On the kinetics of Hadamard-type fractional differential systems

References

[1] K.B. Oldham, J. Spanier, The Fractional Calculus. Academic Press, New York (1974).Suche in Google Scholar

[2] W.H. Deng, C.P. Li, J.H. Lu, Stability analysis of linear fractional differential system with multiple time delays. Nonlinear Dynam. 48, No 4 (2006), 409–416.10.1007/s11071-006-9094-0Suche in Google Scholar

[3] R. Gorenflo, A.A. Kilbas, F. Mainardi, S.V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications. Springer, Berlin (2014).10.1007/978-3-662-43930-2Suche in Google Scholar

[4] C. Yin, Y.H. Cheng, Y.Q. Chen, B. Stark, S.M. Zhong, Adaptive fractional-order switching-type control method design for 3D fractional-order nonlinear systems. Nonlinear Dynam. 82, No 1-2 (2015), 39–52.10.1007/s11071-015-2136-8Suche in Google Scholar

[5] L. Ma, C.P. Li, Center manifold of fractional dynamical system. J. Comput. Nonlinear Dynam. 11, No 2 (2016), Art. 021010.10.1115/1.4031120Suche in Google Scholar

[6] A. Hajipour, M. Hajipour, D. Baleanu, On the adaptive sliding mode controller for a hyperchaotic fractional-order financial system. Physica A497 (2018), 139–153.10.1016/j.physa.2018.01.019Suche in Google Scholar

[7] J. Hadamard, Essai sur l’étude des fonctions données par leur développment de Taylor. J. Math. Pure. Appl. 8 (1892), 101–186.Suche in Google Scholar

[8] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier Science, Amsterdam (2006).Suche in Google Scholar

[9] C. Lomnitz, Creep measurements in igneous rocks. J. Geol. 64, No 5 (1956), 473–479.10.1086/626379Suche in Google Scholar

[10] H. Jeffreys, A modification of Lomnitz’s law of creep in rocks. Geophys. J. R. Astron. Soc. 1, No 1 (1958), 92–95.10.1111/j.1365-246X.1958.tb00037.xSuche in Google Scholar

[11] F. Mainardi, G. Spada, On the viscoelastic characterization of the Jeffreys-Lomnitz law of creep. Rheol. Acta51, No 9 (2012), 783–791.10.1007/s00397-012-0634-xSuche in Google Scholar

[12] B. Ahmad, A. Alsaedi, S.K. Ntouyas, J. Tariboon, Hadamard-Type Fractional Differential Equations, Inclusions and Inequalities. Springer International Publishing, Switzerland (2017).10.1007/978-3-319-52141-1Suche in Google Scholar

[13] A.A. Kilbas, Hadamard-type fractional calculus. J. Korean Math. Soc. 38, No 6 (2001), 1191–1204.Suche in Google Scholar

[14] P.L. Butzer, A.A. Kilbas, J.J. Trujillo, Fractional calculus in the Mellin setting and Hadamard-type fractional integrals. J. Math. Anal. Appl. 269, No 1 (2002), 1–27.10.1016/S0022-247X(02)00001-XSuche in Google Scholar

[15] M. Klimek, Sequential fractional differential equations with Hadamard derivative. Commun. Nonlinear Sci. Numer. Simul. 16, No 12 (2011), 4689–4697.10.1016/j.cnsns.2011.01.018Suche in Google Scholar

[16] J.R. Wang, Y. Zhou, M. Medved’, Existence and stability of fractional differential equations with Hadamard derivative. Topol. Method. Nonlinear Anal. 41, No 1 (2013), 113–133.Suche in Google Scholar

[17] S. Abbas, M. Benchohra, J.E. Lazreg, Y. Zhou, A survey on Hadamard and Hilfer fractional differential equations: analysis and stability. Chaos Soliton. Fractals102 (2017), 47–71.10.1016/j.chaos.2017.03.010Suche in Google Scholar

[18] L. Ma, C.P. Li, On Hadamard fractional calculus. Fractals25, No 3 (2017), Art. 1750033.10.1142/S0218348X17500335Suche in Google Scholar

[19] Y.P. Wu, K. Yao, X. Zhang, On the Hadamard fractional calculus of a fractal function. Fractals26, No 3 (2018), 1850025.10.1142/S0218348X18500251Suche in Google Scholar

[20] L. Ma, C.P. Li, On finite part integrals and Hadamard-type fractional derivatives. J. Comput. Nonlinear Dynam. 13, No 9 (2018), 090905.10.1115/1.4037930Suche in Google Scholar

[21] L. Ma, Blow-up phenomena profile for Hadamard fractional differential systems in finite time. Fractals27, No 6 (2019), 1950093.10.1142/S0218348X19500932Suche in Google Scholar

[22] M. Kirane, B.T. Torebek, Extremum principle for the Hadamard derivatives and its application to nonlinear fractional partial differntial equations. Fract. Calc. Appl. Anal. 22, No 2 (2019), 358–378; 10.1515/fca-2019-0022; https://www.degruyter.com/view/journals/fca/22/2/fca.22.issue-2.xml.Suche in Google Scholar

[23] J.R. Graef, S.R. Grace, E. Tunç, Asymptotic behavior of solutions of nonlinear fractional equations with Caputo-type Hadamard derivatives. Fract. Calc. Appl. Anal. 20, No 1 (2017), 71–87; 10.1515/fca-2017-0004; https://www.degruyter.com/view/journals/fca/20/1/fca.20.issue-1.xml.Suche in Google Scholar

[24] S. Abbas, M. Benchohra, N. Hamidi, J. Henderson, Caputo-Hadamard fractional differential equations in Banach spaces. Fract. Calc. Appl. Anal. 21, No 4 (2018), 1027–1045; 10.1515/fca-2018-0056; https://www.degruyter.com/view/journals/fca/21/4/fca.21.issue-4.xml.Suche in Google Scholar

[25] L. Ma, Comparison theorems for Caputo-Hadamard fractional differential equations. Fractals27, No 3 (2019), Art. 1950036.10.1142/S0218348X19500361Suche in Google Scholar

[26] V.I. Oseledec, A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19 (1968), 197–231.Suche in Google Scholar

[27] C.P. Li, Z.Q. Gong, D.L. Qian, Y.Q. Chen, On the bound of the Lyapunov exponents for the fractional differential systems. Chaos20, No 1 (2010), Art. 013127.10.1063/1.3314277Suche in Google Scholar PubMed

[28] Z.Q. Gong, D.L. Qian, C.P. Li, P. Guo, On the Hadamard type fractional differential system. In: D. Baleanu et al. (Eds.) Fractional Dynamics and Control, 159–171. Springer, New York (2012).10.1007/978-1-4614-0457-6_13Suche in Google Scholar

[29] N.D. Cong, T.S. Doan, H.T. Tuan, On fractional Lyapunov exponent for solutions of linear fractional differential equations. Fract. Calc. Appl. Anal. 17, No 2 (2014), 285–306; 10.2478/s13540-014-0169-1; https://www.degruyter.com/view/journals/fca/17/2/fca.17.issue-2.xml.Suche in Google Scholar

[30] I. Podlubny, Fractional Differential Equations. Academic Press, New York (1999).Suche in Google Scholar

[31] L. Beghin, R. Garra, C. Macci, Correlated fractional counting processes on a finite-time interval. J. Appl. Prob. 52, No 4 (2015), 1045–1061.10.1239/jap/1450802752Suche in Google Scholar

[32] R. Garra, F. Mainardi, G. Spada, A generalization of the Lomnitz logarithmic creep law via Hadamard fractional calculus. Chaos Soliton. Fractals102 (2017), 333–338.10.1016/j.chaos.2017.03.032Suche in Google Scholar

[33] F. Mainardi, A. Mura, R. Gorenflo, M. Stojanović, The two forms of fractional relaxation of distributed order. J. Vib. Control13, No 9 (2007), 1249–1268.10.1177/1077546307077468Suche in Google Scholar

[34] V. Pandey, S. Holm, Linking the fractional derivative and the lomnitz creep law to non-newtonian time-varying viscosity. Phys. Rev. E94, No 3 (2016), Art. 032606.10.1103/PhysRevE.94.032606Suche in Google Scholar PubMed

[35] W. Chen, Y.J. Liang, X.D. Hei, Structural derivative based on inverse Mittag-Leffler function for modeling ultraslow diffusion. Fract. Calc. Appl. Anal. 19, No 5 (2016), 1250–1261; 10.1515/fca-2016-0064; https://www.degruyter.com/view/journals/fca/19/5/fca.19.issue-5.xml.Suche in Google Scholar