# zbMATH — the first resource for mathematics

Symmetry group classification of ordinary differential equations: Survey of some results. (English) Zbl 1135.34029
After the famous results of Sophus Lie on symmetry analysis of ordinary differential equations several results on point symmetry group analysis have been obtained, particularly by P. Leach. This article presents a review on the point symmetry group properties of linear $$n$$th order ($$n\geq 1$$) differential equations as well as the point symmetry group classification of scalar second order ODEs both in the real and complex domains. Many references are given to the papers of P. Leach on well-researched equations and related results on classification and integrability together with some open problem in this domain.

##### MSC:
 34C14 Symmetries, invariants of ordinary differential equations 34C20 Transformation and reduction of ordinary differential equations and systems, normal forms 34A30 Linear ordinary differential equations and systems 70H33 Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics
##### Keywords:
ordinary differential equation; symmetry; transformations
Full Text:
##### References:
 [1] Lie, Mathematishe Annalen 16 pp 441– (1880) [2] Lie, Archiv der Mathematik pp 187– (1883) [3] Lectures on Differential Equations with Known Infinitesimal Transformations. Teubner: Leipzig, 1891 (in German, Lie’s lectures by G. Sheffers). [4] Theorie der Transformationsgruppen, vol. III. Teubner: Leipzig, 1893. [5] Differential invariants and invariant differential equations. Modern group analysis V. Lie Groups and their Applications, vol. 1. Istanbul Technical University: Turkey, 1994; 177. [6] Kummer, Progr. Evang. Königl. Stadtgymnasium Liegnitz (1834) [7] Laguerre, Comptes Rendus de l’ Académie des Sciences, Paris 88 pp 224– (1879) [8] Comptes Rendus de l’ Académie des Sciences, Paris 88 pp 116– (1879) [9] Brioschi, Bulletin de la Société Mathématique de France 7 pp 105– (1879) [10] Halphen, Acta Mathematica 3 pp 325– (1883) [11] Forsyth, Philosophical Transactions of the Royal Society of London, Series A 179 pp 377– (1888) [12] Global Properties of Linear Ordinary Differential Equations. Kluwer: Dordrecht, 1991. [13] Dickson, Annals of Mathematics 25 pp 287– (1924) [14] Anderson, Journal of Mathematical Analysis and Applications 48 pp 301– (1974) [15] González-Gascón, Journal of Mathematical Physics 24 pp 2006– (1983) [16] Aguirre, Journal of Mathematical Physics 29 pp 9– (1988) [17] Ibragimov, Uspekhi Matematicheskikh Nauk 47 pp 83– (1992) [18] Mahomed, Journal of Mathematical Analysis and Applications 178 pp 116– (1993) [19] Govinder, SIAM Review 40 pp 945– (1998) [20] Wulfman, Journal of Physics A–Mathematical and General 9 pp 507– (1976) [21] Leach, Journal of Mathematical Physics 21 pp 300– (1980) [22] Leach, Journal of Physics A–Mathematical and General 13 pp 1991– (1980) [23] Leach, Journal of the Australian Mathematical Society (Series B) 22 pp 12– (1980) · Zbl 0429.70017 [24] Leach, Journal of Australian Mathematical Society (Series B) 23 pp 173– (1981) · Zbl 0479.70020 [25] Prince, Journal of Physics A–Mathematical and General 13 pp 815– (1980) [26] Mahomed, Journal of Mathematical Analysis and Applications 151 pp 80– (1990) [27] Krause, Lecture Notes in Physics 382 pp 251– (1991) [28] Krause, Comptes Rendus de l’ Académie des Sciences, Paris, Série I 307 pp 905– (1988) [29] Leach, Journal of Mathematical Analysis and Applications 252 pp 840– (2000) [30] Malet, Philosophical Transactions of the Royal Society of London 173 pp 751– (1882) [31] Tressé, Acta Mathematica 18 pp 1– (1894) [32] Elementary Lie Group Analysis and Ordinary Differential Equations. Wiley: Chichester, 1999. [33] Canonical forms of ordinary differential equations of order N with power nonlinearity. ISCM HERL’ANY Proceedings, University of Technology Kosice, 1999; 31. [34] Leach, Journal of Mathematical Physics 29 pp 1807– (1988) [35] Flessas, Bulletin of the Greek Mathematical Society 36 pp 63– (1994) [36] Govinder, Journal of Mathematical Analysis and Applications 193 pp 114– (1995) [37] Flessas, Journal of Mathematical Analysis and Applications 212 pp 349– (1997) [38] Leach, Journal of Mathematical Analysis and Applications 235 pp 58– (1999) [39] Kara, Journal of Nonlinear Mathematical Physics 9 pp 60– (2002) [40] . Symmetry properties of autonomous integrating factors. SIGMA 2005, vol. 1, Paper 024, 2005; 12. · Zbl 1106.34022 [41] Krause, Journal of Mathematical Physics 35 pp 5734– (1994) [42] Andriopoulos, Journal of Mathematical Analysis and Applications 262 pp 256– (2001) [43] Andriopoulos, Journal of Nonlinear Mathematical Physics 9 pp 10– (2002) [44] Mahomed, Journal of Mathematical Physics 30 pp 2770– (1989) [45] . Ordinary differential equations. CRC Handbook of Lie Group Analysis of Differential Equations, vol. 3. (ed.). CRC Press: Boca Raton, 1996; 191. [46] Lezioni sulla teoria dei Gruppi Continui Finiti di Transformazioni. Enrico Spoerri: Pisa, 1918. [47] Patera, Journal of Mathematical Physics 18 pp 1449– (1977) [48] Mubarakzyanov, Izvestiya Vysshikh Uchebnykh Zavedeni Matematika 32 pp 114– (1963) [49] Popovych, Journal of Physics A–Mathematical and General 36 pp 7337– (2003) [50] González-Lopéz, Proceedings of the London Mathematical Society 64 pp 339– (1992) [51] Equivalence, Invariants, and Symmetry. Cambridge University Press: Cambridge, MA, 1995. [52] Bouquet, Journal of Mathematical Physics 32 pp 1480– (1991) [53] Leach, International Journal of Nonlinear Mechanics 27 pp 575– (1992) [54] Mellin Conrad, International Journal of Nonlinear Mechanics 29 pp 529– (1994) [55] Détermination des Invariants Ponctuels de l’Équation Différentielle Ordinaire du Second Ordre y”=w(x, y, y’). Hirzel: Leipzig, 1896. [56] Berth, Applicable Algebra in Engineering Communication and Computing 11 pp 359– (2001) [57] Ibragimov, Communications in Nonlinear Science and Numerical Simulations 12 pp 1370– (2007) [58] Hsu, Proceedings of the London Mathematical Society 58 pp 387– (1989) [59] Kamran, Journal of Differential Geometry 22 pp 139– (1985) [60] , . On some equivalence problems for differential equations. Preprint ESI 54, Erwin Schrödinger International Institute for Mathematical Physics, Vienna, 1993. [61] Babich, Journal of Differential Equations 157 pp 452– (1999) [62] Cotsakis, Journal of Physics A–Mathematical and General 27 pp 1625– (1994) [63] Janet bases of 2nd order ordinary differential equations. Proceedings of the ISSAC’96, (ed.). ACM: New York, 1996; 179. · Zbl 0990.37502 [64] Mahomed, Quaestiones Mathematicae 8 pp 241– (1985) [65] Sarlet, Journal of Physics A–Mathematical and General 20 pp 277– (1987) [66] Mahomed, Quaestiones Mathematicae 12 pp 121– (1989) [67] Grissom, Journal of Differential Equations 77 pp 1– (1989) [68] Ibragimov, Nonlinear Dynamics 36 pp 41– (2004) [69] Leach, Journal of Mathematical Physics 29 pp 2563– (1988) [70] Leach, Journal of Mathematical Physics 26 pp 2510– (1985) [71] Wafo, Nonlinear Dynamics 28 pp 213– (2002) [72] Leach, Journal of Mathematical Analysis and Applications 287 pp 337– (2003) [73] Leach, Journal of Mathematical Physics 22 pp 679– (1981) [74] Leach, Journal of Physics A–Mathematical and General 23 pp 2765– (1990) [75] Govinder, Journal of Physics A–Mathematical and General 27 pp 4153– (1994) [76] Maharaj, General Relativity and Gravity 28 pp 35– (1996) [77] Moyo, Journal of Physics A–Mathematical and General 35 pp 5333– (2002) [78] Chern, Tensor N.S 28 pp 173– (1940) [79] . Normal forms for third order equations. Proceedings of the Workshop on Finite Dimensional Integrable Nonlinear Dynamical Systems, Johannesburg, (eds). World Scientific: Singapore, January 1988; 178. [80] Gat, Journal of Mathematical Physics 33 pp 2966– (1992) · Zbl 0777.34011 [81] Grebot, Journal of Mathematical Analysis and Applications 206 pp 364– (1997) · Zbl 0869.34007 [82] Ibragimov, Journal of Mathematical Analysis and Applications 308 pp 266– (2005) [83] Neut, Comptes Rendus de l’ Académie des Sciences, Paris, Série I 335 pp 515– (2002) · Zbl 1016.34007 [84] Berkovich, Programming and Computer Software 26 pp 38– (2000) [85] Euler, Acta Applicandae Mathematicae 76 pp 89– (2003) [86] Gorringe, Quaestiones Mathematicae 11 pp 95– (1988) [87] Wafo, Nonlinear Dynamics 22 pp 121– (2000) [88] Wafo, Nonlinear Dynamics 23 pp 377– (2000) [89] Wafo, International Journal of Nonlinear Mechanics 36 pp 671– (2001) [90] Mahomed, Nonlinear Dynamics 48 pp 417– (2007) [91] . Invariant linearization criteria for systems of cubically semi-linear second-order ordinary differential equations. Preprint, University of the Witwatersrand, Johannesburg. [92] Wafo, Journal of Physics A–Mathematical and General 34 pp 2883– (2001) [93] Fels, Proceedings of the London Mathematical Society 71 pp 221– (1995) [94] Cerquetelli, Journal of Nonlinear Mathematical Physics 9 pp 24– (2002) [95] Continuous Groups of Transformations. Princeton University Press: Princeton, NJ, 1933. · Zbl 0008.10801 [96] Group Analysis of Differential Equations. Academic Press: New York, 1982. [97] Differential Equations: Their Solutions Using Symmetries. Cambridge University Press: New York, 1989. [98] . Symmetry and Integration Methods for Differential Equations. Springer: New York, 2002. [99] . Symmetries of differential equations. An introduction to the how, the why and the wherefore. In Ordinary Differential Equations to Deterministic Chaos, , University of Durban-Westville Press: Durban, 1994; 321–384. [100] Ibragimov, Lie Groups and their Applications 1 pp 49– (1994) [101] Applications of Lie Groups to Differential Equations (2nd edn). Graduate Texts in Mathematics, vol. 107. Springer: New York, 1993. [102] Symmetry Methods for Differential Equations: A Beginner’s Guide. Cambridge Texts in Applied Mathematics. Cambridge University Press: Cambridge, MA, 2000. [103] Wafo, Journal of Nonlinear Mathematical Physics 11 pp 13– (2004)
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.