Properties of solutions for fractional-order linear system with differential equations

Shuo Wang; Juan Liu; Xindong Zhang; Shuo Wang; Juan Liu; Xindong Zhang

doi:10.3934/math.2022860

AIMS Mathematics

2022, Volume 7, Issue 8: 15704-15713. doi: 10.3934/math.2022860

Previous Article Next Article

Research article

Properties of solutions for fractional-order linear system with differential equations

1.
School of Mathematical Sciences, Xinjiang Normal University, Urumqi, Xinjiang 830017, China
2.
College of Big Data Statistics, Guizhou University of Finance and Economics, Guiyang 550025, China

Received: 22 April 2022 Revised: 09 June 2022 Accepted: 15 June 2022 Published: 24 June 2022
MSC : 39A70, 45P05

In this paper, we study the analytical solutions of two-dimensional fractional-order linear system $ \mathcal{D}^{\alpha}_{t}X(t) = AX(t) $ described by fractional differential equations, where $ \mathcal{D} $ is the fractional derivative in the Caputo-Fabrizio sense and $ A = (a_{ij})_{2\times2} $ is nonsingular coefficient matrix with $ a_{ij}\in\mathbb{R} $. The analytical solutions of fractional-order linear system will be compared to the solution of classical linear system. Examples are provided to characterize the behavior of the solutions for fractional-order linear system.
- linear system,
- Caputo-Fabrizio derivative,
- analytical solution,
- fractional differential,
- fractional integral
Citation: Shuo Wang, Juan Liu, Xindong Zhang. Properties of solutions for fractional-order linear system with differential equations[J]. AIMS Mathematics, 2022, 7(8): 15704-15713. doi: 10.3934/math.2022860

Related Papers:

Abstract

In this paper, we study the analytical solutions of two-dimensional fractional-order linear system $ \mathcal{D}^{\alpha}_{t}X(t) = AX(t) $ described by fractional differential equations, where $ \mathcal{D} $ is the fractional derivative in the Caputo-Fabrizio sense and $ A = (a_{ij})_{2\times2} $ is nonsingular coefficient matrix with $ a_{ij}\in\mathbb{R} $. The analytical solutions of fractional-order linear system will be compared to the solution of classical linear system. Examples are provided to characterize the behavior of the solutions for fractional-order linear system.

References

[1]	I. Podlubny, Fractional differential equations, San Diego: Academic Press, 1999.
[2]	Y. M. Lin, C. J. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), 1533–1552. https://doi.org/10.1016/j.jcp.2007.02.001 doi: 10.1016/j.jcp.2007.02.001
[3]	P. Zhuang, F. Liu, V. Anh, I. Turner, Numerical methods for the variable-order fractional advection diffusion equation with a nonlinear source term, SIAM J. Numer. Anal., 47 (2009), 1760–1781. https://doi.org/10.1137/080730597 doi: 10.1137/080730597
[4]	J. Losada, J. J. Nieto, Properties of a new fractional derivative without singular kernel, Prog. Fract. Differ. Appl., 1 (2015), 87–92.
[5]	M. Fardi, Y. Khan, A novel finite difference-spectral method for fractal mobile/immobiletransport model based on Caputo-Fabrizio derivative, Chaos Solitons Fract., 143 (2021), 110573. https://doi.org/10.1016/j.chaos.2020.110573 doi: 10.1016/j.chaos.2020.110573
[6]	J. J. Nieto, Solution of a fractional logistic ordinary differential equation, Appl. Math. Lett., 123 (2022), 107568. https://doi.org/10.1016/j.aml.2021.107568 doi: 10.1016/j.aml.2021.107568
[7]	M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Prog. Fract. Differ. Appl., 1 (2015), 73–85.
[8]	O. J. J. Algahtani, Comparing the Atangana-Baleanu and Caputo-Fabrizio derivative with fractional order: Allen Cahn model, Chaos Solitons Fract., 89 (2016), 552–559. https://doi.org/10.1016/j.chaos.2016.03.026 doi: 10.1016/j.chaos.2016.03.026
[9]	T. Akman, B. Yıldız, D. Baleanu, New discretization of Caputo-Fabrizio derivative, Comput. Appl. Math., 37 (2018), 3307–3333. https://doi.org/10.1007/s40314-017-0514-1 doi: 10.1007/s40314-017-0514-1
[10]	D. Baleanu, H. Mohammadi, S. Rezapour, A fractional differential equation model for the COVID-19 transmission by using the Caputo-Fabrizio derivative, Adv. Differ. Equ., 2020 (2020), 299. https://doi.org/10.1186/s13662-020-02762-2 doi: 10.1186/s13662-020-02762-2
[11]	D. Baleanu, A. Jajarmi, H. Mohammadi, S. Rezapour, A new study on the mathematical modelling of human liver with Caputo-Fabrizio fractional derivative, Chaos. Solitons Fract., 134 (2020), 109705. https://doi.org/10.1016/j.chaos.2020.109705 doi: 10.1016/j.chaos.2020.109705
[12]	S. M. Aydogan, D. Baleanu, H. Mohammadi, S. Rezapour, On the mathematical model of Rabies by using the fractional Caputo-Fabrizio derivative, Adv. Differ. Equ., 2020 (2020), 382. https://doi.org/10.1186/s13662-020-02798-4 doi: 10.1186/s13662-020-02798-4
[13]	M. Z. Xu, Y. J. Jian, Unsteady rotating electroosmotic flow with time-fractional Caputo-Fabrizio derivative, Appl. Math. Lett., 100 (2020), 106015. https://doi.org/10.1016/j.aml.2019.106015 doi: 10.1016/j.aml.2019.106015
[14]	M. Caputo, M. Fabrizio, On the singular kernels for fractional derivatives. Some applications to partial differential equations, Prog. Fract. Differ. Appl., 7 (2021), 79–82. https://doi.org/10.18576/pfda/070201 doi: 10.18576/pfda/070201
[15]	J. Losada, J. J. Nieto, Fractional integral associated to fractional derivatives with nonsingular kernels, Prog. Fract. Differ. Appl., 7 (2021), 137–143.
[16]	F. Haq, I. Mahariq, T. Abdeljawad, N. Maliki, A new approach for the qualitative study of vector born disease using Caputo-Fabrizio derivative, Numer. Methods Part. Differ. Equ., 37 (2021), 1809–1818. https://doi.org/10.1002/num.22728 doi: 10.1002/num.22728
[17]	N. Harrouche, S. Momani, S. Hasan, M. Al-Smadi, Computational algorithm for solving drug pharmacokinetic model under uncertainty with nonsingular kernel type Caputo-Fabrizio fractional derivative, Alex. Eng. J., 60 (2021), 4347–4362. https://doi.org/10.1016/j.aej.2021.03.016 doi: 10.1016/j.aej.2021.03.016
[18]	T. W. Zhang, Y. K. Li, Exponential Euler scheme of multi-delay Caputo-Fabrizio fractional-order differential equations, Appl. Math. Lett., 124 (2022), 107709. https://doi.org/10.1016/j.aml.2021.107709 doi: 10.1016/j.aml.2021.107709
[19]	W. Walter, Ordinary differential equations, New York: Springer-verlag, 1998.
[20]	S. P. Bhaty, D. S. Bernstein, Finite-time stability of continuous autonomous systems, SIAM J. Control Optim., 38 (2000), 751–766. https://doi.org/10.1137/S0363012997321358 doi: 10.1137/S0363012997321358
[21]	R. P. Agarwal, D. O'Regan, An introduction to ordinary differential equations, Springer Science Business Media LLC, 2008.
[22]	J. C. Cortés, A. Navarro-Quiles, J. V. Romero, M. D. Roselló, Full solution of random autonomous first-order linear systems of difference equations. Application to construct random phase portrait for planar systems, Appl. Math. Lett., 68 (2017), 150–156. https://doi.org/10.1016/j.aml.2016.12.015 doi: 10.1016/j.aml.2016.12.015
[23]	W. M. Ahmad, W. M. Harb, On nonlinear control design for autonomous chaotic systems of integer and fractional orders, Chaos Solitons Fract., 18 (2003), 693–701. https://doi.org/10.1016/S0960-0779(02)00644-6 doi: 10.1016/S0960-0779(02)00644-6
[24]	C. G. Li, G. R. Chen, Chaos in the fractional order Chen system and its control, Chaos Solitons Fract., 22 (2004), 549–554. https://doi.org/10.1016/j.chaos.2004.02.035 doi: 10.1016/j.chaos.2004.02.035
[25]	S. T. Kingni, V. T. Pham, S. Jafari, G. R. Kol, P. Woafo, Three-dimensional chaotic autonomous system with a circular equilibrium: Analysis, circuit implementation and its fractional-order form, Circuits Syst. Signal Process., 35 (2016), 1933–1948. https://doi.org/10.1007/s00034-016-0259-x doi: 10.1007/s00034-016-0259-x

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)