zbMATH — the first resource for mathematics

Lie symmetry analysis to the time fractional generalized fifth-order KdV equation. (English) Zbl 1304.35624
Summary: In this paper, using the Lie group analysis method, we study the invariance properties of the time fractional generalized fifth-order KdV equation. It shows that this equation can be reduced to an equation which is related to the Erdélyi-Kober fractional derivative. Of course, this method can also be applied to other nonlinear fractional partial differential equations.

35Q53 KdV equations (Korteweg-de Vries equations)
35A30 Geometric theory, characteristics, transformations in context of PDEs
35R11 Fractional partial differential equations
Full Text: DOI
[1] Olver, P. J., Application of Lie group to differential equation, (1986), Springer New York
[2] Ovsiannikov, L. V., Group analysis of differential equations, (1982), Academic Press New York · Zbl 0485.58002
[3] Lie, S., On integration of a class of linear partial differential equations by means of definite integrals, Arch Math, VI, 3, 328-368, (1881) · JFM 13.0298.01
[4] Bluman, G. W.; Kumei, S., Symmetries and differential equations, (1989), Springer New York · Zbl 0698.35001
[5] (Ibragimov, N. H., CRC handbook of Lie group analysis of differential equations, vol. 1-3, (1994), CRC Press Boca Raton, FL) · Zbl 0864.35001
[6] Liu, N.; Liu, X. Q., Similarity reductions and similarity solutions of the (3+1)-dimensional Kadomtsev-Petviashvili equation, Chin Phys Lett, 25, 3527-3530, (2008)
[7] Yan, Z. L.; Liu, X. Q., Symmetry and similarity solutions of variable coefficients generalized Zakharov-Kuznetsov equation, Appl Math Comput, 180, 288-294, (2006) · Zbl 1109.35101
[8] Xu, B.; Liu, X. Q., Classification, reduction, group invariant solutions and conservation laws of the gardner-KP equation, Appl Math Comput, 215, 1244-1250, (2009) · Zbl 1172.76006
[9] Wang, H.; Tian, Y. H., Non-Lie symmetry groups and new exact solutions of a (2+1)-dimensional generalized Broer-Kaup system, Commun Nonlinear Sci Numer Simul, 16, 3933-3940, (2011) · Zbl 1219.35263
[10] Kumar, S.; Singh, K.; Gupta, R. K., Painlevé analysis Lie symmetries and exact solutions for (2+1)-dimensional variable coefficients Broer-Kaup equations, Commun Nonlinear Sci Numer Simul, 17, 1529-1541, (2012) · Zbl 1245.35096
[11] Vaneeva, O., Lie symmetries and exact solutions of variable coefficient mkdv equations: an equivalence based approach, Commun Nonlinear Sci Numer Simul, 17, 611-618, (2012) · Zbl 1245.35114
[12] Johnpillai, A. G.; Khalique, C. M., Group analysis of KdV equation with time dependent coefficients, Appl Math Comput, 216, 3761-3771, (2010) · Zbl 1197.35231
[13] Adem, A. R.; Khalique, C. M., Symmetry reduction exact solutions and conservation laws of a new coupled KdV system, Commun Nonlinear Sci Numer Simul, 17, 9, C3465-C3475, (2012)
[14] Johnpillai, A. G.; Khalique, C. M., Lie group classification and invariant solutions of mkdv equation with time-dependent coefficients, Commun Nonlinear Sci Numer Simul, 16, 1207-1215, (2011) · Zbl 1221.35338
[15] Liu, H.; Li, J.; Liu, L., Lie symmetry analysis optimal systems and exact solutions to the fifth-order KdV types of equations, J Math Anal Appl, 368, 551-558, (2010) · Zbl 1192.35011
[16] Diethelm, Kai, The analysis of fractional differential equations, (2010), Springer · Zbl 1215.34001
[17] Liang, Y. J.; Chen, Wen, A survey on numerical evaluation of lvy stable distributions and a new MATLAB toolbox, Signal Process, 93, 242-251, (2013)
[18] Hu, S.; Chen, W.; Gou, X., Modal analysis of fractional derivative damping model of frequency-dependent viscoelastic soft matter, Adv Vib Eng, 10, 187-196, (2011)
[19] El-Sayed, A. M.A.; Gaber, M., The Adomian decomposition method for solving partial differential equations of fractal order in finite domains, Phys Lett A, 359, 175-182, (2006) · Zbl 1236.35003
[20] Chen, Y.; An, H. L., Numerical solutions of a new type of fractional coupled nonlinear equations, Commun Theor Phys, 49, 839-844, (2008) · Zbl 1392.35328
[21] Chen, Y.; An, H. L., Numerical solutions of coupled Burgers equations with time- and space-fractional derivatives, Appl Math Comput, 200, 87-95, (2008) · Zbl 1143.65102
[22] Odibat, Z.; Momani, S., A generalized differential transform method for linear partial differential equations of fractional order, Appl Math Lett, 21, 194-199, (2008) · Zbl 1132.35302
[23] Li, X.; Chen, W., Analytical study on the fractional anomalous diffusion in a half-plane, J Phys A: Math Theor, 43, 49, 11, (2010) · Zbl 1205.82108
[24] He, J. H., A coupling method of a homotopy technique and a perturbation technique for non-linear problems, J Non-Linear Mech, 35, 37-43, (2000) · Zbl 1068.74618
[25] Wu, G.; Lee, E. W.M., Fractional variational iteration method and its application, Phys Lett A, 374, 2506-2509, (2010) · Zbl 1237.34007
[26] Zhang, S.; Zhang, H. Q., Fractional sub-equation method and its applications to nonlinear fractional pdes, Phys Lett A, 375, 1069-1073, (2011) · Zbl 1242.35217
[27] Guo, S.; Mei, L. Q.; Li, Y.; Sun, Y. F., The improved fractional sub-equation method and its applications to the space-time fractional differential equations in fluid mechanics, Phys Lett A, 376, 407-411, (2012) · Zbl 1255.37022
[28] Lu, B., Báklund transformation of fractional Riccati equation and its applications to nonlinear fractional partial differential equations, Phys Lett A, 376, 2045-2048, (2012) · Zbl 1266.35139
[29] Gazizov, R. K.; Kasatkin, A. A.; Lukashchuk, S. Yu., Continuous transformation groups of fractional differential equations vestnik, USATU, 9, 125-135, (2007), [in Russian]
[30] Gazizov, R. K.; Kasatkin, A. A.; Lukashchuk, S. Yu., Symmetry properties of fractional diffusion equations, Phys Scr T, 136, 014016, (2009) · Zbl 1299.34014
[31] 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
[32] Djordjevic, V. D.; Atanackovic, T. M., Similarity solutions to nonlinear heat conduction and Burgers/Korteweg-devries fractional equations, J Comput Appl Math, 212, 701-714, (2008) · Zbl 1157.35470
[33] 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
[34] Jumarie, G., Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results, Comput Math Appl, 51, 1367-1376, (2006) · Zbl 1137.65001
[35] Jumarie, G., Cauchy’s integral formula via the modified Riemann-Liouville derivative for analytic functions of fractional order, Appl Math Lett, 23, 1444-1450, (2010) · Zbl 1202.30068
[36] Miller, K. S.; Ross, B., An introduction to the fractional calculus and fractional differential equations, (1993), Wiley New York · Zbl 0789.26002
[37] Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego, CA · Zbl 0918.34010
[38] Oldham, K. B.; Spanier, J., The fractional calculus, (1974), Academic Press · Zbl 0428.26004
[39] Kiryakova V. Generalised fractional calculus and applications. In: Pitman research notes in mathematics, vol. 301; 1994. · Zbl 0882.26003
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.