Symmetries, Differential Equations, and Applications: Galois Bicentenary
View this Special IssueResearch Article  Open Access
A Note on FourDimensional Symmetry Algebras and FourthOrder Ordinary Differential Equations
Abstract
We provide a supplementation of the results on the canonical forms for scalar fourthorder ordinary differential equations (ODEs) which admit fourdimensional Lie algebras obtained recently. Together with these new canonical forms, a complete list of scalar fourthorder ODEs that admit fourdimensional Lie algebras is available.
1. Introduction
The integrability of scalar ordinary differential equations (ODEs) by use of the Lie symmetry method depends on their symmetrical Lie algebra if the Lie algebra is solvable and of sufficient dimension. There exists two different approaches to the integrability of differential equations using Lie point symmetries. One is the direct method in which Lie point symmetries are utilized to perform integrability by successive reduction of order of the equation using ideals of the algebra. The other approach is the canonical form method if the equations are classified into different types according to the canonical forms of the corresponding Lie algebra.
Lie [1] classified scalar secondorder ODEs into four types on the basis of their admitted twodimensional Lie algebras and also performed integration of the representative equations corresponding to the canonical forms of the twodimensional symmetry algebra. Therefore, scalar secondorder ODEs can be integrated by using canonical variables which map the symmetry generators to Lie’s canonical forms. Also here one can use successive reduction of order by using ideals of the symmetry algebra (see, e.g., Olver [2]). The Noether equivalence approach for Lagrangians corresponding to scalar secondorder ODEs is discussed in Kara et al. [3].
The canonical forms for scalar thirdorder ODEs that admit three symmetries were obtained by Mahomed and Leach [4]. Then Ibragimov and Nucci [5] provided the integrability of these canonical forms.
In their paper, Cerquetelli et al. [6] constructed realizations in the plane of fourdimensional Lie algebras listed by Patera and Winternitz [7]. Moreover, the classification of subalgebras of all real Lie algebras of dimension ≤4 was discussed in [7]. The construction of Cerquetelli et al. [6] was based on the threedimensional subalgebras provided in [7]. They invoked the realizations of threedimensional Lie algebras in the plane derived earlier by Mahomed and Leach [8] for this purpose. They then determined the fourthorder ODEs admitting the realizations of the obtained fourdimensional algebras as their Lie symmetry algebra. Finally, they provided the route to the integration of the classified fourthorder ODEs. However, the derived realizations of fourdimensional Lie algebras need supplementation in the light of recent work by Popovych et al. [9]. Recently, these authors constructed a complete set of inequivalent realizations of real Lie algebras of dimension not greater than four in vector fields in the space of an arbitrary (finite) numbers of variables. We use the results of [9] to complete the classification of fourthorder ODEs in terms of their fourdimensional algebras presented in [6].
Apart from scalar fourthorder ODEs arising in the symmetry reductions of partial differential equations such as the linear wave equation in an inhomogeneous medium (see [10]), they occur prominently as model equations in the form of the static EulerBernoulli beam (see, e.g., [11]) and EmdenFowler equations. Such equations have been investigated for symmetry properties in [12, 13].
Firstly, we provide a comparison of the results of [9] and that of [6] related to the realizations of fourdimensional Lie algebras as vector fields in the plane. Then we list the new canonical forms of scalar fourthorder ODEs which possess fourdimensional algebras.
2. Comparison of the Results of [6, 9]
We show here that the results on realizations of fourdimensional algebras in the plane given in [6] are a special case of the corresponding set of realizations given in [9]. We make a comparison of the lists of realizations given in [6] and [9]. It should be remarked that in general a result of classification of realizations may contain errors of two types, namely,(i)missing of some inequivalent cases and(ii)mutually equivalent cases.
In the following comparison, some first type of errors exist in [6]. Five cases are missing. There are some other cases which can be combined in a compact form and also some arbitrary parameters and functions need modification according to the results of [9] related to realizations in the plane. Below we keep the notations of both: on the left hand side the notations of [6] and on right hand side that of [9]. However, for the final results and further utilization, we keep the notations of [9].
2.1. FourDimensional Algebras
We use the nomenclature of Patera and Winternitz [7] in the naming of the algebras such as . Thus we do not provide a table of the abstract algebras of dimension four as this is easily available.
Here the refers to the realizations given in the work [6] and to that of [9].(i). .(ii). No realization exists in dimension.(iii), , , . , , whereas is missing in [6].(iv). No realization exists in dimension.(v). (vi). No realization exists in dimension.(vii). .(viii). (ix). (x). (xi). whereas , is missing in [6].(xii). No realization exists in dimension.(xiii). (xiv). .(xv). No realization exists in dimension.(xvi). .(xvii). (xviii), , , , , , , . , , , , , , , .(xix), , . No realization exists in dimension.(xx), . No realization exists in dimension.(xxi). .(xxii). (xxiii). (xxiv). (xxv). , whereasis missing in [6].(xxvi). (xxvii). whereas, is missing in [6].(xxviii). No realization exists in dimension.(xxix). No realization exists in dimension.(xxx). whereas , is missing in [6].
Remarks for Table 1(i): , , and form a linearly independent set.

Remarks for Tables 1 and 2. In both tables we have(i) are arbitrary functions with specified conditions mentioned in the corresponding realizations.(ii), and are parameters and arbitrary constants, whose range and values are mentioned in each of the realizations.(iii): these are the cases of realizations which are missing in [6].

Remarks for Table 2(i): these are the canonical forms of fourthorder ODEs which are missing in [6]. Only these are given here with their corresponding algebras and realizations.
3. Concluding Remarks
In this contribution we have supplemented the work [6] for the canonical forms of scalar fourthorder ODEs and have obtained four new forms as listed in Table 2. The integrability of these equations has the same route as the others which are discussed at length in [6].
Acknowledgments
A. Fatima gratefully acknowledges the financial support and scholarship from School of Computational and Applied Mathematics, University of the Witwatersrand.
References
 S. Lie, “Classification und Integration von gewöhnlichen Differentialgleichungen zwischen x, y, die eine Gruppe von Transformationen gestatten,” Mathematische Annalen, vol. 8, no. 9, p. 187, 1888. View at: Google Scholar
 P. J. Olver, Applications of Lie Groups to Differential Equations, vol. 107, Springer, New York, NY, USA, 2nd edition, 1993. View at: Publisher Site  Zentralblatt MATH  MathSciNet
 A. H. Kara, F. M. Mahomed, and P. G. L. Leach, “Noether equivalence problem for particle Lagrangians,” Journal of Mathematical Analysis and Applications, vol. 188, no. 3, pp. 867–884, 1994. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 F. M. Mahomed and P. G. L. Leach, “Normal forms for thirdorder equations,” in Proceedings of the Workshop on Finite Dimensional Integrable Nonlinear Dynamical Systems, P. G. L. Leach and W. H. Steeb, Eds., p. 178, World Scientic, Johannesburg, South Africa, 1988. View at: Google Scholar
 N. H. Ibragimov and M. C. Nucci, “Integration of third order ordinary differential equations by lies method: equations admitting threedimensional lie algebras,” Lie Groups and Their Applications, vol. 1, p. 4964, 1994. View at: Google Scholar
 T. Cerquetelli, N. Ciccoli, and M. C. Nucci, “Four dimensional Lie symmetry algebras and fourth order ordinary differential equations,” Journal of Nonlinear Mathematical Physics, vol. 9, supplement 2, pp. 24–35, 2002. View at: Publisher Site  Google Scholar  MathSciNet
 J. Patera and P. Winternitz, “Subalgebras of real three and fourdimensional Lie algebras,” Journal of Mathematical Physics, vol. 18, no. 7, pp. 1449–1455, 1977. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 F. M. Mahomed and P. G. L. Leach, “Lie algebras associated with scalar secondorder ordinary differential equations,” Journal of Mathematical Physics, vol. 30, no. 12, pp. 2770–2777, 1989. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 R. O. Popovych, V. M. Boyko, M. O. Nesterenko, and M. W. Lutfullin, “Realizations of real lowdimensional Lie algebras,” Journal of Physics A, vol. 36, no. 26, pp. 7337–7360, 2003. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 G. Bluman and S. Kumei, “On invariance properties of the wave equation,” Journal of Mathematical Physics, vol. 28, no. 2, pp. 307–318, 1987. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 S. P. Timoshenko, Theory of Elastic Stability, McGrawHill, New York, NY, USA, 2nd edition, 1961. View at: MathSciNet
 A. H. Bokhari, F. M. Mahomed, and F. D. Zaman, “Symmetries and integrability of a fourthorder EulerBernoulli beam equation,” Journal of Mathematical Physics, vol. 51, no. 5, Article ID 053517, 2010. View at: Publisher Site  Google Scholar  MathSciNet
 F. I. Leite, P. D. Lea, and T. Mariano, “Symmetry and integrability of a fourthorder EmdenFowler equation,” Technical Report CMCCUFABC113, 2012. View at: Google Scholar
Copyright
Copyright © 2013 A. Fatima et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.