Lie symmetry analysis and exact solution of certain fractional ordinary differential equations. (English) Zbl 1374.34016
Summary: A systematic investigation of finding Lie point symmetries of certain fractional linear and nonlinear ordinary differential equations is presented. More precisely, Lie point symmetries of fractional Riccati equation, nonhomogeneous fractional linear ordinary differential equation with variable coefficients and quadratic fractional Liénard-type equation in the sense of Riemann-Liouville fractional derivative are derived. Using the obtained Lie point symmetries, we derive exact solution of the above-mentioned fractional ordinary differential equations wherever possible.
MSC:
| 34A08 | Fractional ordinary differential equations |
| 34C14 | Symmetries, invariants of ordinary differential equations |
Keywords:
fractional ordinary differential equations; Lie group formalism; Riemann-Liouville fractional derivative; Laplace transformation techniqueSoftware:
FracSymReferences:
| [1] | Podlubny, I.: Fractional Differential Equations. Acadmic Press, New York (1999) · Zbl 0924.34008 |
| [2] | Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006) · Zbl 1092.45003 |
| [3] | Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000) · Zbl 0998.26002 · doi:10.1142/3779 |
| [4] | Herrmann, R.: Fractional Calculus: An Introduction For Physicists. World Scientific, Singapore (2011) · Zbl 1232.26006 · doi:10.1142/8072 |
| [5] | Li, C., Zeng, F.: Numerical Methods for Fractional Calculus. Chapman and Hall/CRC, New York (2015) · Zbl 1326.65033 |
| [6] | Das, S.: Functional Fractional Calculus. Springer, Berlin (2011) · Zbl 1225.26007 · doi:10.1007/978-3-642-20545-3 |
| [7] | Kiryakova, V.: Generalized Fractional Calculus and Applications. Pitman Research Notes in Mathematics, vol. 301. Longman, Wiley, Harlow, New York (1994) · Zbl 0882.26003 |
| [8] | Atangana, A., Secer, A.: A note on fractional order derivatives and table of fractional derivatives of some special functions. Abstr. Appl. Anal. (2013). doi:10.1155/2013/279681 · Zbl 1276.26010 · doi:10.1155/2013/279681 |
| [9] | El-Nabulsi, R.A.: A cosmology governed by a fractional differential equation and the generalized Kilbas-Saigo-Mittag-Leffler function. Int. J. Theor. Phys. 55, 625-635 (2016) · Zbl 1336.83049 · doi:10.1007/s10773-015-2700-5 |
| [10] | El-Nabulsi, R.A.: Fractional functional with two occurrences of integrals and asymptotic optimal change of drift in the Black-Scholes model. Acta Math. Vietnam. 40, 689-703 (2015) · Zbl 1335.91118 · doi:10.1007/s40306-014-0079-7 |
| [11] | El-Nabulsi, R.A.: The fractional white dwarf hydrodynamical nonlinear differential equation and emergence of quark stars. Appl. Math. Comput. 218, 2837-2849 (2011) · Zbl 1243.85006 |
| [12] | Magin, R.L., Royston, T.J.: Fractional-order elastic models of cartilage: a multi-scale approach. Commun. Nonlinear Sci. Numer. Simulat. 15, 657-664 (2010) · Zbl 1221.74055 · doi:10.1016/j.cnsns.2009.05.008 |
| [13] | El-Nabulsi, R.A.: Modifications at large distances from fractional and fractal arguments. Fractals 18(2), 185-190 (2010) · Zbl 1196.28014 · doi:10.1142/S0218348X10004828 |
| [14] | Herrmann, R.: Common aspects of q-deformed Lie algebras and fractional calculus. Phys. A 389, 4613-4622 (2010) · doi:10.1016/j.physa.2010.07.004 |
| [15] | Herrmann, R.: Infrared spectroscopy of diatomic molecules: a fractional calculus approach. Int. J. Mod. Phys. B 27, 1350019(17p) (2013) · doi:10.1142/S0217979213500197 |
| [16] | Michelitsch, T.M., Collet, B.A., Riascos, A.P., Nowakowski, A.F., Nicolleau, F.C.G.A.: A fractional generalization of the classical lattice dynamics approach. Chaos Solitons Fractals 92, 43-50 (2016) · Zbl 1357.82059 · doi:10.1016/j.chaos.2016.09.009 |
| [17] | El-Nabulsi, R.A.: The fractional Boltzmann transport equation. Comput. Math. Appl. 62, 1568-1575 (2011) · Zbl 1228.82075 · doi:10.1016/j.camwa.2011.03.040 |
| [18] | Silva, M.F., Machado, J.A.T., Lopes, A.M.: Fractional order control of a hexapod robot. Nonlinear Dyn. 38, 417-433 (2004) · Zbl 1096.70004 · doi:10.1007/s11071-004-3770-8 |
| [19] | El-Nabulsi, R.A.: Fractional quantum Euler-Cauchy equation in the Schrödinger picture, complexified harmonic oscillators and emergence of complexified Lagrangian and Hamiltonian dynamics. Mod. Phys. Lett. B 23(28), 3369-3386 (2009) · Zbl 1185.81089 · doi:10.1142/S0217984909021387 |
| [20] | Zhang, Y.: Particle-tracking simulation of fractional diffusion-reaction processes. Phys. Rev. E 84, 066704 (2011) · doi:10.1103/PhysRevE.84.066704 |
| [21] | Zhang, Y., Benson, D.A., Reeves, D.M.: Time and space nonlocalities underlying fractional-derivative models: distinction and literature review of field applications. Adv. Water Resour. 32, 561-581 (2009) · doi:10.1016/j.advwatres.2009.01.008 |
| [22] | Zhang, Y.: Moments for tempered fractional advection-diffusion equations. J. Stat. Phys. 139, 915-939 (2010) · Zbl 1301.82044 · doi:10.1007/s10955-010-9965-0 |
| [23] | El-Nabulsi, R.A.: Fractional variational symmetries of Lagrangians, the fractional Galilean transformation and the modified Schrödinger equation. Nonlinear Dyn. 81, 939-948 (2015) · Zbl 1347.26020 · doi:10.1007/s11071-015-2042-0 |
| [24] | Kiryakova, V., Luchko, Y.: Riemann-Liouville and Caputo type multiple Erdélyi-Kober operators. Cent. Eur. J. Phys. 11(10), 1314-1336 (2013) |
| [25] | El-Nabulsi, R.A.: Generalized fractal space from Erdélyi-Kober fractional integral. Fizika A 19(3), 125-132 (2010) |
| [26] | El-Nabulsi, R.A.: The fractional calculus of variations from extended Erdélyi-Kober operator. Int. J. Mod. Phys. B 23(16), 3349-3361 (2009) · doi:10.1142/S0217979209052923 |
| [27] | El-Nabulsi, R.A.: Calculus of variations with hyperdifferential operators from Tabasaki-Takebe-Toda lattice arguments. RACSAM 107, 419-436 (2013) · Zbl 1275.49034 · doi:10.1007/s13398-012-0086-2 |
| [28] | El-Nabulsi, R.A.: Glaeske-Kilbas-Saigo fractional integration and fractional Dixmier trace. Acta Math. Vietnam. 37(2), 149-160 (2012) · Zbl 1254.26013 |
| [29] | El-Nabulsi, R.A.: Universal fractional Euler-Lagrange equation from a generalized fractional derivative operator. Cent. Eur. J. Phys. 9(1), 250-256 (2011) |
| [30] | El-Nabulsi, R.A.: Fractional variational problems from extended exponentially fractional integral. Appl. Math. Comput. 217, 9492-9496 (2011) · Zbl 1220.26004 |
| [31] | Lakshmanan, M., Rajasekar, S.: Nonlinear Dynamics: Integrability, Chaos, and Patterns. Springer, Berlin (2003) · Zbl 1038.37001 · doi:10.1007/978-3-642-55688-3 |
| [32] | Sahadevan, R., Prakash, P.: Exact solution of certain time fractional nonlinear partial differential equations. Nonlinear Dyn. 85, 659-673 (2016) · Zbl 1349.35406 · doi:10.1007/s11071-016-2714-4 |
| [33] | 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 · doi:10.1016/j.cnsns.2016.05.017 |
| [34] | Daftardar-Gejji, V., Jafari, H.: Adomian decomposition: a tool for solving a system of fractional differential equations. J. Math. Anal. Appl. 301, 508-518 (2005) · Zbl 1061.34003 · doi:10.1016/j.jmaa.2004.07.039 |
| [35] | Harris, P.A., Garra, R.: Analytic solution of nonlinear fractional Burgers-type equation by invariant subspace method. Nonlinear Stud. 20(4), 471-481 (2013) · Zbl 1305.35151 |
| [36] | Garra, R.: On the generalized Hardy-Hardy-Maurer model with memory effects. Nonlinear Dyn. 86, 861-868 (2016) · doi:10.1007/s11071-016-2928-5 |
| [37] | 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 · doi:10.1016/j.jmaa.2012.04.006 |
| [38] | Bakkyaraj, T., Sahadevan, R.: Group formalism of Lie transformations to time fractional partial differential equations. Pramana 85, 849-860 (2015) · Zbl 1499.35682 · doi:10.1007/s12043-015-1103-8 |
| [39] | El Kinani, E.H., Ouhadan, A.: Lie symmetry analysis of some time fractional partial differential equations. Int. J. Mod. Phys. Conf. Ser. 38, 1560075(8p) (2015) · doi:10.1142/S2010194515600757 |
| [40] | Bakkyaraj, T., Sahadevan, R.: Invariant analysis of nonlinear fractional ordinary differential equations with Riemann-Liouville derivative. Nonlinear Dyn. 80, 447-455 (2015) · Zbl 1345.34003 · doi:10.1007/s11071-014-1881-4 |
| [41] | Buckwar, E., Luchko, Y.: Invariance of a partial differential equation of fractional order under the Lie group of scaling transformations. J. Math. Anal. Appl. 227, 81-97 (1998) · Zbl 0932.58038 · doi:10.1006/jmaa.1998.6078 |
| [42] | Jefferson, G.F., Carminati, J.: FracSym: automated symbolic computation of Lie symmetries of fractional differential equations. Comput. Phys. Commun. 185, 430-441 (2014) · Zbl 1344.35003 · doi:10.1016/j.cpc.2013.09.019 |
| [43] | Gazizov, R.K., Kasatkin, A.A., Lukashchuk, SYu.: Continuous transformation groups of fractional differential equations. Vestnik USATU 9 3(21), 125-135 (2007). (in Russian) |
| [44] | Gazizov, R.K., Kasatkin, A.A., Lukashchuk, SYu.: Symmetry properties of fractional diffusion equations. Phys. Scr. T136, 014016 (2009) · doi:10.1088/0031-8949/2009/T136/014016 |
| [45] | Gazizov, RK; Kasatkin, AA; Lukashchuk, SYu; Machado, JAT (ed.); Luo, ACJ (ed.); Barbosa, RS (ed.); Silva, MF (ed.); Figueiredo, LB (ed.), Group invariant solutions of fractional differential equations, 51-58 (2011), Heidelberg · Zbl 1217.37066 · doi:10.1007/978-90-481-9884-9_5 |
| [46] | Ovsiannikov, L.V.: Group Analysis of Differential Equations. Academic Press, New York (1982) · Zbl 0485.58002 |
| [47] | Hydon, P.E.: Symmetry Methods for Differential Equations. Cambridge University Press, Cambridge (2000) · Zbl 0951.34001 · doi:10.1017/CBO9780511623967 |
| [48] | Lakshmanan, M., Kaliappan, P.: Lie transformations, nonlinear evolution equations, and Painlevé forms. J. Math. Phys. 24, 795-806 (1983) · Zbl 0525.35022 · doi:10.1063/1.525752 |
| [49] | Olver, P.J.: Applications of Lie Groups to Differential Equations. Springer, Heidelberg (1986) · Zbl 0588.22001 · doi:10.1007/978-1-4684-0274-2 |
| [50] | Yasar, E., Yildirim, Y., Khalique, C.M.: Lie symmetry analysis, conservation laws and exact solutions of the seventh-order time fractional Sawada-Kotera-Ito equation. Results Phys. 6, 322-328 (2016) · doi:10.1016/j.rinp.2016.06.003 |
| [51] | Gaur, M., Singh, K.: Symmetry analysis of time-fractional potential Burgers’ equation. Math. Commun. 22, 1-11 (2017) · Zbl 1373.35024 |
| [52] | Gaur, M., Singh, K.: Symmetry classification and exact solutions of a variable coefficient space-time fractional potential Burgers’ equation. Int. J. Diff. Equ. (2016). doi:10.1155/2016/4270724 · Zbl 1349.35399 · doi:10.1155/2016/4270724 |
| [53] | Mathai, A.M., Haubold, H.J.: Special Functions for Applied Scientists. Springer, New York (2008) · Zbl 1151.33001 |
| [54] | Mathai, A.M., Saxena, R.K., Haubold, H.J.: A certain class of Laplace transforms with applications to reaction and reaction-diffusion equations. Astrophys. Space Sci. 305, 283-288 (2006) · Zbl 1105.35301 · doi:10.1007/s10509-006-9188-7 |
| [55] | Jordan, D.W., Smith, P.: Nonlinear Ordinary Differential Equations: Problems and Solutions. Oxford University Press, New York (2007) · Zbl 1130.34002 |
| [56] | Tiwari, A.K., Pandey, S.N., Senthilvelan, M., Lakshmanan, M.: Classification of Lie point symmetries for quadratic Liénard type equation \[\ddot{x}+f(x)\dot{x}^2+g(x)=0\] x¨+f(x)x˙2+g(x)=0. J. Math. Phys. 54, 053506 (2013) · Zbl 1295.34049 · doi:10.1063/1.4803455 |
| [57] | Mimura, F., Nâno, T.: A new conservation law for a system of second-order differential equations. Bull. Kyushu Inst. Tech. 41, 1-10 (1994) · Zbl 0805.34006 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
