# Exact solution for commensurate and incommensurate linear systems of fractional differential equations

Volume 28, Issue 2, pp 123--136
Publication Date: May 05, 2022 Submission Date: January 28, 2022 Revision Date: March 11, 2022 Accteptance Date: March 18, 2022
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### Authors

A. Al-Habahbeh - Department of Mathematics, Tafila Technical University, Tafila, Jordan.

### Abstract

In this paper, we introduce exact solutions for the initial value problems of two classes of a linear system of fractional ordinary differential equations with constant coefficients. This article concerns a linear system of fractional order, where the orders are equal or different rational numbers between zero and one. The conformable fractional derivative presented by [R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, J. Comput. Appl. Math., $\textbf{264}$ (2014), 65--70] is considered. Two different approaches are adopted to give analytic solutions for fractional order systems. The presented methods are illustrated by analyzing some numerical examples that show the effectiveness of the implemented methods.

### Share and Cite

##### ISRP Style

A. Al-Habahbeh, Exact solution for commensurate and incommensurate linear systems of fractional differential equations, Journal of Mathematics and Computer Science, 28 (2023), no. 2, 123--136

##### AMA Style

Al-Habahbeh A., Exact solution for commensurate and incommensurate linear systems of fractional differential equations. J Math Comput SCI-JM. (2023); 28(2):123--136

##### Chicago/Turabian Style

Al-Habahbeh, A.. "Exact solution for commensurate and incommensurate linear systems of fractional differential equations." Journal of Mathematics and Computer Science, 28, no. 2 (2023): 123--136

### Keywords

• Conformable fractional derivative
• fractional Laplace transform
• commensurate and incommensurate fractional order systems
• asymptotically stable

•  26A33
•  34A08
•  14C20
•  93C05

### References

• [1] T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57--66

• [2] M. Abu Hammad, R. Khalil, Abel’s Formula and Wronskian for Conformable Fractional Differential Equations, Int. J. Differ. Equ. Appl., 13 (2014), 11 pages

• [3] I. Abu Hammad, R. Khalil, Fractional fourier series with applications, Amer. J. Comput. Appl. Math., 4 (2014), 187--191

• [4] A. Akbulut, M. Kaplan, Auxiliary equation method for time-fractional differential equations with conformable derivative, Comput. Math. Appl., 75 (2018), 876--882

• [5] M. Al-Horani, R. Khalil, I. Aldarawi, Fractional Cauchy Euler Differential Equation, J. Comput. Anal. Appl., 28 (2020), 13 pages

• [6] M. Alsauodi, M. Alhorani, R. Khalil, Solutions of Certain Fractional Partial Differential Equations, WSEAS Trans. Math., 20 (2021), 504--507

• [7] D. R. Anderson, E. Camrud, D. J. Ulness, On the nature of the conformable derivative and its applications to physics, J. Fract. Calc. Appl., 10 (2019), 92--135

• [8] D. Baleanu, H. Mohammadi, S. Rezapour, A mathematical theoretical study of a particular system of Caputo–Fabrizio fractional differential equations for the Rubella disease model, Adv. Difference Equ., 2020 (2020), 19 pages

• [9] I. Benkemache, M. Al-Horani, R. Khalil, Tensor Product and Certain Solutions of Fractional Wave Type Equation, Eur. J. Pure Appl. Math., 14 (2021), 942--948

• [10] S. Buedo-Fernandez, J. J. Nieto, Basic control theory for linear fractional differential equations with constant coefficients, Frontiers in Phys., 8 (2020), 12 pages

• [11] L. Debnath, Recent applications of fractional calculus to science and engineering, Int. J. Math. Math. Sci., 2003 (2003), 3413--3442

• [12] K. Diethelm, N. J. Ford, Analysis of freactional differential equations, J. Math. Anal. Appl., 265 (2002), 229--248

• [13] K. Diethelm, N. J. Ford, A. D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynam., 29 (2002), 3--22

• [14] S. M. Guo, L. Q. Mei, Y. Li, Y. F. Sun, https://www.sciencedirect.com/science/article/pii/S0375960111013776, Phys. Lett. A, 376 (2012), 407--411

• [15] I. Kadri, M. Horani, R. Khalil, Tensor product technique and fractional differential equations, J. Semigroup Theory Appl., 2020 (2020), 10 pages

• [16] A. Khader, The conformable Laplace transform of the fractional Chebyshev and Legendre polynnomials, M.Sc. Thesis, Zarqa University (2017)

• [17] R. Khalil, M. Abu Hammad, Conformable Fractional Heat Differential Equation, Int. J. Pure Appl. Math., 94 (2014), 215--217

• [18] R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65--70

• [19] K. D. Kucche, S. T. Sutar, Analysis of nonlinear fractional differential equations involving Atangana-Baleanu-Caputo derivative, Chaos Solitons Fractles, 143 (2021), 9 pages

• [20] R. L. Magin, Fractional calculus in bioengineering, Crit. Rev. Biomed. Eng., 32 (2004), 1--104

• [21] D. Matignon, Stability result on fractional differential equations with applications to control processing, Comput. Eng. Syst. Appl., 2 (1996), 963--968

• [22] T. Matsuzaki, M. Nakagawa, A chaos neuron model with fractional differential equation, J. Phys. Soc. Japan, 72 (2003), 2678--2684

• [23] R. Metzler, J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339 (2000), 77 pages

• [24] M. Mhailan, M. Abu Hammad, M. Al Horani, R. Khalil, On fractional vector analysis, J. Math. Comput. Sci., 10 (2020), 2320--2326

• [25] S. Momani, Z. Odibat, Numerical approach to differential equations of fractional order, J. Comput. Appl. Math., 207 (2007), 96--110

• [26] Z. M. Odibat, Analytic study on linear systems of fractional differential equations, Comput. Math. Appl., 59 (2010), 1171--1183

• [27] K. B. Oldham, Fractional differential equations in electrochemistry, Adv. Eng. software, 41 (2010), 9--12

• [28] H. Rezazadeh, H. Aminikhah, S. Refahi, Stability analysis of conformable fractional systems, Iran. J. Numer. Anal. Optim., 7 (2017), 13--32

• [29] J. Ribeiro, J. de Castro, M. Meggiolaro, Modeling concrete and polymer creep using fractional calculus, J. Mater. Res. Tech., 12 (2021), 1184--1193

• [30] A. Saadatmandi, M. Dehghan, A new operational matrix for solving fractional-order differential equations, Comput. Math. Appl., 59 (2010), 1326--1336

• [31] V. Tarasov, V. Tarasove, Economic Dynamics with Memory: Fractional Calculus Approach, Walter de Gruyter GmbH & Co., New York (2021)