. F. Luchko, The hypergeometric approach to integral transforms and convolutions, Boston: Kluwer Academic, 1994. https://doi.org/10.1007/978-94-011-1196-6
[17]	G. W. Wang, X. Q. Liu, Y. Y. Zhang, Lie symmetry analysis to the time fractional generalized fifth-order KdV equation, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 2321–2326. https://doi.org/10.1016/j.cnsns.2012.11.032 doi: 10.1016/j.cnsns.2012.11.032
[18]	B. Kour, S. Kumar, Symmetry analysis, explicit power series solutions and conservation laws of the space-time fractional variant Boussinesq system, Eur. Phys. J. Plus, 133 (2018), 520. https://doi.org/10.1140/epjp/i2018-12297-1 doi: 10.1140/epjp/i2018-12297-1
[19]	V. S. Kiryakova, B. N. Al-Saqabi, Transmutation method for solving Erdélyi-Kober fractional differintegral equations, J. Math. Anal. Appl., 221 (1997), 347–364. https://doi.org/10.1006/jmaa.1997.5469 doi: 10.1006/jmaa.1997.5469
[20]	Q. H. Ma, J. Pečarić, Some new explicit bounds for weakly singular integral inequalities with applications to fractional differential and integral equations, J. Math. Anal. Appl., 341 (2008), 894–905. https://doi.org/10.1016/j.jmaa.2007.10.036 doi: 10.1016/j.jmaa.2007.10.036
[21]	J. R. Wang, X. W. Dong, Y. Zhou, Analysis of nonlinear integral equations with Erdélyi-Kober fractional operator, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 3129–3139. https://doi.org/10.1016/j.cnsns.2011.12.002 doi: 10.1016/j.cnsns.2011.12.002
[22]	B. Ahmad, A. Alsaedi, S. K. Ntouyas, J. Tariboom, Hadamard-type fractional differential equations, inclusions and inequalities, Switzerland: Springer, 2017. https://doi.org/10.1007/978-3-319-52141-1
[23]	K. Diethelm, The analysis of fractional differential equations, Heidelberg: Springer, 2010. https://doi.org/10.1007/978-3-642-14574-2
[24]	U. N. Katugampola, Existence and uniqueness results for a class of generalized fractional differential equations, arXiv, 2016. Available from: https://arXiv.org/abs/1411.5229.
[25]	S. M. Momani, Local and global uniqueness theorems on differential equations of non-integer order via Bihari's and Gronwall's inequalities, Rev. Téc. Fac. Ing., 23 (2000), 66–69.
[26]	S. B. Hadid, Local and global existence theorems on differential equations of non-integer order, J. Fract. Calc., 7 (1995), 101–105.
[27]	C. P. Li, S. Sarwar, Existence and continuation of solutions for Caputo type fractional differential equations, Electron. J. Diff. Equ., 2016 (2016), 1–14.
[28]	R. Almeida, A. B. Malinowska, M. T. T. Monteiro, Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications, Math. Method. Appl. Sci., 41 (2018), 336–352. https://doi.org/10.1002/mma.4617 doi: 10.1002/mma.4617
[29]	M. Gohar, C. P. Li, C. T. Yin, On Caputo-Hadamard fractional differential equations, Int. J. Comput. Math., 97 (2020), 1459–1483. https://doi.org/10.1080/00207160.2019.1626012 doi: 10.1080/00207160.2019.1626012
[30]	R. W. Ibrahim, S. Momani, On the existence and uniqueness of solutions of a class of fractional differential equations, J. Math. Anal. Appl., 334 (2007), 1–10. https://doi.org/10.1016/j.jmaa.2006.12.036 doi: 10.1016/j.jmaa.2006.12.036
[31]	Y. Li, Y. Q. Chen, I. Podlubny, Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability, Comput. Math. Appl., 59 (2010), 1810–1821. https://doi.org/10.1016/j.camwa.2009.08.019 doi: 10.1016/j.camwa.2009.08.019
[32]	I. Petráš, Stability of fractional-order systems, In: Fractional-order nonlinear systems, Heidelberg: Springer, Berlin, 2011. https://doi.org/10.1007/978-3-642-18101-6_4
[33]	Y. Q. Chen, K. L. Moore, Analytical stability bound for a class of delayed fractional-order dynamic systems, Nonlinear Dynam., 29 (2002), 191–200. https://doi.org/10.1023/A:1016591006562 doi: 10.1023/A:1016591006562
[34]	D. Matignon, Stability results for fractional differential equations with applications to control processing, Comput. Eng. Syst. Appl., 2 (1996), 963–968.
[35]	W. H. Deng, C. P. Li, J. H. Lü, Stability analysis of linear fractional differential system with multiple time delays, Nonlinear Dynam., 48 (2006), 409–416. https://doi.org/10.1007/s11071-006-9094-0 doi: 10.1007/s11071-006-9094-0
[36]	D. L. Qian, C. P. Li, R. P. Agarwal, P. J. Y. Wong, Stability analysis of fractional differential system with Riemann-Liouville derivative, Math. Comput. Model., 52 (2010), 862–874. https://doi.org/10.1016/j.mcm.2010.05.016 doi: 10.1016/j.mcm.2010.05.016
[37]	C. P. Li, Y. T. Ma, Fractional dynamical system and its linearization theorem, Nonlinear Dynam., 71 (2013), 621–633. https://doi.org/10.1007/s11071-012-0601-1 doi: 10.1007/s11071-012-0601-1
[38]	C. P. Li, Z. Q. Li, Stability and logarithmic decay of the solution to Hadamard-type fractional differential equation, J. Nonlinear Sci., 31 (2021), 31. https://doi.org/10.1007/s00332-021-09691-8 doi: 10.1007/s00332-021-09691-8
[39]	C. P. Li, Z. Q. Li, Stability and $ \psi $ -algebraic decay of the solution to $ \psi $ -fractional differential system, Int. J. Nonlinear Sci. Numer. Simul., 2021. https://doi.org/10.1515/ijnsns-2021-0189

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)