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

Volume 28, Issue 2, pp 123--136 http://dx.doi.org/10.22436/jmcs.028.02.01

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.

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Keywords

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

MSC

•  26A33
•  34A08
•  14C20
•  93C05

References

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


• [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
  • View Article
  • Google Scholar


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


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


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


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


• [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
  • View Article
  • MathSciNet


• [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
  • View Article
  • MathSciNet


• [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
  • View Article
  • MathSciNet


• [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
  • View Article
  • Google Scholar


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


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


• [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
  • View Article
  • MathSciNet


• [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
  • View Article
  • MathSciNet


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


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


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


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


• [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
  • View Article
  • MathSciNet


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


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


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


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


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


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


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


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


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


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


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


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