On Lie symmetry analysis of certain coupled fractional ordinary differential equations. (English) Zbl 1497.34015
Summary: In this article, we explain how to extend the Lie symmetry analysis method for \(n\)-coupled system of fractional ordinary differential equations in the sense of Riemann-Liouville fractional derivative. Also, we systematically investigated how to derive Lie point symmetries of scalar and coupled fractional ordinary differential equations namely (i) fractional Thomas-Fermi equation, (ii) Bagley-Torvik equation, (iii) two-coupled system of fractional quartic oscillator, (iv) fractional type coupled equation of motion and (v) fractional Lotka-Volterra ABC system. The dimensions of the symmetry algebras for the Bagley-Torvik equation and its various cases are greater than 2 and for this reason we construct optimal system of one-dimensional subalgebras. In addition, the exact solutions of the above mentioned fractional ordinary differential equations are explicitly derived wherever possible using the obtained symmetries.
MSC:
| 34A08 | Fractional ordinary differential equations |
| 70G65 | Symmetries, Lie group and Lie algebra methods for problems in mechanics |
| 26A33 | Fractional derivatives and integrals |
Keywords:
fractional ODEs; Lie group formalism; Laplace transform technique; Mittag-Leffler function; exact solutionReferences:
| [1] | Adomian, G., Solution of the Thomas-Fermi equation, App. Math. Lett., 11, 131-133, 1998 · Zbl 0947.34501 |
| [2] | Arrigo, DJ, Symmetry Analysis of Differential Equations: An Introduction, 192, 2015, New Jersey: John Wiley & Sons, New Jersey · Zbl 1312.00002 |
| [3] | Bagley, RL; Torvik, PJ, On the appearance of the fractional derivative in the behavior of real materials, ASME J. Appl. Mech., 51, 294-298, 1984 · Zbl 1203.74022 |
| [4] | Bakkyaraj, T., Lie symmetry analysis of system of nonlinear fractional partial differential equations with Caputo fractional derivative, Eur. Phys. J. Plus, 135, 126, 2020 |
| [5] | Bakkyaraj, T.; Sahadevan, R., Group formalism of Lie transformations to time-fractional partial differential equations, Pramana - J. Phys., 85, 849-860, 2015 |
| [6] | Bakkyaraj, T.; Sahadevan, R., Invariant analysis of nonlinear fractional ordinary differential equations with Riemann-Liouville fractional derivative, Nonlinear Dyn., 80, 447-455, 2015 · Zbl 1345.34003 |
| [7] | Baleanu, D.; Inc, M.; Yusuf, A.; Aliyu, AI, Lie symmetry analysis, exact solutions and conservation laws for the time fractional Caudrey-Dodd-Gibbon—Sawada-Kotera equation, Commun. Nonlinear Sci. Numer. Simulat., 59, 222-234, 2018 · Zbl 1510.35365 |
| [8] | G.W. Bluman, S.C. Anco, Symmetry and Integration Methods for Differential Equations, Springer, New York, 2002. · Zbl 1013.34004 |
| [9] | Chauhan, A.; Arora, R., Time fractional Kupershmidt equation: symmetry analysis and explicit series solution with convergence analysis, Commun. Math., 27, 171-185, 2019 · Zbl 1464.34018 |
| [10] | Chauhan, A.; Sharm, K.; Arora, R., Lie symmetry analysis, optimal system, and generalized group invariant solutions of the (2 + 1)-dimensional Date-Jimbo-Kashiwara-Miwa equation, Math. Meth. Appl. Sci., 43, 8823-8840, 2020 · Zbl 1455.35215 |
| [11] | Choudhary, S.; Prakash, P.; Daftardar-Gejji, V., Invariant subspaces and exact solutions for a system of fractional PDEs in higher dimensions, Comp. Appl. Math, 38, 126, 2019 · Zbl 1449.35435 |
| [12] | Dahmani, Z.; Anber, A., Two numerical methods for solving the fractional Thomas-Fermi equation, J. Interdisciplinary Math., 18, 35-41, 2015 |
| [13] | Das, S.; Gupta, PK, A mathematical model on fractional Lotka-Volterra equations, J. Theor. Biol., 277, 1-6, 2011 · Zbl 1405.92227 |
| [14] | Diethelm, K.; Ford, NJ, Numerical solution of the Bagley-Torvik equation, BIT Numer. Math., 42, 490-507, 2002 · Zbl 1035.65067 |
| [15] | Du, XX; Tian, B.; Qu, QX; Yuan, YQ; Zhao, XH, Lie group analysis, solitons, self-adjointness and conservation laws of the modified Zakharov-Kuznetsov equation in an electron-positron-ion magnetoplasma, Chaos Solitons Fractals, 134, 109709, 2020 · Zbl 1483.35177 |
| [16] | El-Nabulsi, RA, The fractional Boltzmann transport equation, Comput. Math. Appl., 62, 1568-1575, 2011 · Zbl 1228.82075 |
| [17] | Feng, W.; Sun, S.; Sun, Y., Existence of positive solutions for a generalized and fractional ordered Thomas-Fermi theory of neutral atoms, Adv. Differ. Equ, 2015, 350, 2015 · Zbl 1422.34031 |
| [18] | R.K. Gazizov, A.A. Kasatkin, S.Y. Lukashchuk, Continuous transformation groups of fractional differential equations, Vestnik USATU 9 (2007), 125-135. (in Russian) |
| [19] | R.K. Gazizov, A.A. Kasatkin, S.Y. Lukashchuk, Group-invariant solutions of fractional differential equations, in: J.A.T. Machado, A.C.J. Luo, R.S. Barbosa, M.F. Silva, L.B. Figueiredo, (eds.), Nonlinear Science and Complexity, Springer, Dordrecht, Heidelberg, 2011, pp. 51-59. · Zbl 1217.37066 |
| [20] | Gazizov, RK; Kasatkin, AA; Lukashchuk, SY, Symmetry properties of fractional diffusion equations, Phys. Scr, 136, 014016, 2009 |
| [21] | Gazizov, RK; Kasatkin, AA, Construction of exact solutions for fractional order differential equations by the invariant subspace method, Comput. Math. Appl., 66, 576-584, 2013 · Zbl 1348.34012 |
| [22] | A. Goriely, Integrability and Nonintegrability of Dynamical Systems, World Scientific, Singapore, 2001, p. 436. · Zbl 1002.34001 |
| [23] | Górka, P.; Prado, H.; Trujillo, J., The time fractional Schrödinger equation on Hilbert space, Integr. Equ. Oper. Theory, 87, 1-14, 2017 · Zbl 1368.35231 |
| [24] | Grigorenko, I.; Grigorenko, E., Chaotic dynamics of the fractional Lorenz system, Phys. Rev. Lett, 91, 034101, 2003 |
| [25] | Herrmann, R., Infrared spectroscopy of diatomic molecules — a fractional calculus approach, Int. J. Mod. Phys. B, 27, 1350019, 2013 |
| [26] | R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000, p. 472. · Zbl 0998.26002 |
| [27] | P.E. Hydon, Symmetry Methods for Differential Equations, Cambridge University Press, Cambridge, 2000. · Zbl 0951.34001 |
| [28] | N.H. Ibragimov, CRC Handbook of Lie group Analysis of Differential Equations, vol. 1: Symmetries, Exact Solutions and Conservation Laws, CRC Press, Boca Raton, FL, USA, 1994. · Zbl 0864.35001 |
| [29] | Jefferson, GF; Carminati, J., FracSym: automated symbolic computation of Lie symmetries of fractional differential equations, Comput. Phys. Commun., 185, 430-441, 2014 · Zbl 1344.35003 |
| [30] | Kasatkin, AA, Symmetry properties for systems of two ordinary fractional differential equations, Ufa Mathematical J., 4, 65-75, 2012 |
| [31] | Kaur, B.; Gupta, RK, Multiple types of exact solutions and conservation laws of new coupled (2 + 1)-dimensional Zakharov-Kuznetsov system with time-dependent coefficients, Pramana - J. Phys, 93, 59, 2019 |
| [32] | A.A. Kilbas, J.J. Trujillo, H.M. Srivastava, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006, p. 540. · Zbl 1092.45003 |
| [33] | Y. Kosmann-Schwarzbach, B. Grammaticos, K.M. Tamizhmani, Integrability of Nonlinear systems, Springer-Verlag, Berlin, 2004. · Zbl 1034.93002 |
| [34] | Lakshmanan, M.; Kaliappan, P., Lie transformations, nonlinear evolution equations, and Painlevé forms, J. Math. Phys., 24, 795-806, 1983 · Zbl 0525.35022 |
| [35] | M. Lakshmanan, S. Rajasekar, Nonlinear Dynamics: Integrability, Chaos, and Patterns, Springer-Verlag, Berlin, 2003. · Zbl 1038.37001 |
| [36] | N. Laskin, Fractional quantum mechanics, World Scientific, Singapore, 2018, p. 360. · Zbl 1425.81007 |
| [37] | Li, C.; Zhang, J., Lie symmetry analysis and exact solutions of generalized fractional Zakharov-Kuznetsov equations, Symmetry, 11, 601, 2019 · Zbl 1425.35216 |
| [38] | Ma, WX, Conservation laws by symmetries and adjoint symmetries, Discrete and Continuous Dynamical Systems-Series S, 11, 707-721, 2018 · Zbl 1386.70041 |
| [39] | Ma, WX; Liu, Y., Invariant subspaces and exact solutions of a class of dispersive evolution equations, Commun. Nonlinear Sci. Numer. Simul., 17, 3795-3801, 2012 · Zbl 1250.35057 |
| [40] | Ma, WX; Mousa, MM; Ali, MR, Application of a new hybrid method for solving singular fractional Lane-Emden-type equations in astrophysics, Mod. Phys. Lett. B, 34, 2050049, 2020 |
| [41] | Maier, RS, The integration of three-dimensional Lotka-Volterra systems, Proc. R. Soc. A, 469, 20120693, 2013 · Zbl 1371.34003 |
| [42] | F. Mainardi, Fractional calculus: some basic problems in continuum and statistical mechanics, in: F. Mainardi (ed.), Fractals and fractional calculus in continuum mechanics, International Centre for Mechanical Sciences, Springer-Verlag, New York, 1997, pp. 291-348. · Zbl 0917.73004 |
| [43] | A.M. Mathai, H.J. Haubold, Special Functions for Applied Scientists, Springer, New York, 2008. · Zbl 1151.33001 |
| [44] | Momani, S.; Odibat, Z., Analytical approach to linear fractional partial differential equations arising in fluid mechanics, Phys. Lett. A, 355, 271-279, 2006 · Zbl 1378.76084 |
| [45] | Momani, S.; Odibat, Z., Homotopy perturbation method for nonlinear partial differential equations of fractional order, Phys. Lett. A, 365, 345-350, 2007 · Zbl 1203.65212 |
| [46] | Nass, AM, Symmetry analysis of space-time fractional Poisson equation with a delay, Quaestiones Mathematicae, 42, 1221-1235, 2019 · Zbl 1428.35017 |
| [47] | Nass, AM, Lie symmetry analysis and exact solutions of fractional ordinary differential equations with neutral delay, Appl. Math. Comput., 347, 370-380, 2019 · Zbl 1428.34117 |
| [48] | P.J. Olver, Applications of Lie Groups to Differential Equations, Springer, Heidelberg, 1986. · Zbl 0588.22001 |
| [49] | L.V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, New York, 1982, p. 432. · Zbl 0485.58002 |
| [50] | I. Podlubny, Fractional Differential Equations, Acadmic Press, New York, 1999. · Zbl 0924.34008 |
| [51] | Prakash, P., New exact solutions of generalized convection-reaction-diffusion equation, Eur. Phys. J. Plus, 134, 261, 2019 |
| [52] | Prakash, P., Invariant subspaces and exact solutions for some types of scalar and coupled time-space fractional diffusion equations, Pramana - J. Phys, 94, 103, 2020 |
| [53] | Prakash, P.; Choudhary, S.; Daftardar-Gejji, V., Exact solutions of generalized nonlinear time-fractional reaction-diffusion equations with time delay, Eur. Phys. J. Plus, 135, 490, 2020 |
| [54] | Prakash, P.; Sahadevan, R., Lie symmetry analysis and exact solution of certain fractional ordinary differential equations, Nonlinear Dyn., 89, 305-319, 2017 · Zbl 1374.34016 |
| [55] | Saadatmandi, A.; Dehghan, M., A new operational matrix for solving fractional-order differential equations, Comput. Math. Appl., 59, 1326-1336, 2010 · Zbl 1189.65151 |
| [56] | Sadat, R.; Kaseem, MM; Ma, WX, Families of analytic solutions for (2+1) model in unbounded domain via optimal Lie vectors with integrating factors, Mod. Phys. Lett. B, 33, 1950229, 2019 |
| [57] | Sahadevan, R.; Prakash, P., Exact solution of certain time fractional nonlinear partial differential equations, Nonlinear Dyn., 85, 659-673, 2016 · Zbl 1349.35406 |
| [58] | Sahadevan, R.; Prakash, P., Exact solutions and maximal dimension of invariant subspaces of time fractional coupled nonlinear partial differential equations, Commun. Nonlinear Sci. Numer. Simulat., 42, 158-177, 2017 · Zbl 1473.35636 |
| [59] | Sahadevan, R.; Bakkyaraj, T., Invariant analysis of time fractional generalized Burgers and Korteweg-de Vries equations, J. Math. Anal. Appl., 393, 341-347, 2012 · Zbl 1245.35142 |
| [60] | Sahadevan, R.; Prakash, P., On Lie symmetry analysis and invariant subspace methods of coupled time fractional partial differential equations, Chaos Solitons Fractals, 104, 107-120, 2017 · Zbl 1380.35161 |
| [61] | Sahadevan, R.; Prakash, P., Lie symmetry analysis and conservation laws of certain time fractional partial differential equations, Int. J. Dynamical Systems and Differential Equations, 9, 44-64, 2019 · Zbl 1428.35671 |
| [62] | Senthilvelan, M.; Chandrasekar, VK; Mohanasubha, R., Symmetries of nonlinear ordinary differential equations: the modified Emden equation as a case study, Pramana - J. Phys., 85, 755-787, 2015 |
| [63] | Silva, MF; Machado, JAT; Lopes, AM, Fractional order control of a hexapod robot, Nonlinear Dyn., 38, 417-433, 2004 · Zbl 1096.70004 |
| [64] | Tarasov, VE, Fractional hydrodynamic equations for fractal media, Ann. Phys., 318, 286-307, 2005 · Zbl 1071.76002 |
| [65] | Tian, J.; Yu, Y.; Wang, H., Stability and bifurcation of two kinds of three-dimensional fractional Lotka-Volterra systems, Math. Prob. Eng, 2014, 695871, 2014 · Zbl 1407.34016 |
| [66] | Wang, ZH; Wang, X., General solution of the Bagley-Torvik equation with fractional-order derivative, Commun. Nonlinear Sci. Numer. Simulat., 15, 1279-1285, 2010 · Zbl 1221.34020 |
| [67] | Wulfman, CE; Wybourne, BG, The Lie group of Newton’s and Lagrange’s equations for the harmonic oscillator, J. Phys. A: Math. Gen., 9, 507-518, 1976 · Zbl 0321.34034 |
| [68] | Yaşar, E., The (G′ /G, 1/G)-expansion method for solving nonlinear space-time fractional differential equations, Pramana, J. Phys, 87, 17, 2016 |
| [69] | Yaşar, E.; Yıldırım, Y.; Yaşar, E., New optical solitons of space-time conformable fractional perturbed Gerdjikov-Ivanov equation by sine-Gordon equation method, Results Phys., 9, 1666-1672, 2018 |
| [70] | Yaşar, E.; Yıldırım, Y.; Khalique, CM, Lie symmetry analysis, conservation laws and exact solutions of the seventh-order time fractional Sawada-Koter-a-Ito equation, Results Phys., 6, 322-328, 2016 |
| [71] | Ye, Y.; Ma, WX; Shen, S.; Zhang, D., A class of third-order nonlinear evolution equations admitting invariant subspaces and associated reductions, J. Nonlinear Math. Phys., 21, 132-148, 2014 · Zbl 1420.35072 |
| [72] | Zaitsev, NA; Matyushkin, IV; Shamonov, DV, Numerical solution of Thomas-Fermi equation for the centrally symmetric atom, Russ. Microelectron., 33, 303-309, 2004 |
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