Symmetries, Differential Equations, and Applications: Galois Bicentenary
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Ben Muatjetjeja, Chaudry Masood Khalique, Fazal Mahmood Mahomed, "Group Classification of a Generalized LaneEmden System", Journal of Applied Mathematics, vol. 2013, Article ID 305032, 12 pages, 2013. https://doi.org/10.1155/2013/305032
Group Classification of a Generalized LaneEmden System
Abstract
We perform the group classification of the generalized LaneEmden system , which occurs in many applications of physical phenomena such as pattern formation, population evolution, and chemical reactions. We obtain four cases depending on the values of n.
1. Introduction
The celebrated LaneEmden equation where is a real constant and is a realvalued function of the variable , has many applications in mathematical physics and astrophysics. Equation (1), for certain fixed values of and , models several phenomena such as the theory of stellar structure, the thermal behavior of a spherical cloud of gas, isothermal gaseous sphere, and the theory of thermionic currents [1â€“3]. Several methods for the solution and many applications of the LaneEmden Equation (1) can be found in the literature. The interested reader is referred to [4] and the references therein. It is worth mentioning that Wong [5], in his review paper of 1975, presented more than 140 references on this topic.
A natural extension of (1), called the generalized LaneEmden system [6], is given by
Such systems arise in the modeling of several physical phenomena, such as pattern formation, population evolution, chemical reactions, and so on [7], and in the past few years have attracted much attention. Various researchers have worked on existence and uniqueness results for the LaneEmden systems [8, 9] and other related systems [10â€“12].
In [6] the authors studied Noether operators with respect to the standard Lagrangian of the generalized coupled LaneEmden system (2). They obtained seven cases out of which six cases resulted in Noether point symmetries. The first integrals corresponding to the Noether operators in each case were also constructed.
The objective of this paper is to perform the Lie group classification of the generalized LaneEmden system (2). The paper is organized as follows. In Section 2, we calculate the equivalence transformations of the LaneEmden system (2). We determine the principal Lie algebra and perform the group classification of system (2) in Section 3. Finally, concluding remarks are presented in Section 4.
2. Equivalence Transformations
An equivalence transformation (see, e.g., [13]) of the system (2) is an invertible transformation involving the variables , , and that map system (2) into itself, with possibly the form of the transformed functions being different from that of the original functions and . We write system (2) as where and are differential variables with independent variable , and is a differential function of the independent variables and , whereas is a differential function of the independent variables and . We obtain the generators of the group of equivalence transformations as
We apply Lie's infinitesimal approach by using the prolongation of to involve the derivatives in system (3) as, for example, in [14].
We summarize our results below.
Case 1 (). In this case system (3) has the ninedimensional equivalence Lie algebra spanned by the equivalence generators
and hence the nineparameter equivalence group is given by
Thus the composition of these transformations gives
Case 2 (). In this case system (3) has the ninedimensional equivalence Lie algebra spanned by the equivalence generators
and hence the nineparameter equivalence group is given by
Hence the composition of these transformations gives
Case 3 (). In this case system (3) has the ninedimensional equivalence Lie algebra spanned by the equivalence generators
and hence the nineparameter equivalence group is given by
and so the composition of these transformations gives
Case 4 (). In this case system (3) has the tendimensional equivalence Lie algebra spanned by the equivalence generators
and hence the tenparameter equivalence group is given by
Therefore the composition of these transformations gives
3. Principal Lie Algebra and Lie Group Classification
The generalized LaneEmden system (2) admits a Lie point symmetry if
After some albeit tedious and lengthy calculations, the above determining equation gives
Consequently, we conclude that the principal Lie algebra of (2) is trivial and the classifying relations are where , and are constants.
These classifying relations are invariant under the equivalence transformation (7) if
The classifying relations (20) are also invariant under the equivalence transformation (10) if
It is also noted that the classifying relations (20) are invariant under the equivalence transformation (13) if
The classifying relations (20) are also invariant under the equivalence transformation (16) if
The above relations are now used to find the nonequivalence forms of and and their corresponding Lie point symmetry. Several cases arise and are presented in Tables 1, 2, 3, and 4.
