Generalized Mittag-Leffler stability of nonlinear fractional regularized Prabhakar differential systems

Document Type : Research Paper


1 Department of Applied Mathematics and Computer Sciences, Faculty of Mathematical Sciences, Shahrekord University, P.O.Box 115, Shahrekord, Iran

2 Department of Applied Mathematics and Computer Sciences, Faculty of Mathematical Sciences, Shahrekord University, P.O. Box 115, Shahrekord, Iran


This work is devoted to study of  the stability analysis of generalized fractional nonlinear system including the regularized Prabhakar derivative. We present several criteria for the generalized Mittag-Leffler stability and the asymptotic stability of this system by using the Lyapunov direct method. Further, we provide two test cases to illustrate the effectiveness  of  results. We apply the numerical method to solve the generalized fractional system with the regularized Prabhakar fractional systems and reveal asymptotic stability behavior of the presented systems by employing numerical simulation.


[1] R. Agarwal, S. Jain and RP. Agarwal, Analytic solution of generalized space time fractional reaction diffusion equation, Fractional Differ. Calc. 7 (2017) 169–184.
[2] R. Bellman, Introduction to matrix analysis, Society for Industrial and Applied Mathematics (SIAM), USA, 1997.
[3] VM. Bulavatsky, Mathematical modeling of fractional differential filtration dynamics based on models of HilferPrabhakar derivatives, Cybern. Syst. Anal. 53(2) (2017) 204–216.
[4] J. Devi, F. Mc Rae and Z. Drici, Variational Lyapunov method for fractional differential equations, Appl. Math. Comput. 64(298) (2012) 2–9.
[5] M. D’Ovidio and F. Polito, Fractional diffusion-telegraph equations and their associated stochastic solutions, Theory Probab. its Appl. 62(4) (2018) 552–574.
[6] MA. Duarte-Mermoud, N. Aguila-Camacho, JA. Gallegos and R. Castro-Linares, Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems, Commun. Nonlinear Sci. Numer. Simulat. 22 (2015) 650–659.
[7] S. Eshaghi and A. Ansari, Autoconvolution equations and generalized Mittag-Leffler functions, Int. J. Ind. Math. 7(4) (2015) 335–341.
[8] S. Eshaghi and A. Ansari, Lyapunov inequality for fractional differential equations with Prabhakar derivative, Math. Inequal. Appl. 19(1) (2016) 349–358.
[9] S. Eshaghi and A. Ansari, Finite fractional Storm-Liouville transforms for generalized fractional derivatives, Iran. J. Sci. Technol. Trans. A Sci. 41(4) (2017) 931–937.
[10] S. Eshaghi, A. Ansari, R. Khoshsiar Ghaziani and M. Ahmadi Darani, Fractional Black-Scholes model with regularized Prabhakar derivative, Publ. de l’Institut Math. 102(116) (2017) 121–132.
[11] S. Eshaghi, A. Ansari and R. Khoshsiar Ghaziani, Lyapunov-type inequality for nonlinear systems with RiemannLiouville fractional derivatives, Novi Sad J. Math. 49(2) (2019) 17–34.
[12] S. Eshaghi, R. Khoshsiar Ghaziani and A. Ansari, Stability and chaos control of regularized Prabhakar fractional dynamical systems without and with delay, Math. Methods Appl. Sci. 42(7) (2019) 2302–2323.
[13] S. Eshaghi, R. Khoshsiar Ghaziani and A. Ansari, Hopf bifurcation, chaos control and synchronization of a chaotic fractional-order system with chaos entanglement function, Math. Comput. Simul. 172(C) (2020) 321–340.
[14] S. Eshaghi, R. Khoshsiar Ghaziani and A. Ansari, Stability and dynamics of neutral and integro-differential regularized Prabhakar fractional differential systems, Comput. Appl. Math. 39(4) (2020) 1–21.
[15] R. Garra and R. Garrappa, The Prabhakar or three parameter Mittag-Leffler function: Theory and application, Commun. Nonlinear Sci. Numer. Simulat. 56 (2018) 314–329.
[16] R. Garra, R. Gorenflo, F. Polito and Z. Tomovski, Hilfer-Prabhakar derivatives and some applications, Appl. Math. Comput. 242 (2014) 576–589.
[17] R. Garrappa, Gr¨unwald-Letnikov operators for fractional relaxation in Havriliak-Negami models, Commun. Nonlinear Sci. Numer. Simul. 38 (2016) 178–191.
[18] R. Garrappa, F. Mainardi and G. Maione, Models of dielectric relaxation based on completely monotone functions, Fract. Calc. Appl. Anal. 19(5) (2016) 1105–1160.
[19] A. Giusti and I. Colombaro, Prabhakar-like fractional viscoelasticity, Commun. Nonlinear Sci. Numer. Simulat. 56 (2018) 138–143.
[20] R. Gorenflo, AA. Kilbas, F. Mainardi and SV. Rogosin, Mittag-Leffler Functions: Related Topics and Applications, Springer Monographs in Mathematics, New York, 2014.
[21] RK. Gupta, BS. Shaktawat and D. Kumar, Certain relation of generalized fractional calculus associated with the generalized Mittag-Leffler function, J. Rajasthan Acad. Phys. Sci. 15(3) (2016) 117–126.
[22] HK. Khalil, Nonlinear Systems, third edition, Prentice Hall, Upper Saddle River, 2002.
[23] R. Hilfer and H. Seybold, Computation of the generalized Mittag-Leffler function and its inverse in the complex plane, Integral Transforms Spec. Funct. 17 (2006) 637–652.
[24] AA. Kilbas, M. Saigo and RK. Saxena, Generalized Mittag-Leffler function and generalized fractional calculus operators, Integral Transforms Spec. Funct. 15(1) (2004) 31–49.
[25] AA. Kilbas, HM. Srivastava and JJ. Trujillo, Theory and Applications of Fractional Differential Equations, NorthHolland Mathematical Studies, 204, Elsevier (North-Holland) Science Publishers, Amsterdam, 2006.
[26] Y. Li, YQ. Chen and I. Podlubny, Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability, Appl. Math. Comput. 59 (2010) 1810–1821.
[27] S. Liu, W. Jiang, X. Li and XF Zhou, Lyapunov stability analysis of fractional nonlinear systems, Appl. Math. Lett. 51 (2016) 13–19.
[28] GP. Lu and DWC. Ho, Generalized quadratic stability for continuous-time singular systems with nonlinear perturbation, Trans. Automat. Contr. 51 (2006) 818–823.
[29] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, London: Imperial College Press, 2010.
[30] F. Mainardi and R. Garrappa, On complete monotonicity of the Prabhakar function and non-Debye relaxation in dielectrics, J. Comput. Phys. 293 (2015) 70–80.
[31] SC. Pandey, The Lorenzo-Hartley’s function for fractional calculus and its applications pertaining to fractionalorder modelling of anomalous relaxation in dielectrics, Comput. Appl. Math. 37(3) (2017) 2648–2666.
[32] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
[33] F. Polito and Z. Tomovski, Some properties of Prabhakar-type fractional calculus operators, Fractional Differ. Calc. 6(1) (2016) 73–94.
[34] TR. Prabhakar, A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math. J. 19 (1971) 7–15.
[35] HJ. Seybold and R. Hilfer, Numerical results for the generalized Mittag-Leffler function, Fract. Calc. Appl. Anal. 8 (2005) 127–139.
[36] S. Momani and S. Hadid, Lyapunov stability solutions of fractional integrodifferential equations, Int. J. Math. Math. Sci. 47 (2004) 2503–2507.
[37] HM. Srivastava and Z. Tomovski, Fractional calculus with an integral operator containing a generalized MittagLeffler function in the kernel, Appl. Math. Comput. 211 (2009) 198–210.
[38] Z. Tomovski, R. Hilfer and HM. Srivastava, Fractional and operational calculus with generalized fractional derivative operators and Mittag-Leffler type functions, Fract. Calc. Appl. Anal. 21 (2010) 797–814.
[39] J. Xu, Time-fractional particle deposition in porous media, J. Phys. A Math. Theor. 50(19) (2017) 195002.
[40] F. Zhang, Ch. Li and YQ. Chen, Asymptotical stability of nonlinear differential systems with Caputo derivative, Int. J. Differ. Equ. 2011 (2011) 12 pages.
[41] L. Zhang, J. Li and G. Chen, Extension of Lyapunov second method by fractional calculus, Pure Appl. Math. 3 (2005) 1008–5513.
Volume 12, Issue 2
November 2021
Pages 665-678
  • Receive Date: 11 December 2019
  • Revise Date: 22 December 2019
  • Accept Date: 08 January 2020