Research article

Analytical solutions of incommensurate fractional differential equation systems with fractional order $ 1 < \alpha, \beta < 2 $ via bivariate Mittag-Leffler functions

  • Received: 03 September 2021 Accepted: 01 November 2021 Published: 10 November 2021
  • MSC : 34A08, 34A25, 26A33

  • In this paper, we derive the explicit analytical solution of incommensurate fractional differential equation systems with fractional order $ 1 < \alpha, \beta < 2 $. The derivation is extended from a recently published paper by Huseynov et al. in [1], which is limited for incommensurate fractional order $ 0 < \alpha, \beta < 1 $. The incommensurate fractional differential equation systems were first converted to Volterra integral equations. Then, the Mittag-Leffler function and Picard's successive approximations were used to obtain the analytical solution of incommensurate fractional order systems with $ 1 < \alpha, \beta < 2 $. The solution will be simplified via some combinatorial concepts and bivariate Mittag-Leffler function. Some special cases will be discussed, while some examples will be given at the end of this paper.

    Citation: Yong Xian Ng, Chang Phang, Jian Rong Loh, Abdulnasir Isah. Analytical solutions of incommensurate fractional differential equation systems with fractional order $ 1 < \alpha, \beta < 2 $ via bivariate Mittag-Leffler functions[J]. AIMS Mathematics, 2022, 7(2): 2281-2317. doi: 10.3934/math.2022130

    Related Papers:

  • In this paper, we derive the explicit analytical solution of incommensurate fractional differential equation systems with fractional order $ 1 < \alpha, \beta < 2 $. The derivation is extended from a recently published paper by Huseynov et al. in [1], which is limited for incommensurate fractional order $ 0 < \alpha, \beta < 1 $. The incommensurate fractional differential equation systems were first converted to Volterra integral equations. Then, the Mittag-Leffler function and Picard's successive approximations were used to obtain the analytical solution of incommensurate fractional order systems with $ 1 < \alpha, \beta < 2 $. The solution will be simplified via some combinatorial concepts and bivariate Mittag-Leffler function. Some special cases will be discussed, while some examples will be given at the end of this paper.



    加载中


    [1] I. T. Huseynov, A. Ahmadova, A. Fernandez, N. I. Mahmudov, Explicit analytical solutions of incommensurate fractional differential equation systems, Appl. Math. Comput., 390 (2021), 125590. doi: 10.1016/j.amc.2020.125590. doi: 10.1016/j.amc.2020.125590
    [2] A. Hajipour, H, Tavakoli, Analysis and circuit simulation of a novel nonlinear fractional incommensurate order financial system, Optik, 127 (2016), 10643–10652. doi: 10.1016/j.ijleo.2016.08.098. doi: 10.1016/j.ijleo.2016.08.098
    [3] I. Pan, S. Das, S. Das, Multi-objective active control policy design for commensurate and incommensurate fractional order chaotic financial systems, Appl. Math. Model., 39 (2015), 500–514. doi: 10.1016/j.apm.2014.06.005. doi: 10.1016/j.apm.2014.06.005
    [4] K. Zourmba, A. A. Oumate, B. Gambo, J. Y. Effa, A. Mohamadou, Chaos in the incommensurate fractional order system and circuit simulations, Int. J. Dyn. Control, 7 (2019), 94–111. doi: 10.1007/s40435-018-0442-y. doi: 10.1007/s40435-018-0442-y
    [5] X. Wang, Z. Wang, J. Xia, Stability and bifurcation control of a delayed fractional-order eco-epidemiological model with incommensurate orders, J. Franklin Inst., 356 (2019), 8278–8295. doi: 10.1016/j.jfranklin.2019.07.028. doi: 10.1016/j.jfranklin.2019.07.028
    [6] B. Daşbaşi. Stability analysis of the HIV model through incommensurate fractional-order nonlinear system, Chaos, Solitons Fractals, 137 (2020), 109870. doi: 10.1016/j.chaos.2020.109870.
    [7] N. Debbouche, A. O. Almatroud, A. Ouannas, I. M. Batiha, Chaos and coexisting attractors in glucose-insulin regulatory system with incommensurate fractional-order derivatives, Chaos, Solitons Fractals, 143 (2021), 110575. doi: 10.1016/j.chaos.2020.110575. doi: 10.1016/j.chaos.2020.110575
    [8] M. Tavazoei, M. H. Asemani, On robust stability of incommensurate fractional-order systems, Commun. Nonlinear Sci. Numer. Simul., 90 (2020), 105344. doi: 10.1016/j.cnsns.2020.105344. doi: 10.1016/j.cnsns.2020.105344
    [9] M. Tavazoei, M. H. Asemani, Robust stability analysis of incommensurate fractional-order systems with time-varying interval uncertainties, J. Franklin Inst., 357 (2020), 13800–13815. doi: 10.1016/j.jfranklin.2020.09.044. doi: 10.1016/j.jfranklin.2020.09.044
    [10] Y. Shen, Y. Wang, N. Yuan, A graphical approach for stability and robustness analysis in commensurate and incommensurate fractional-order systems, Asian J. Control, 22 (2020), 1241–1252. doi: 10.1002/asjc.1980. doi: 10.1002/asjc.1980
    [11] R. Luo, H. Su, The stability of impulsive incommensurate fractional order chaotic systems with Caputo derivative, Chinese J. Phys., 56 (2018), 1599–1608. doi: 10.1016/j.cjph.2018.06.017. doi: 10.1016/j.cjph.2018.06.017
    [12] C. M. Chang, H. K. Chen, Chaos and hybrid projective synchronization of commensurate and incommensurate fractional-order Chen-Lee systems, Nonlinear Dyn., 62 (2010), 851–858. doi: 10.1007/s11071-010-9767-6. doi: 10.1007/s11071-010-9767-6
    [13] C. Ma, J. Mou, J. Liu, F. Yang, H. Yan, X. Zhao, Coexistence of multiple attractors for an incommensurate fractional-order chaotic system, Eur. Phys. J. Plus, 135 (2020), 1–21. doi: 10.1140/epjp/s13360-019-00093-0. doi: 10.1140/epjp/s13360-019-00093-0
    [14] C. Huang, J. Cao, M. Xiao, A. Alsaedi, F. E. Alsaadi, Controlling bifurcation in a delayed fractional predator-prey system with incommensurate orders, Appl. Math. Comput., 293 (2017), 293–310. doi: 10.1016/j.amc.2016.08.033. doi: 10.1016/j.amc.2016.08.033
    [15] K. Diethelm, N. J. Ford, A. D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dyn., 29 (2002), 3–22. doi: 10.1023/A:1016592219341. doi: 10.1023/A:1016592219341
    [16] K. Diethelm, N. J. Ford, Multi-order fractional differential equations and their numerical solution, Appl. Math. Comput., 154 (2004), 621–640. doi: 10.1016/S0096-3003(03)00739-2. doi: 10.1016/S0096-3003(03)00739-2
    [17] H. Liao, Y. Ding, L. Wang, Adomian decomposition algorithm for studying incommensurate fractional-order memristor-based Chua's system, Int. J. Bifurcat. Chaos, 28 (2018), 1850134. doi: 10.1142/S0218127418501341. doi: 10.1142/S0218127418501341
    [18] H. N. Soloklo, N. Bigdeli, Direct approximation of fractional order systems as a reduced integer/fractional-order model by genetic algorithm, Sadhana, 45 (2020), 1–15. doi: 10.1007/s12046-020-01503-1.
    [19] A. Ahmadova, I. T. Huseynov, A. Fernandez, N. I. Mahmudov, Trivariate Mittag-Leffler functions used to solve multi-order systems of fractional differential equations, Commun. Nonlinear Sci. Numer. Simul., 97 (2021), 105735. doi: 10.1016/j.cnsns.2021.105735. doi: 10.1016/j.cnsns.2021.105735
    [20] N. I. Mahmudov, I. T. Huseynov, N. A. Aliev, F. A. Aliev, Analytical approach to a class of Bagley-Torvik equations, TWMS J. Pure Appl. Math., 11 (2020), 238–258.
    [21] I. T. Huseynov, A. Ahmadova, N. I. Mahmudov, Fractional Leibniz integral rules for Riemann-Liouville and Caputo fractional derivatives and their applications, 2020. Available from: https://arXiv.org/abs/2012.11360.
    [22] M. A. Özarslan, A. Fernandez, On the fractional calculus of multivariate Mittag-Leffler functions, Int. J. Comput. Math., 2021, 1–27. doi: 10.1080/00207160.2021.1906869.
    [23] B. Bira, H. Mandal, D. Zeidan, Exact solution of the time fractional variant Boussinesq-Burgers equations, Appl. Math., 66 (2021), 437–449. doi: 10.21136/AM.2021.0269-19. doi: 10.21136/AM.2021.0269-19
    [24] D. Zeidan, C. K. Chau, T. T. Lu, W. Q. Zheng, Mathematical studies of the solution of Burgers' equations by Adomian decomposition method, Math. Methods Appl. Sci., 43 (2020), 2171–2188. doi: 10.1002/mma.5982. doi: 10.1002/mma.5982
    [25] J. R. Loh, C. Phang, Numerical solution of Fredholm fractional integro-differential equation with right-sided Caputo's derivative using Bernoulli polynomials operational matrix of fractional derivative, Mediterr. J. Math., 16 (2019), 1–25. doi: 10.1007/s00009-019-1300-7. doi: 10.1007/s00009-019-1300-7
    [26] J. R. Loh, C. Phang, K. G. Tay, New method for solving fractional partial integro-differential equations by combination of Laplace transform and resolvent kernel method, Chinese J. Phys., 67 (2020), 666–680. doi: 10.1016/j.cjph.2020.08.017. doi: 10.1016/j.cjph.2020.08.017
    [27] M. A. Ebadi, E. Hashemizadeh, A new approach based on the Zernike radial polynomials for numerical solution of the fractional diffusion-wave and fractional Klein-Gordon equations, Phys. Scripta, 93 (2018), 125202. doi: 10.1088/1402-4896/aae726
    [28] L. N. Kaharuddin, C. Phang, S. S. Jamaian, Solution to the fractional logistic equation by modified Eulerian numbers, Eur. Phys. J. Plus, 135 (2020), 1–11. doi: 10.1140/epjp/s13360-020-00135-y. doi: 10.1140/epjp/s13360-020-00135-y
    [29] P. Roul, V. M. K. Prasad Goura, A high order numerical scheme for solving a class of non-homogeneous time-fractional reaction diffusion equation, Numer. Methods Partial Differ. Equ., 37 (2021), 1506–1534. doi: 10.1002/num.22594. doi: 10.1002/num.22594
    [30] P. Roul, A high accuracy numerical method and its convergence for time-fractional Black-Scholes equation governing European options, Appl. Numer. Math., 151 (2020), 472–493. doi: 10.1016/j.apnum.2019.11.004. doi: 10.1016/j.apnum.2019.11.004
    [31] S. Kumar, D. Zeidan, An efficient Mittag-Leffler kernel approach for time-fractional advection-reaction-diffusion equation, Appl. Numer. Math., 170 (2021), 190–207. doi: 10.1016/j.apnum.2021.07.025. doi: 10.1016/j.apnum.2021.07.025
    [32] F. Sultana, D. Singh, R. K. Pandey, D. Zeidan, Numerical schemes for a class of tempered fractional integro-differential equations, Appl. Numer. Math., 157 (2020), 110–134. doi: 10.1016/j.apnum.2020.05.026. doi: 10.1016/j.apnum.2020.05.026
    [33] A. Fernandez, C. Kürt, M. A. Özarslan, A naturally emerging bivariate Mittag-Leffler function and associated fractional-calculus operators, Comput. Appl. Math., 39 (2020), 1–27. doi: 10.1007/s40314-020-01224-5. doi: 10.1007/s40314-020-01224-5
    [34] H. Seybold, R. Hilfer, Numerical algorithm for calculating the generalized Mittag-Leffler function, SIAM J. Numer. Anal., 47 (2009), 69–88. doi: 10.1137/070700280. doi: 10.1137/070700280
    [35] R. Garrappa, Numerical evaluation of two and three parameter Mittag-Leffler functions, SIAM J. Numer. Anal., 53 (2015), 1350–1369. doi: 10.1137/140971191. doi: 10.1137/140971191
    [36] C. Kürt, M. A. Özarslan, A. Fernandez, On a certain bivariate Mittag-Leffler function analysed from a fractional-calculus point of view, Math. Methods Appl. Sci., 44 (2021), 2600–2620. doi: 10.1002/mma.6324. doi: 10.1002/mma.6324
    [37] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Vol. 204, Elsevier, 2006.
    [38] A. A. Hamou, E. Azroul, Z. Hammouch, A. Lamrani alaoui, On dynamics of fractional incommensurate model of Covid-19 with nonlinear saturated incidence rate, medRxiv, 2021. doi: 10.1101/2021.07.18.21260711.
  • Reader Comments
  • © 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)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1627) PDF downloads(111) Cited by(4)

Article outline

Figures and Tables

Figures(1)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog