Research Article Group Classification of a Generalized Lane-Emden System Ben Muatjetjeja,

Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2013, Article ID 305032, 12 pages http://dx.doi.org/10.1155/2013/305032 Research Article Group Classification of a Generalized Lane-Emden System Ben Muatjetjeja,1 Chaudry Masood Khalique,1 and Fazal Mahmood Mahomed2 1 Department of Mathematical Sciences, International Institute for Symmetry Analysis and Mathematical Modelling, North-West University, Mafikeng Campus, Private Bag X2046, Mmabatho 2735, South Africa 2 Centre for Differential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics, University of the Witwatersrand, (Wits), Johannesburg 2050, South Africa Correspondence should be addressed to Chaudry Masood Khalique; masood.khalique@nwu.ac.za Received 11 September 2012; Accepted 25 November 2012 Academic Editor: Mehmet Pakdemirli Copyright © 2013 Ben Muatjetjeja 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. We perform the group classification of the generalized Lane-Emden system 𝑥𝑢󸀠󸀠 + 𝑛𝑢󸀠 + 𝑥𝐻(𝑣) = 0, 𝑥𝑣󸀠󸀠 + 𝑛𝑣󸀠 + 𝑥𝑔(𝑢) = 0, 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 Lane-Emden equation 𝑑2 𝑦 𝑛 𝑑𝑦 + + 𝑓 (𝑦) = 0, 𝑑𝑥2 𝑥 𝑑𝑥 (1) where 𝑛 is a real constant and 𝑓(𝑦) is a real-valued function of the variable 𝑦, has many applications in mathematical physics and astrophysics. Equation (1), for certain fixed values of 𝑛 and 𝑓(𝑦), 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 [1–3]. Several methods for the solution and many applications of the Lane-Emden 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 2. Equivalence Transformations 2 𝑑 𝑢 𝑛 𝑑𝑢 + + 𝐻 (𝑣) = 0, 𝑑𝑥2 𝑥 𝑑𝑥 𝑑2 𝑣 𝑛 𝑑𝑣 + + 𝐺 (𝑢) = 0. 𝑑𝑥2 𝑥 𝑑𝑥 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 [10–12]. 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 Lane-Emden system (2). The paper is organized as follows. In Section 2, we calculate the equivalence transformations of the Lane-Emden 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) An equivalence transformation (see, e.g., [13]) of the system (2) is an invertible transformation involving the variables 𝑥, 𝑢, and 𝑣 that map system (2) into itself, with possibly the form of the transformed functions being different from that 2 Journal of Applied Mathematics of the original functions 𝐻(𝑣) and 𝐺(𝑢). We write system (2) as 𝑋1 : 𝑥 = 𝑒𝑎1 𝑥, 2 𝑑 𝑢 𝑛 𝑑𝑢 + + 𝐻 (𝑣) = 0, 𝑑𝑥2 𝑥 𝑑𝑥 𝐻𝑢 = 0, 𝐺𝑥 = 0, (3) 𝐺𝑣 = 0, 𝑋3 : 𝑥 = 𝑥, 𝜕 𝜕 + 𝜇1 (𝑥, 𝑢, 𝑣, 𝐻, 𝐺) 𝜕𝑣 𝜕𝐻 𝐻 = 𝐻, 𝑋4 : 𝑥 = 𝑥, 𝑣 = 𝑣, Case 1 (𝑛 ≠ − 1, 1, 3). In this case system (3) has the ninedimensional equivalence Lie algebra spanned by the equivalence generators 𝑣 = 𝑒𝑎3 𝑣, 𝐺 = 𝑒𝑎3 𝐺, 𝐻 = 𝐻, 𝐺 = 𝐺, 𝑢 = 𝑢, 𝑣 = 𝑣 + 𝑎5 , 𝐺 = 𝐺, 𝐻 = 𝐻, 𝑋7 : 𝑥 = 𝑥, We apply Lie’s infinitesimal approach by using the prolongation of 𝑌 to involve the derivatives in system (3) as, for example, in [14]. We summarize our results below. 𝐺 = 𝐺, 𝑢 = 𝑢 + 𝑎6 𝑥1−𝑛 , 𝑋6 : 𝑥 = 𝑥, 𝑣 = 𝑣, 𝜕 + 𝜇 (𝑥, 𝑢, 𝑣, 𝐻, 𝐺) . 𝜕𝐺 2 𝑣 = 𝑣, 𝑢 = 𝑢 + 𝑎4 , 𝐻 = 𝐻, (4) 𝐺 = 𝑒−2𝑎1 𝐺, 𝑢 = 𝑢, 𝑋5 : 𝑥 = 𝑥, 𝜕 𝜕 + 𝜂1 (𝑥, 𝑢, 𝑣) 𝑌 = 𝜉 (𝑥, 𝑢, 𝑣) 𝜕𝑥 𝜕𝑢 𝑣 = 𝑣, 𝑢 = 𝑒𝑎2 𝑢, 𝑋2 : 𝑥 = 𝑥, 𝐻 = 𝑒𝑎2 𝐻, where 𝑢 and 𝑣 are differential variables with independent variable 𝑥, and 𝐻 is a differential function of the independent variables 𝑥 and 𝑣, whereas 𝐺 is a differential function of the independent variables 𝑥 and 𝑢. We obtain the generators of the group of equivalence transformations as + 𝜂2 (𝑥, 𝑢, 𝑣) 𝑢 = 𝑢, 𝐻 = 𝑒−2𝑎1 𝐻, 𝑑2 𝑣 𝑛 𝑑𝑣 + + 𝐺 (𝑢) = 0, 𝑑𝑥2 𝑥 𝑑𝑥 𝐻𝑥 = 0, and hence the nine-parameter equivalence group is given by 𝑢 = 𝑢, 𝐻 = 𝐻, 𝐺 = 𝐺, 𝑣 = 𝑣 + 𝑎7 𝑥1−𝑛 , 𝐺 = 𝐺, 𝑢 = 𝑢 + 𝑎8 𝑥2 , 𝑋8 : 𝑥 = 𝑥, 𝐻 = 𝐻 − 2 (1 + 𝑛) 𝑎8 , 𝑋9 : 𝑥 = 𝑥, 𝑢 = 𝑢, 𝐻 = 𝐻, 𝑣 = 𝑣, 𝐺 = 𝐺, 𝑣 = 𝑣 + 𝑎9 𝑥2 , 𝐺 = 𝐺 − 2 (1 + 𝑛) 𝑎9 . (6) 𝑋1 = 𝑥 𝜕 𝜕 𝜕 − 2𝐻 − 2𝐺 , 𝜕𝑥 𝜕𝐻 𝜕𝐺 𝑋2 = 𝑢 𝜕 𝜕 +𝐻 , 𝜕𝑢 𝜕𝐻 𝑥 = 𝑒𝑎1 𝑥, 𝑋3 = 𝑣 𝜕 𝜕 +𝐺 , 𝜕𝑣 𝜕𝐺 𝑢 = 𝑒𝑎2 (𝑢 + 𝑎8 𝑥2 + 𝑎6 𝑥1−𝑛 + 𝑎4 ) , Thus the composition of these transformations gives 𝑣 = 𝑒𝑎3 (𝑣 + 𝑎9 𝑥2 + 𝑎7 𝑥1−𝑛 + 𝑎5 ) , 𝜕 𝑋4 = , 𝜕𝑢 𝑋5 = (7) 𝐻 = 𝑒𝑎2 −2𝑎1 (𝐻 − 2 (1 + 𝑛) 𝑎8 ) , 𝜕 , 𝜕𝑣 (5) 𝑋6 = 𝑥1−𝑛 𝜕 , 𝜕𝑢 𝑋7 = 𝑥1−𝑛 𝜕 , 𝜕𝑣 𝐺 = 𝑒𝑎3 −2𝑎1 (𝐺 − 2 (1 + 𝑛) 𝑎9 ) . Case 2 (𝑛 = −1). In this case system (3) has the ninedimensional equivalence Lie algebra spanned by the equivalence generators 𝑋8 = 𝑥2 𝜕 𝜕 − 2 (1 + 𝑛) , 𝜕𝑢 𝜕𝐻 𝑋1 = 𝑥 𝜕 𝜕 𝜕 − 2𝐻 − 2𝐺 , 𝜕𝑥 𝜕𝐻 𝜕𝐺 𝑋9 = 𝑥2 𝜕 𝜕 − 2 (1 + 𝑛) 𝜕𝑣 𝜕𝐺 𝑋2 = 𝑢 𝜕 𝜕 +𝐻 , 𝜕𝑢 𝜕𝐻 Journal of Applied Mathematics 𝑋3 = 𝑣 3 𝜕 𝜕 +𝐺 , 𝜕𝑣 𝜕𝐺 𝑋4 = 𝜕 , 𝜕𝑢 𝑋5 = 𝜕 , 𝜕𝑣 Hence the composition of these transformations gives 𝑥 = 𝑒𝑎1 𝑥, 𝑢 = 𝑒𝑎2 (𝑢 + 𝑎6 𝑥2 ln 𝑥 + 𝑥2 𝑎8 + 𝑎4 ) , 𝑣 = 𝑒𝑎3 (𝑣 + 𝑎7 𝑥2 ln 𝑥 + 𝑥2 𝑎9 + 𝑎5 ) , 𝜕 𝜕 𝑋6 = 𝑥2 ln 𝑥 −2 , 𝜕𝑢 𝜕𝐻 𝑋7 = 𝑥2 ln 𝑥 𝑋8 = 𝑥2 𝜕 , 𝜕𝑢 𝑋9 = 𝑥2 𝜕 𝜕𝑣 𝐻 = 𝑒𝑎2 −2𝑎1 (𝐻 − 2𝑎6 ) , 𝜕 𝜕 −2 , 𝜕𝑣 𝜕𝐺 𝐺 = 𝑒𝑎3 −2𝑎1 (𝐺 − 2𝑎7 ) . Case 3 (𝑛 = 1). In this case system (3) has the ninedimensional equivalence Lie algebra spanned by the equivalence generators (8) and hence the nine-parameter equivalence group is given by 𝑋1 : 𝑥 = 𝑒𝑎1 𝑥, 𝑢 = 𝑢, 𝐻 = 𝑒−2𝑎1 𝐻, 𝑋2 : 𝑥 = 𝑥, 𝑋3 : 𝑥 = 𝑥, 𝐻 = 𝐻, 𝑋4 : 𝑥 = 𝑥, 𝐻 = 𝐻, 𝑋5 : 𝑥 = 𝑥, 𝐻 = 𝐻, 𝑋6 : 𝑥 = 𝑥, 𝑣 = 𝑣, 𝐺 = 𝑒−2𝑎1 𝐺, 𝑢 = 𝑒𝑎2 𝑢, 𝐻 = 𝑒𝑎2 𝐻, 𝑣 = 𝑣, 𝐺 = 𝐺, 𝑣 = 𝑒𝑎3 𝑣, 𝑢 = 𝑢, 𝑎3 𝐺 = 𝑒 𝐺, 𝑢 = 𝑢 + 𝑎4 , 𝑣 = 𝑣, 𝐺 = 𝐺, 𝑢 = 𝑢, 𝑣 = 𝑣 + 𝑎5 , 𝐺 = 𝐺, 𝑢 = 𝑢 + 𝑎6 𝑥2 ln 𝑥, 𝐻 = 𝐻, 𝑋8 : 𝑥 = 𝑥, 𝐻 = 𝐻, 𝑋9 : 𝑥 = 𝑥, 𝐻 = 𝐻, 𝑢 = 𝑢, 𝑋2 = 𝑢 𝜕 𝜕 +𝐻 , 𝜕𝑢 𝜕𝐻 𝑋3 = 𝑣 𝜕 𝜕 +𝐺 , 𝜕𝑣 𝜕𝐺 𝑋4 = 𝜕 , 𝜕𝑢 𝑋5 = 𝜕 , 𝜕𝑣 (11) 𝑋6 = ln 𝑥 𝜕 , 𝜕𝑢 𝑋7 = ln 𝑥 𝜕 , 𝜕𝑣 𝑋8 = 𝑥2 𝜕 𝜕 −4 , 𝜕𝑢 𝜕𝐻 𝑋9 = 𝑥2 𝜕 𝜕 −4 𝜕𝑣 𝜕𝐺 𝑢 = 𝑢, 𝐻 = 𝑒−2𝑎1 𝐻, 𝑋2 : 𝑥 = 𝑥, 𝐺 = 𝐺 − 2𝑎7 , 𝑋3 : 𝑥 = 𝑥, 𝐻 = 𝐻, 𝑣 = 𝑣 + 𝑎9 𝑥2 , 𝑋4 : 𝑥 = 𝑥, 𝐺 = 𝐺. (9) 𝐻 = 𝐻, 𝑣 = 𝑣, 𝐺 = 𝑒−2𝑎1 𝐺, 𝑢 = 𝑒𝑎2 𝑢, 𝐻 = 𝑒𝑎2 𝐻, 𝑣 = 𝑣, 𝐺 = 𝐺, 𝑢 = 𝑢, 𝜕 𝜕 𝜕 − 2𝐻 − 2𝐺 , 𝜕𝑥 𝜕𝐻 𝜕𝐺 𝑋1 : 𝑥 = 𝑒𝑎1 𝑥, 𝑣 = 𝑣 + 𝑎7 𝑥2 ln 𝑥, 𝑢 = 𝑢 + 𝑎8 𝑥2 , 𝑋1 = 𝑥 and hence the nine-parameter equivalence group is given by 𝑣 = 𝑣, 𝐻 = 𝐻 − 2𝑎6 , 𝐺 = 𝐺, 𝑋7 : 𝑥 = 𝑥, (10) 𝑣 = 𝑣, 𝐺 = 𝐺, 𝑢 = 𝑢, 𝑣 = 𝑒𝑎3 𝑣, 𝐺 = 𝑒𝑎3 𝐺, 𝑢 = 𝑢 + 𝑎4 , 𝐺 = 𝐺, 𝑣 = 𝑣, 4 Journal of Applied Mathematics Table 1: Lie symmetries for 𝑛 ≠ − 1, 1, 3, for various functions 𝐻(𝑣) and 𝐺(𝑢). 𝐻(𝑣) arbitrary, 𝐺(𝑢) = 𝑐, 𝑐 a constant 𝜕 𝜕 𝜕 𝑋1 = , 𝑋 = 𝑥(1−𝑛) , 𝑋3 = (𝑐𝑥2 + 2𝑛𝑣 + 2𝑣) 𝜕𝑢 2 𝜕𝑢 𝜕𝑢 𝐻(𝑣) = 𝑑, 𝑑 a constant, 𝐺(𝑢) arbitrary 𝜕 𝜕 𝜕 𝑋1 = , 𝑋 = 𝑥(1−𝑛) , 𝑋3 = (𝑑𝑥2 + 2𝑛𝑢 + 2𝑢) 𝜕𝑣 2 𝜕𝑣 𝜕𝑣 𝐻(𝑣) = 𝑑, 𝑑 a constant, 𝐺(𝑢) = 𝛼 + 𝛽𝑢, 𝛼 and 𝛽 constants (𝛽 ≠ 0) 𝜕 𝜕 𝜕 𝑋1 = , 𝑋 = 𝑥(1−𝑛) , 𝑋3 = (𝑑𝑥2 + 2𝑛𝑢 + 2𝑢) , 𝜕𝑣 2 𝜕𝑣 𝜕𝑣 𝜕 𝜕 + (2𝑛 − 6)𝑥(1−𝑛) , 𝑋4 = 𝛽𝑥(3−𝑛) 𝜕𝑣 𝜕𝑢 𝜕 𝜕 𝑋5 = (2𝑛 + 2) − 𝛽𝑥2 , 𝜕𝑢 𝜕𝑣 𝜕 𝜕 𝜕 𝑋6 = (2𝑛2 + 8𝑛 + 6) 𝑥 − 4 (𝑛2 𝑢 + 3𝑑𝑥2 + 4𝑛𝑢 + 3𝑢 + 𝑛𝑑𝑥2 ) + (𝛽𝑑𝑥2 − 2𝛼𝑛 − 6𝛼)𝑥2 , 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝜕 𝜕 𝜕 − (4𝑛 + 12)𝑑𝑥2 + (8𝑛2 𝑣 + 32𝑛𝑣 + 𝛽𝑑𝑥4 + 24𝑣) 𝑋7 = (16𝑛𝑥 + 4𝑛2 𝑥 + 12𝑥) 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑑, 𝑑 a constant, 𝐺(𝑢) = 𝛼 + 𝛽𝑢−𝑝 , 𝛼, 𝑝, and 𝛽 constants (𝛽, 𝑝 ≠ 0) 𝜕 𝜕 𝑋1 = , 𝑋 = 𝑥(1−𝑛) , 𝜕𝑣 2 𝜕𝑣 𝜕 𝜕 𝜕 + (2𝑛𝑢 + 2𝑢) − (2𝑛𝑝𝑣 − 2𝑛𝑣 + 𝛼𝑝𝑥2 + 2𝑝𝑣 − 2𝑣) , 𝑋3 = (𝑛𝑥 + 𝑥) 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝜕 𝑋4 = (2𝑢 + 2𝑛𝑢 + 𝑑𝑥2 ) 𝜕𝑣 𝐻(𝑣) = 𝑑, 𝑑 a constant, 𝐺(𝑢) = 𝛼 + 𝛽𝑒−𝑘𝑢 , 𝛼, 𝑘, and 𝛽 constants (𝛽, 𝑘 ≠ 0) 𝜕 𝜕 𝜕 𝑋1 = , 𝑋 = 𝑥(1−𝑛) , 𝑋3 = (𝑑𝑥2 + 2𝑛𝑢 + 2𝑢) , 𝜕𝑣 2 𝜕𝑣 𝜕𝑣 𝜕 𝜕 − (𝛼𝑘𝑥2 + 2𝑘𝑛𝑣 + 2𝑘𝑣) 𝑋4 = (2𝑛 + 2) 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑑, 𝑑 a constant, 𝐺(𝑢) = 𝛼 + 𝛽 ln 𝑢, 𝛼 and 𝛽 constants (𝛽 ≠ 0) 𝜕 𝜕 𝜕 𝑋1 = , 𝑋 = 𝑥(1−𝑛) , 𝑋3 = (𝑑𝑥2 + 2𝑛𝑢 + 2𝑢) , 𝜕𝑣 2 𝜕𝑣 𝜕𝑣 𝜕 𝜕 𝜕 + (2𝑢 + 2𝑛𝑢) + (2𝑛𝑣 − 𝛽𝑥2 + 2𝑣) 𝑋4 = (𝑥𝑛 + 𝑥) 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑎 + 𝑏𝑣, 𝑎 and 𝑏 constants (𝑏 ≠ 0), 𝐺(𝑢) = 𝑐, 𝑐 constant 𝜕 𝜕 𝜕 𝑋1 = , 𝑋2 = 𝑥(1−𝑛) , 𝑋3 = (𝑐𝑥2 + 2𝑛𝑣 + 2𝑣) , 𝜕𝑢 𝜕𝑢 𝜕𝑢 𝜕 𝜕 + (2𝑛 − 6)𝑥(1−𝑛) , 𝑋4 = 𝑏𝑥(3−𝑛) 𝜕𝑢 𝜕𝑣 𝜕 𝜕 − (2𝑛 + 2) , 𝑋5 = 𝑏𝑥2 𝜕𝑢 𝜕𝑣 𝜕 𝜕 − (4𝑐𝑛𝑥2 + 24𝑣 + 8𝑛2 𝑣 + 32𝑛𝑣 + 12𝑥2 ) , 𝑋6 = (𝑏𝑐𝑥4 − 32𝑛𝑢 − 12𝑎𝑥2 − 24𝑢 − 8𝑛2 𝑢 − 4𝑎𝑛𝑥2 ) 𝜕𝑢 𝜕𝑣 𝜕 𝜕 𝜕 + (8𝑛2 𝑢 + 24𝑢 + 32𝑛𝑢 + 𝑏𝑐𝑥4 ) − (12𝑐𝑥2 + 4𝑐𝑛𝑥2 ) 𝑋7 = (16𝑛𝑥 + 4𝑛2 𝑥 + 12𝑥) 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑎 + 𝑏𝑣, 𝑎 and 𝑏 constants (𝑏 ≠ 0), 𝐺(𝑢) = 𝛼 + 𝛽𝑢, 𝛼 and 𝛽 constants (𝛽 ≠ 0) 𝜕 𝜕 𝜕 𝜕 𝑋1 = 𝐹(𝑥) , 𝑋2 = 𝑊(𝑥) , 𝑋3 = 𝑢 +𝑣 , 𝜕𝑣 𝜕𝑢 𝜕𝑢 𝜕𝑣 𝜕 𝜕 + 𝛽𝑢 𝑋4 = 𝑏𝑣 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑎 + 𝑏𝑣, 𝑎 and 𝑏 constants (𝑏 ≠ 0), 𝐺(𝑢) = 𝛽𝑢−𝑝 , 𝛽 and 𝑝 constants (𝛽, 𝑝 ≠ 0) 𝜕 𝜕 𝜕 𝑋1 = (𝑏𝑥 + 𝑏𝑝𝑥) + 4𝑏𝑢 + (2𝑎 + 2𝑏𝑣 − 2𝑎𝑝 − 2𝑏𝑝𝑣) 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑎 + 𝑏𝑣, 𝑎 and 𝑏 constants (𝑏 ≠ 0), 𝐺(𝑢) = 𝛽𝑒−𝑘𝑢 , 𝛽 and 𝑘 constants (𝛽, 𝑘 ≠ 0) 𝜕 𝜕 𝜕 𝑋1 = 𝑏𝑘𝑥 + 4𝑏 − (2𝑎𝑘 + 2𝑏𝑘𝑣) 𝜕𝑥 𝜕𝑢 𝜕𝑣 Journal of Applied Mathematics 5 Table 1: Continued. −𝑚 𝐻(𝑣) = 𝑎 + 𝑏𝑣 , 𝑎, 𝑚, and 𝑏 constants (𝑏, 𝑚 ≠ 0), 𝐺(𝑢) = 𝑐, 𝑐 a constant 𝜕 𝜕 𝜕 𝑋1 = , 𝑋 = 𝑥(1−𝑛) , 𝑋3 = (𝑐𝑥2 + 2𝑛𝑣 + 2𝑣) , 𝜕𝑢 2 𝜕𝑢 𝜕𝑢 𝜕 𝜕 𝜕 + (2𝑢 − 2𝑛𝑚𝑢 + 2𝑛𝑢 − 𝑎𝑚𝑥2 − 2𝑚𝑢) + (2𝑛𝑣 + 2𝑣) 𝑋4 = (𝑛𝑥 + 𝑥) 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑏𝑣−𝑚 , 𝑚 and 𝑏 constants (𝑏, 𝑚 ≠ 0), 𝐺(𝑢) = 𝛼 + 𝛽𝑢, 𝛼 and 𝛽 constants (𝛽 ≠ 0) 𝜕 𝜕 𝜕 𝑋1 = (𝛽𝑥 + 𝛽𝑚𝑥) + (2𝛼 − 2𝛼𝑚 + 2𝛽𝑢 − 2𝛽𝑚𝑢) + 4𝛽𝑣 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑏𝑣−𝑚 , 𝑚 and 𝑏 constants (𝑏, 𝑚 ≠ 0), 𝐺(𝑢) = 𝛽𝑢−𝑝 , 𝛽 and 𝑝 constants (𝛽, 𝑝 ≠ 0) 𝜕 𝜕 𝜕 𝑋1 = (𝑝𝑚𝑥 − 𝑥) + (2𝑚𝑢 − 2𝑢) + (2𝑝𝑣 − 2𝑣) 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑏𝑣−𝑚 , 𝑚 and 𝑏 constants (𝑏, 𝑚 ≠ 0), 𝐺(𝑢) = 𝛽𝑒−𝑘𝑢 , 𝛽 and 𝑘 constants (𝛽, 𝑘 ≠ 0) 𝜕 𝜕 𝜕 𝑋1 = 𝑘𝑚𝑥 + (2𝑚 − 2) + 2𝑘𝑣 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑎 + 𝑏𝑒−𝑚𝑣 , 𝑎, 𝑚 and 𝑏 constants (𝑏, 𝑚 ≠ 0), 𝐺(𝑢) = 𝑐, 𝑐 constant 𝜕 𝜕 𝜕 𝑋1 = , 𝑋 = 𝑥(1−𝑛) , 𝑋3 = (𝑐𝑥2 + 2𝑛𝑣 + 2𝑣) , 𝜕𝑢 2 𝜕𝑢 𝜕𝑢 𝜕 𝜕 − 2(𝑛 + 1) 𝑋4 = (𝑎𝑥2 + 2𝑢 + 2𝑛𝑢) 𝑚 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑏𝑒−𝑚𝑣 , 𝑚 and 𝑏 constants (𝑏, 𝑚 ≠ 0), 𝐺(𝑢) = 𝛼 + 𝛽𝑢 (𝛽 ≠ 0) 𝜕 𝜕 𝜕 𝑋1 = 𝛽𝑚𝑥 − (2𝛼𝑚 + 2𝛽𝑚𝑢) + 4𝛽 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑏𝑒−𝑚𝑣 , 𝑚 and 𝑏 constants (𝑏, 𝑚 ≠ 0), 𝐺(𝑢) = 𝛽𝑢−𝑝 , 𝛽 and 𝑝 constants (𝛽, 𝑝 ≠ 0) 𝜕 𝜕 𝜕 𝑋1 = 𝑚𝑝𝑥 + 2𝑚𝑢 + (2𝑝 − 2) 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑏𝑒−𝑚𝑣 , 𝑚 and 𝑏 constants (𝑏, 𝑚 ≠ 0), 𝐺(𝑢) = 𝛽𝑒−𝑘𝑢 , 𝛽 and 𝑘 constants (𝛽, 𝑘 ≠ 0) 𝜕 𝜕 𝜕 𝑋1 = 𝑚𝑘𝑥 + 2𝑚 + 2𝑘 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑎 + 𝑏 ln 𝑣, 𝑎, 𝑚 and 𝑏 constants (𝑏 ≠ 0), 𝐺(𝑢) = 𝑐, 𝑐 constant 𝜕 𝜕 𝜕 𝑋1 = , 𝑋 = 𝑥(1−𝑛) , 𝑋3 = (𝑐𝑥2 + 2𝑣 + 2𝑛𝑣) , 𝜕𝑢 2 𝜕𝑢 𝜕𝑢 𝜕 𝜕 𝜕 + (2𝑛𝑢 − 𝑏𝑥2 + 2𝑢) + (2𝑛𝑣 + 2𝑣) 𝑋4 = (𝑛 + 1)𝑥 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝑋5 : 𝑥 = 𝑥, 𝐻 = 𝐻, 𝑋6 : 𝑥 = 𝑥, 𝐻 = 𝐻, 𝑋7 : 𝑥 = 𝑥, 𝐻 = 𝐻, 𝑋8 : 𝑥 = 𝑥, 𝑢 = 𝑢, 𝐻 = 𝐻, and so the composition of these transformations gives 𝑥 = 𝑒𝑎1 𝑥, 𝐺 = 𝐺, 𝑢 = 𝑢 + 𝑎6 ln 𝑥, 𝑢 = 𝑒𝑎2 (𝑢 + 𝑎6 ln 𝑥 + 𝑎8 𝑥2 + 𝑎4 ) , 𝑣 = 𝑣, 𝑣 = 𝑒𝑎3 (𝑣 + 𝑎7 ln 𝑥 + 𝑎9 𝑥2 + 𝑎5 ) , 𝐺 = 𝐺, 𝑢 = 𝑢, 𝐺 = 𝑒𝑎3 −2𝑎1 (𝐺 − 4𝑎9 ) . 𝑢 = 𝑢 + 𝑎8 𝑥2 , 𝑢 = 𝑢, (13) 𝐻 = 𝑒𝑎2 −2𝑎1 (𝐻 − 4𝑎8 ) , 𝑣 = 𝑣 + 𝑎7 ln 𝑥, 𝐺 = 𝐺, 𝐻 = 𝐻 − 4𝑎8 , 𝑋9 : 𝑥 = 𝑥, 𝑣 = 𝑣 + 𝑎5 , Case 4 (𝑛 = 3). In this case system (3) has the ten-dimensional equivalence Lie algebra spanned by the equivalence generators 𝑣 = 𝑣, 𝐺 = 𝐺, 𝑣 = 𝑣 + 𝑎9 𝑥2 , 𝑋1 = 𝑥 𝐺 = 𝐺 − 4𝑎9 (12) 𝜕 𝜕 𝜕 − 2𝐻 − 2𝐺 , 𝜕𝑥 𝜕𝐻 𝜕𝐺 𝑋2 = 𝑥−1 𝜕 𝜕 𝜕 − 2𝑥−2 𝑢 − 2𝑥−2 𝑣 , 𝜕𝑥 𝜕𝑢 𝜕𝑣 6 Journal of Applied Mathematics Table 2: Lie symmetries for 𝑛 = −1, for various functions 𝐻(𝑣) and 𝐺(𝑢). 𝐻(𝑣) arbitrary, 𝐺(𝑢) = 𝑐, 𝑐 a constant 𝜕 𝜕 𝜕 𝑋1 = , 𝑋 = 𝑥2 , 𝑋3 = (𝑐𝑥2 ln 𝑥 + 2𝑣) 𝜕𝑢 2 𝜕𝑢 𝜕𝑢 𝐻(𝑣) = 𝑑, 𝑑 a constant, 𝐺(𝑢) arbitrary 𝜕 𝜕 𝜕 𝑋1 = , 𝑋 = 𝑥2 , 𝑋3 = (𝑑𝑥2 ln 𝑥 + 2𝑢) 𝜕𝑣 2 𝜕𝑣 𝜕𝑣 𝐻(𝑣) = 𝑑, 𝑑 a constant, 𝐺(𝑢) = 𝛼 + 𝛽𝑢, 𝛼 and 𝛽 constants (𝛽 ≠ 0) 𝜕 𝜕 𝜕 𝜕 𝜕 𝑋1 = , 𝑋 = 𝑥2 , 𝑋3 = (𝑑𝑥2 ln 𝑥 + 2𝑢) , 𝑋4 = 𝛽𝑥4 − 8𝑥2 , 𝜕𝑣 2 𝜕𝑣 𝜕𝑣 𝜕𝑣 𝜕𝑢 𝜕 𝜕 𝑋5 = 𝛽𝑥2 ln 𝑥 − 2 , 𝜕𝑣 𝜕𝑢 𝜕 𝜕 𝑋6 = (4𝑑𝛽𝑥4 ln 𝑥 − 32𝛼𝑥2 ln 𝑥 − 3𝛽𝑑𝑥4 − 64𝑣) − (32𝑥2 ln 𝑥 + 64𝑢) , 𝜕𝑣 𝜕𝑢 𝜕 𝜕 𝜕 + (4𝑑𝛽𝑥4 ln 𝑥 − 16𝛼𝑥2 ln 𝑥 − 3 𝛽𝑑𝑥4 ) − (32 𝑑𝑥2 ln 𝑥 + 32𝑢) 𝑋7 = 16𝑥 𝜕𝑥 𝜕𝑣 𝜕𝑢 𝐻(𝑣) = 𝑑, 𝑑 a constant, 𝐺(𝑢) = 𝛼 + 𝛽𝑢−𝑝 , 𝛼, 𝑝, and 𝛽 constants (𝛽, 𝑝 ≠ 0) 𝜕 𝜕 𝜕 𝑋1 = , 𝑋 = 𝑥2 , 𝑋3 = (𝑑𝑥2 ln 𝑥 + 2𝑢) , 𝜕𝑣 2 𝜕𝑣 𝜕𝑣 𝜕 𝜕 𝜕 + 2𝑢 + (2𝑣 − 2𝑝𝑣 − 𝛼𝑝𝑥2 ln 𝑥) 𝑋4 = 𝑥 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑑, 𝑑 a constant, 𝐺(𝑢) = 𝛼 + 𝛽𝑒−𝑘𝑢 , 𝛼, 𝑘, and 𝛽 constants (𝛽, 𝑘 ≠ 0) 𝜕 𝜕 𝜕 𝑋1 = , 𝑋 = 𝑥2 , 𝑋3 = (𝑑𝑥2 ln 𝑥 + 2𝑢) , 𝜕𝑣 2 𝜕𝑣 𝜕𝑣 𝜕 𝜕 − (𝛼𝑘𝑥2 ln 𝑥 + 2𝑘𝑣) 𝑋4 = 2 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑑, 𝑑 a constant, 𝐺(𝑢) = 𝛼 + 𝛽 ln 𝑢, 𝛼 and 𝛽 constants (𝛽 ≠ 0) 𝜕 𝜕 𝜕 𝑋1 = , 𝑋 = 𝑥2 , 𝑋3 = (𝑑𝑥2 ln 𝑥 + 2𝑢) , 𝜕𝑣 2 𝜕𝑣 𝜕𝑣 𝜕 𝜕 𝜕 + 2𝑢 + (2𝑣 − 𝛽𝑥2 ln 𝑥) 𝑋4 = 𝑥 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑎 + 𝑏𝑣, 𝑎 and 𝑏 constants (𝑏 ≠ 0), 𝐺(𝑢) = 𝑐, 𝑐 constant 𝜕 𝜕 𝜕 𝜕 𝜕 𝑋1 = , 𝑋 = 𝑥2 , 𝑋3 = (𝑐𝑥2 ln 𝑥 + 2𝑣) , 𝑋4 = 𝑐𝑥2 ln 𝑥 −2 , 𝜕𝑢 2 𝜕𝑢 𝜕𝑢 𝜕𝑢 𝜕𝑣 𝜕 𝜕 − 8𝑥2 , 𝑋5 = 𝑏𝑥4 ln 𝑥 𝜕𝑢 𝜕𝑣 𝜕 𝜕 − (32𝑐𝑥2 ln 𝑥 + 64𝑣) , 𝑋6 = (4𝑏𝑐𝑥4 ln 𝑥 − 32𝑎𝑥2 ln 𝑥 − 3𝑏𝑐𝑥4 − 64𝑢) 𝜕𝑢 𝜕𝑣 𝜕 𝜕 𝜕 + (4𝑐𝑑𝑥4 ln 𝑥 − 16𝑎𝑥2 ln 𝑥 − 3𝑏𝑐𝑥4 ) − (32𝑐𝑥2 ln 𝑥 + 32𝑣) 𝑋7 = 16𝑥 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑎 + 𝑏𝑣, 𝑎 and 𝑏 constants (𝑏 ≠ 0), 𝐺(𝑢) = 𝛼 + 𝛽𝑢, 𝛼 and 𝛽 constants (𝛽 ≠ 0) 𝜕 𝜕 𝜕 𝜕 𝑋1 = 𝐹(𝑥) , 𝑋2 = 𝑊(𝑥) , 𝑋3 = 𝑢 +𝑣 , 𝜕𝑣 𝜕𝑢 𝜕𝑢 𝜕𝑣 𝜕 𝜕 + 𝛽𝑢 𝑋4 = 𝑏𝑣 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑎 + 𝑏𝑣, 𝑎 and 𝑏 constants (𝑏 ≠ 0), 𝐺(𝑢) = 𝛽𝑢−𝑝 , 𝛽 and 𝑝 constants (𝛽, 𝑝 ≠ 0) 𝜕 𝜕 𝜕 𝑋1 = (𝑏𝑥 + 𝑏𝑝𝑥) + 4𝑏𝑢 + (2𝑎 + 2𝑏𝑣 − 2𝑎𝑝 − 2𝑏𝑝𝑣) 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑎 + 𝑏𝑣, 𝑎 and 𝑏 constants (𝑏 ≠ 0), 𝐺(𝑢) = 𝛽𝑒−𝑘𝑢 , 𝛽 and 𝑘 constants (𝛽, 𝑘 ≠ 0) 𝜕 𝜕 𝜕 𝑋1 = 𝑏𝑘𝑥 + 4𝑏 − (2𝑎𝑘 + 2𝑏𝑘𝑣) 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑎 + 𝑏𝑣−𝑚 , 𝑎, 𝑚, and 𝑏 constants (𝑏, 𝑚 ≠ 0), 𝐺(𝑢) = 𝑐, 𝑐 a constant 𝜕 𝜕 𝜕 𝑋1 = , 𝑋 = 𝑥2 , 𝑋3 = (𝑐𝑥2 ln 𝑥 + 2𝑣) , 𝜕𝑢 2 𝜕𝑢 𝜕𝑢 𝜕 𝜕 𝜕 + (2𝑢 − 𝑎𝑚𝑥2 ln 𝑥 − 2𝑚𝑢) + 2𝑣 𝑋4 = 𝑥 𝜕𝑥 𝜕𝑢 𝜕𝑣 Journal of Applied Mathematics 7 Table 2: Continued. −𝑚 𝐻(𝑣) = 𝑏𝑣 , 𝑚 and 𝑏 constants (𝑏, 𝑚 ≠ 0), 𝐺(𝑢) = 𝛼 + 𝛽𝑢, 𝛼 and 𝛽 constants (𝛽 ≠ 0) 𝜕 𝜕 𝜕 𝑋1 = (𝛽𝑥 + 𝛽𝑚𝑥) + (2𝛼 − 2𝛼𝑚 + 2𝛽𝑢 − 2𝛽𝑚𝑢) + 4𝛽𝑣 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑏𝑣−𝑚 , 𝑚 and 𝑏 constants (𝑏, 𝑚 ≠ 0), 𝐺(𝑢) = 𝛽𝑢−𝑝 , 𝑝 and 𝛽 constants (𝛽, 𝑝 ≠ 0) 𝜕 𝜕 𝜕 𝑋1 = (𝑝𝑚𝑥 − 𝑥) + (2𝑚𝑢 − 2𝑢) + (2𝑝𝑣 − 2𝑣) 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑏𝑣−𝑚 , 𝑚 and 𝑏 constants (𝑏, 𝑚 ≠ 0), 𝐺(𝑢) = 𝛽𝑒−𝑘𝑢 , 𝑘 and 𝛽 constants (𝛽, 𝑘 ≠ 0) 𝜕 𝜕 𝜕 𝑋1 = 𝑘𝑚𝑥 + (2𝑚 − 2) + 2𝑘𝑣 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑎 + 𝑏𝑒−𝑚𝑣 , 𝑎, 𝑚, and 𝑏 constants (𝑏, 𝑚 ≠ 0), 𝐺(𝑢) = 𝑐, 𝑐 constant 𝜕 𝜕 𝜕 𝜕 𝜕 𝑋1 = , 𝑋 = 𝑥2 , 𝑋3 = (𝑐𝑥2 ln 𝑥 + 2𝑣) , 𝑋4 = (𝛼𝑚𝑥2 ln 𝑥 + 2𝑚𝑢) −2 𝜕𝑢 2 𝜕𝑢 𝜕𝑢 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑏𝑒−𝑚𝑣 , 𝑚 and 𝑏 constants (𝑏, 𝑚 ≠ 0), 𝐺(𝑢) = 𝛼 + 𝛽𝑢, 𝛼 and 𝛽 constants (𝛽 ≠ 0) 𝜕 𝜕 𝜕 𝑋1 = 𝛽𝑚𝑥 − (2𝛼𝑚 + 2𝛽𝑚𝑢) + 4𝛽 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑏𝑒−𝑚𝑣 , 𝑚 and 𝑏 constants (𝑏, 𝑚 ≠ 0), 𝐺(𝑢) = 𝛽𝑢−𝑝 , 𝛽 and 𝑝 constants (𝛽, 𝑝 ≠ 0) 𝜕 𝜕 𝜕 𝑋1 = 𝑚𝑝𝑥 + 2𝑚𝑢 + (2𝑝 − 2) 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑏𝑒−𝑚𝑣 , 𝑚 and 𝑏 constants (𝑏, 𝑚 ≠ 0), 𝐺(𝑢) = 𝛽𝑒−𝑘𝑢 , 𝛽 and 𝑘 constants (𝛽, 𝑘 ≠ 0) 𝜕 𝜕 𝜕 𝑋1 = 𝑚𝑘𝑥 + 2𝑚 + 2𝑘 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑎 + 𝑏 ln 𝑣, 𝑎, 𝑚, and 𝑏 constants (𝑏 ≠ 0), 𝐺(𝑢) = 𝑐, 𝑐 constant 𝜕 𝜕 𝜕 𝜕 𝜕 𝜕 𝑋1 = , 𝑋 = 𝑥2 , 𝑋3 = (𝑐𝑥2 ln 𝑥 + 2𝑣) , 𝑋4 = 𝑥 + (2𝑢 − 𝑏𝑥2 ln 𝑥) + 2𝑣 𝜕𝑢 2 𝜕𝑢 𝜕𝑢 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝑋3 = 𝑢 𝜕 𝜕 +𝐻 , 𝜕𝑢 𝜕𝐻 𝑋3 : 𝑥 = 𝑥, 𝐻 = 𝑒𝑎3 𝐻, 𝜕 𝜕 𝑋4 = 𝑣 + 𝐺 , 𝜕𝑣 𝜕𝐺 𝑋4 : 𝑥 = 𝑥, 𝜕 , 𝜕𝑢 𝜕 , 𝑋6 = 𝜕𝑣 𝑋5 = 𝐻 = 𝐻, 𝑋5 : 𝑥 = 𝑥, 𝑋7 = 𝑥2 𝜕 𝜕 −8 , 𝜕𝑢 𝜕𝐻 𝑋8 = 𝑥2 𝜕 𝜕 −8 , 𝜕𝑣 𝜕𝐺 𝑋9 = 𝑥−2 𝜕 , 𝜕𝑢 𝑋10 = 𝑥−2 𝜕 𝜕𝑣 𝑢 = 𝑒𝑎3 𝑢, 𝐻 = 𝐻, 𝑋6 : 𝑥 = 𝑥, 𝐻 = 𝐻, 𝑋7 : 𝑥 = 𝑥, 𝐺 = 𝐺, (14) and hence the ten-parameter equivalence group is given by 𝑋1 : 𝑥 = 𝑒𝑎1 𝑥, 𝑢 = 𝑢, 𝑣 = 𝑣, 𝐻 = 𝑒−2𝑎1 𝐻, 𝐺 = 𝑒−2𝑎1 𝐺, 1/2 𝑋2 : 𝑥 = (𝑥2 + 2𝑎2 ) −1 −1 𝑣 = 𝑣𝑥2 (𝑥2 + 2𝑎2 ) , 𝐻 = 𝐻, 𝑋9 : 𝑥 = 𝑥, 𝐻 = 𝐻, 𝑢 = 𝑢𝑥2 (𝑥2 + 2𝑎2 ) , , 𝐻 = 𝐻, 𝐺 = 𝐺, 𝑋10 : 𝑥 = 𝑥, 𝐻 = 𝐻, 𝑣 = 𝑒𝑎4 𝑣, 𝑢 = 𝑢, 𝐺 = 𝑒𝑎4 𝐺, 𝑢 = 𝑢 + 𝑎5 , 𝑣 = 𝑣, 𝐺 = 𝐺, 𝑢 = 𝑢, 𝑣 = 𝑣 + 𝑎6 , 𝐺 = 𝐺, 𝑢 = 𝑢 + 𝑎7 𝑥2 , 𝐻 = 𝐻 − 8𝑎7 , 𝑋8 : 𝑥 = 𝑥, 𝑣 = 𝑣, 𝑣 = 𝑣, 𝐺 = 𝐺, 𝑢 = 𝑢, 𝑣 = 𝑣 + 𝑎8 𝑥2 , 𝐺 = 𝐺 − 8𝑎8 , 𝑢 = 𝑢 + 𝑎9 𝑥−2 , 𝑣 = 𝑣, 𝐺 = 𝐺, 𝑢 = 𝑢, 𝑣 = 𝑣 + 𝑎10 𝑥−2 , 𝐺 = 𝐺. (15) 8 Journal of Applied Mathematics Table 3: Lie symmetries for 𝑛 = 1, for various functions 𝐻(𝑣) and 𝐺(𝑢). 𝐻(𝑣) arbitrary, 𝐺(𝑢) = 𝑐, 𝑐 a constant 𝜕 𝜕 𝜕 𝑋1 = , 𝑋 = ln 𝑥 , 𝑋3 = (𝑐𝑥2 + 4𝑣) 𝜕𝑢 2 𝜕𝑢 𝜕𝑢 𝐻(𝑣) = 𝑑, 𝑑 a constant, 𝐺(𝑢) arbitrary 𝜕 𝜕 𝜕 𝑋1 = , 𝑋 = ln 𝑥 , 𝑋3 = (𝑑𝑥2 ln 𝑥 + 4𝑢) 𝜕𝑣 2 𝜕𝑣 𝜕𝑣 𝐻(𝑣) = 𝑑, 𝑑 a constant, 𝐺(𝑢) = 𝛼 + 𝛽𝑢, 𝛼 and 𝛽 constants (𝛽 ≠ 0) 𝜕 𝜕 𝜕 𝜕 𝜕 𝑋1 = , 𝑋 = ln 𝑥 , 𝑋3 = (𝑑𝑥2 ln 𝑥 + 4𝑢) , 𝑋4 = 𝛽𝑥2 −4 , 𝜕𝑣 2 𝜕𝑣 𝜕𝑣 𝜕𝑣 𝜕𝑢 𝜕 𝜕 𝑋5 = (𝛽𝑥2 ln 𝑥 − 𝛽𝑥2 ) − 4 ln 𝑥 , 𝜕𝑣 𝜕𝑢 𝜕 𝜕 + (16𝛼𝑥2 + 64𝑣 − 𝛽𝑑𝑥4 ) , 𝑋6 = (16𝑑𝑥2 + 64𝑢) 𝜕𝑢 𝜕𝑣 𝜕 𝜕 𝜕 − (8𝛼𝑥2 − 𝛽𝑑𝑥4 ) − (16𝑑𝑥2 + 32𝑢) 𝑋7 = 16𝑥 𝜕𝑥 𝜕𝑣 𝜕𝑢 𝐻(𝑣) = 𝑑, 𝑑 a constant, 𝐺(𝑢) = 𝛼 + 𝛽𝑢−𝑝 , 𝛼, 𝑝, and 𝛽 constants (𝛽, 𝑝 ≠ 0) 𝜕 𝜕 𝜕 𝑋1 = , 𝑋 = ln 𝑥 , 𝑋3 = (𝑑𝑥2 ln 𝑥 + 4𝑢) , 𝜕𝑣 2 𝜕𝑣 𝜕𝑣 𝜕 𝜕 𝜕 𝑋4 = 2𝑥 + 4𝑢 − (4𝑣𝑝 + 𝛼𝑝𝑥2 − 4𝑣) 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑑, 𝑑 a constant, 𝐺(𝑢) = 𝛼 + 𝛽𝑒−𝑘𝑢 , 𝛼, 𝑘, and 𝛽 constants (𝛽, 𝑘 ≠ 0) 𝜕 𝜕 𝜕 𝑋1 = , 𝑋 = ln 𝑥 , 𝑋3 = (𝑑𝑥2 + 4𝑢) , 𝜕𝑣 2 𝜕𝑣 𝜕𝑣 𝜕 𝜕 − (𝛼𝑘𝑥2 + 4𝑘𝑣) 𝑋4 = 4 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑑, 𝑑 a constant, 𝐺(𝑢) = 𝛼 + 𝛽 ln 𝑢, 𝛼 and 𝛽 constants (𝛽 ≠ 0) 𝜕 𝜕 𝜕 𝑋1 = , 𝑋 = ln 𝑥 , 𝑋3 = (𝑑𝑥2 + 4𝑢) , 𝜕𝑣 2 𝜕𝑣 𝜕𝑣 𝜕 𝜕 𝜕 + 4𝑢 + (4𝑣 − 𝛽𝑥2 ) 𝑋4 = 2𝑥 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑎 + 𝑏𝑣, 𝑎 and 𝑏 constants (𝑏 ≠ 0), 𝐺(𝑢) = 𝑐, 𝑐 constant 𝜕 𝜕 𝜕 𝜕 𝜕 𝑋1 = , 𝑋 = ln 𝑥 , 𝑋3 = (𝑐𝑥2 + 4𝑣) , 𝑋4 = 4 ln 𝑥 − (𝑏𝑥2 ln 𝑥 − 𝑏𝑥2 ) , 𝜕𝑢 2 𝜕𝑢 𝜕𝑢 𝜕𝑣 𝜕𝑢 𝜕 𝜕 −4 , 𝑋5 = 𝑏𝑥2 𝜕𝑢 𝜕𝑣 𝜕 𝜕 𝜕 𝑋6 = 32𝑥 − 16𝑐𝑥2 + (𝑏𝑐𝑥4 + 64𝑢) , 𝜕𝑥 𝜕𝑣 𝜕𝑢 𝜕 𝜕 𝜕 𝑋7 = 16𝑥 − (8𝑎𝑥2 − 𝑏𝑐𝑥4 ) − (16𝑐𝑥2 + 32𝑣) 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑎 + 𝑏𝑣, 𝑎 and 𝑏 constants (𝑏 ≠ 0), 𝐺(𝑢) = 𝛼 + 𝛽𝑢, 𝛼 and 𝛽 constants (𝛽 ≠ 0) 𝜕 𝜕 𝜕 𝜕 𝑋1 = 𝐹(𝑥) , 𝑋2 = 𝑊(𝑥) , 𝑋3 = 𝑢 +𝑣 , 𝜕𝑣 𝜕𝑢 𝜕𝑢 𝜕𝑣 𝜕 𝜕 + 𝛽𝑢 𝑋4 = 𝑏𝑣 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑎 + 𝑏𝑣, 𝑎 and 𝑏 constants (𝑏 ≠ 0), 𝐺(𝑢) = 𝛽𝑢−𝑝 , 𝛽 and 𝑝 constants (𝛽, 𝑝 ≠ 0) 𝜕 𝜕 𝜕 𝑋1 = (𝑏𝑥 + 𝑏𝑝𝑥) + 4𝑏𝑢 + (2𝑎 + 2𝑏𝑣 − 2𝑎𝑝 − 2𝑏𝑝𝑣) 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑎 + 𝑏𝑣, 𝑎 and 𝑏 constants (𝑏 ≠ 0), 𝐺(𝑢) = 𝛽𝑒−𝑘𝑢 , 𝛽 and 𝑘 constants (𝛽, 𝑘 ≠ 0) 𝜕 𝜕 𝜕 𝑋1 = 𝑏𝑘𝑥 + 4𝑏 − (2𝑎𝑘 + 2𝑏𝑘𝑣) 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑎 + 𝑏𝑣−𝑚 , 𝑎, 𝑚, and 𝑏 constants (𝑏, 𝑚 ≠ 0), 𝐺(𝑢) = 𝑐, 𝑐 a constant 𝜕 𝜕 𝜕 𝑋1 = , 𝑋2 = ln 𝑥 , 𝑋3 = (𝑐𝑥2 + 4𝑣) , 𝜕𝑢 𝜕𝑢 𝜕𝑢 𝜕 𝜕 𝜕 + (4𝑢 − 𝑎𝑚𝑥2 − 4𝑚𝑢) + 4𝑣 𝑋4 = 2𝑥 𝜕𝑥 𝜕𝑢 𝜕𝑣 Journal of Applied Mathematics 9 Table 3: Continued. −𝑚 𝐻(𝑣) = 𝑏𝑣 , 𝑚 and 𝑏 constants (𝑏, 𝑚 ≠ 0), 𝐺(𝑢) = 𝛼 + 𝛽𝑢, 𝛼 and 𝛽 constants (𝛽 ≠ 0) 𝜕 𝜕 𝜕 𝑋1 = (𝛽𝑥 + 𝛽𝑚𝑥) + (2𝛼 − 2𝛼𝑚 + 2𝛽𝑢 − 2𝛽𝑚𝑢) + 4𝛽𝑣 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑏𝑣−𝑚 , 𝑚 and 𝑏 constants (𝑏, 𝑚 ≠ 0), 𝐺(𝑢) = 𝛽𝑢−𝑝 , 𝑝 and 𝛽 constants (𝛽, 𝑝 ≠ 0) 𝜕 𝜕 𝜕 𝑋1 = (𝑝𝑚𝑥 − 𝑥) + (2𝑚𝑢 − 2𝑢) + (2𝑝𝑣 − 2𝑣) 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑏𝑣−𝑚 , 𝑚 and 𝑏 constants (𝑏, 𝑚 ≠ 0), 𝐺(𝑢) = 𝛽𝑒−𝑘𝑢 , 𝑘 and 𝛽 constants (𝛽, 𝑘 ≠ 0) 𝜕 𝜕 𝜕 𝑋1 = 𝑘𝑚𝑥 + (2𝑚 − 2) + 2𝑘𝑣 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑎 + 𝑏𝑒−𝑚𝑣 , 𝑎, 𝑚, and 𝑏 constants (𝑏, 𝑚 ≠ 0), 𝐺(𝑢) = 𝑐, 𝑐 constant 𝜕 𝜕 𝜕 𝜕 𝜕 𝑋1 = , 𝑋 = ln 𝑥 , 𝑋3 = (𝑐𝑥2 + 4𝑣) , 𝑋4 = (𝑎𝑚𝑥2 + 4𝑚𝑢) −4 𝜕𝑢 2 𝜕𝑢 𝜕𝑢 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑏𝑒−𝑚𝑣 , 𝑚 and 𝑏 constants (𝑏, 𝑚 ≠ 0), 𝐺(𝑢) = 𝛼 + 𝛽𝑢, 𝛼 and 𝛽 constants (𝛽 ≠ 0) 𝜕 𝜕 𝜕 𝑋1 = 𝛽𝑚𝑥 − (2𝛼𝑚 + 2𝛽𝑚𝑢) + 4𝛽 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑏𝑒−𝑚𝑣 , 𝑚 and 𝑏 constants (𝑏, 𝑚 ≠ 0), 𝐺(𝑢) = 𝛽𝑢−𝑝 , 𝛽 and 𝑝 constants (𝛽, 𝑝 ≠ 0) 𝜕 𝜕 𝜕 𝑋1 = 𝑚𝑝𝑥 + 2𝑚𝑢 + (2𝑝 − 2) 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑏𝑒−𝑚𝑣 , 𝑚 and 𝑏 constants (𝑏, 𝑚 ≠ 0), 𝐺(𝑢) = 𝛽𝑒−𝑘𝑢 , 𝛽 and 𝑘 constants (𝛽, 𝑘 ≠ 0) 𝜕 𝜕 𝜕 𝜕 𝜕 𝜕 𝑋1 = 𝑚𝑘𝑥 + 2𝑚 + 2𝑘 , 𝑋2 = (𝑚𝑘𝑥 ln 𝑥 − 𝑘𝑚𝑥) + 2𝑚 ln 𝑥 + 2𝑘 ln 𝑥 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝐻 (𝑣) = 𝑎 + 𝑏 ln 𝑣, 𝑎, 𝑚, and 𝑏 constants (𝑏 ≠ 0), 𝐺(𝑢) = 𝑐, 𝑐 constant 𝜕 𝜕 𝜕 𝜕 𝜕 𝜕 𝑋1 = , 𝑋 = ln 𝑥 , 𝑋3 = (𝑐𝑥2 + 4𝑣) , 𝑋4 = 2𝑥 + (4𝑢 − 𝑏𝑥2 ) + 4𝑣 𝜕𝑢 2 𝜕𝑢 𝜕𝑢 𝜕𝑥 𝜕𝑢 𝜕𝑣 Therefore the composition of these transformations gives 𝑥 = 𝑒𝑎1 (𝑥2 + 2𝑎2 ) 1/2 After some albeit tedious and lengthy calculations, the above determining equation gives , 𝜉 = 𝑒 (𝑥) , −1 𝑢 = 𝑒𝑎3 (𝑥2 + 2𝑎2 ) (𝑥2 𝑢 + 𝑎9 + 𝑎7 𝑥4 + 𝑎5 𝑥2 ) , 𝑎4 −1 2 2 4 2 𝑣 = 𝑒 (𝑥 + 2𝑎2 ) (𝑥 𝑣 + 𝑎10 + 𝑎8 𝑥 + 𝑎6 𝑥 ) , 𝑎3 −2𝑎1 𝐻=𝑒 (16) (𝐻 − 8𝑎7 ) , 3. Principal Lie Algebra and Lie Group Classification The generalized Lane-Emden system (2) admits a Lie point symmetry 𝜕 𝜕 𝜕 + 𝜂1 (𝑥, 𝑢, 𝑣) + 𝜂2 (𝑥, 𝑢, 𝑣) 𝜕𝑥 𝜕𝑢 𝜕𝑣 (17) if 𝑋[2] ( 𝑋[2] ( 󵄨󵄨 𝑑2 𝑢 𝑛 𝑑𝑢 󵄨󵄨 󵄨󵄨 = 0, + + 𝐻 (𝑣)) 󵄨󵄨 𝑑𝑥2 𝑥 𝑑𝑥 󵄨(2) 󵄨󵄨 𝑑2 𝑣 𝑛 𝑑𝑣 󵄨󵄨 󵄨󵄨 = 0. + + 𝐺 (𝑢)) 󵄨󵄨 𝑑𝑥2 𝑥 𝑑𝑥 󵄨(2) 𝜂2 = 𝑐 (𝑥) 𝑣 + 𝑑 (𝑥) , 𝑛 󸀠 𝑛 𝑒 − 2 𝑒 + 2𝑘󸀠 = 0, 𝑥 𝑥 𝑛 𝑛 −𝑒󸀠󸀠 + 𝑒󸀠 − 2 𝑒 + 2𝑐󸀠 = 0, 𝑥 𝑥 𝑛 󸀠 󸀠 (𝑘𝑢 + 𝑙) 𝐺 (𝑢) + (2𝑒 − 𝑐) 𝐺 (𝑢) + 𝑑󸀠 + 𝑑󸀠󸀠 = 0, 𝑥 𝑛 (𝑐𝑣 + 𝑑) 𝐻󸀠 (𝑣) + (2𝑒󸀠 − 𝑘) 𝐻 (𝑣) + 𝑙󸀠 + 𝑙󸀠󸀠 = 0. 𝑥 −𝑒󸀠󸀠 + 𝐺 = 𝑒𝑎4 −2𝑎1 (𝐺 − 8𝑎8 ) . 𝑋 = 𝜉 (𝑥, 𝑢, 𝑣) 𝜂1 = 𝑘 (𝑥) 𝑢 + 𝑙 (𝑥) , Consequently, we conclude that the principal Lie algebra of (2) is trivial and the classifying relations are (𝛼𝑢 + 𝛽) 𝐺󸀠 (𝑢) + 𝛾𝐺 (𝑢) + 𝛿 = 0, (18) (19) (𝜃𝑣 + 𝜆) 𝐻󸀠 (𝑣) + 𝜑𝐻 (𝑣) + 𝜔 = 0, where 𝛼, 𝛽, 𝛾, 𝛿, 𝜃, 𝜆, 𝜓, and 𝜔 are constants. (20) 10 Journal of Applied Mathematics Table 4: Lie symmetries for 𝑛 = 3, for various functions 𝐻(𝑣) and 𝐺(𝑢). 𝐻(𝑣) arbitrary, 𝐺(𝑢) = 𝑐, 𝑐 a constant 𝜕 𝜕 𝜕 𝑋1 = , 𝑋 = 𝑥−2 , 𝑋3 = (𝑐𝑥2 + 8𝑣) 𝜕𝑢 2 𝜕𝑢 𝜕𝑢 𝐻(𝑣) = 𝑑, 𝑑 a constant, 𝐺(𝑢) arbitrary 𝜕 𝜕 𝜕 𝑋1 = , 𝑋 = 𝑥−2 , 𝑋3 = (𝑑𝑥2 + 8𝑢) 𝜕𝑣 2 𝜕𝑣 𝜕𝑣 𝐻(𝑣) = 𝑑, 𝑑 a constant, 𝐺(𝑢) = 𝛼 + 𝛽𝑢, 𝛼 and 𝛽 constants (𝛽 ≠ 0) 𝜕 𝜕 𝜕 𝜕 𝜕 𝑋1 = , 𝑋 = 𝑥−2 , 𝑋3 = (𝑑𝑥2 + 8𝑢) , 𝑋4 = 𝛽 ln 𝑥 − 2𝑥−2 , 𝜕𝑣 2 𝜕𝑣 𝜕𝑣 𝜕𝑣 𝜕𝑢 𝜕 𝜕 −8 , 𝑋5 = 𝛽𝑥2 𝜕𝑣 𝜕𝑢 𝜕 𝜕 𝜕 − 24𝑑𝑥2 + (𝛽𝑑𝑥4 + 192𝑣) , 𝑋6 = 96𝑥 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝜕 𝜕 𝑋7 = (192𝑢 + 24𝑑𝑥4 ) + (192𝑣 − 𝛽𝑑𝑥2 + 24𝛼𝑥2 ) 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑑, 𝑑 a constant, 𝐺(𝑢) = 𝛼 + 𝛽𝑢−𝑝 , 𝛼, 𝑝, and 𝛽 constants (𝛽, 𝑝 ≠ 0) 𝜕 𝜕 𝜕 𝑋1 = , 𝑋 = 𝑥−2 , 𝑋3 = (𝑑𝑥2 + 8𝑢) , 𝜕𝑣 2 𝜕𝑣 𝜕𝑣 𝜕 𝜕 𝜕 + 8𝑢 − (𝛼𝑝𝑥2 − 8𝑣 + 8𝑝𝑣) 𝑋4 = 4𝑥 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑑, 𝑑 a constant, 𝐺(𝑢) = 𝛼 + 𝛽𝑒−𝑘𝑢 , 𝛼, 𝑘, and 𝛽 constants (𝛽, 𝑘 ≠ 0) 𝜕 𝜕 𝜕 𝑋1 = , 𝑋 = 𝑥−2 , 𝑋3 = (𝑑𝑥2 + 8𝑢) , 𝜕𝑣 2 𝜕𝑣 𝜕𝑣 𝜕 𝜕 𝑋4 = 8 − (𝛼𝑘𝑥2 + 8𝑘𝑣) 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑑, 𝑑 a constant, 𝐺(𝑢) = 𝛼 + 𝛽 ln 𝑢, 𝛼 and 𝛽 constants (𝛽 ≠ 0) 𝜕 𝜕 𝜕 𝑋1 = , 𝑋 = 𝑥−2 , 𝑋3 = (𝑑𝑥2 + 8𝑢) , 𝜕𝑣 2 𝜕𝑣 𝜕𝑣 𝜕 𝜕 𝜕 𝑋4 = 4𝑥 + 8𝑢 + (8𝑣 − 𝛽𝑥2 ) , 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝜕 𝜕 𝜕 𝑋5 = 𝑥−1 − 2𝑥−2 𝑢 − (2𝑥−2 𝑣 − 𝛽 ln 𝑥) 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑎 + 𝑏𝑣, 𝑎 and 𝑏 constants (𝑏 ≠ 0), 𝐺(𝑢) = 𝑐, 𝑐 constant 𝜕 𝜕 𝜕 𝜕 𝜕 𝑋1 = , 𝑋 = 𝑥−2 , 𝑋3 = (𝑐𝑥2 + 8𝑣) , 𝑋4 = 𝑏 ln 𝑥 − 2𝑥−2 , 𝜕𝑢 2 𝜕𝑢 𝜕𝑢 𝜕𝑢 𝜕𝑣 𝜕 𝜕 −8 , 𝑋5 = 𝑏𝑥2 𝜕𝑢 𝜕𝑣 𝜕 𝜕 𝜕 𝑋6 = 96𝑥 + (𝑏𝑐𝑥4 + 192𝑢) − 24𝑐𝑥2 , 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝜕 𝜕 𝑋7 = (24𝑎𝑥2 + 192𝑢 − 𝑏𝑐𝑥4 ) + (24𝑐𝑥2 + 192𝑣) 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑎 + 𝑏𝑣, 𝑎 and 𝑏 constants (𝑏 ≠ 0), 𝐺(𝑢) = 𝛼 + 𝛽𝑢, 𝛼 and 𝛽 constants (𝛽 ≠ 0) 𝜕 𝜕 𝜕 𝜕 𝑋1 = 𝐹(𝑥) , 𝑋2 = 𝑊(𝑥) , 𝑋3 = 𝑢 +𝑣 , 𝜕𝑣 𝜕𝑢 𝜕𝑢 𝜕𝑣 𝜕 𝜕 𝑋4 = 𝑏𝑣 + 𝛽𝑢 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑎 + 𝑏𝑣, 𝑎 and 𝑏 constants (𝑏 ≠ 0), 𝐺(𝑢) = 𝛽𝑢−𝑝 , 𝛽 and 𝑝 constants (𝛽, 𝑝 ≠ 0) 𝜕 𝜕 𝜕 𝑋1 = (𝑏𝑥 + 𝑏𝑝𝑥) + 4𝑏𝑢 + (2𝑎 + 2𝑏𝑣 − 2𝑎𝑝 − 2𝑏𝑝𝑣) 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑎 + 𝑏𝑣, 𝑎 and 𝑏 constants (𝑏 ≠ 0) 𝐺(𝑢) = 𝛽𝑒−𝑘𝑢 , 𝛽 and 𝑘 constants (𝛽, 𝑘 ≠ 0) 𝜕 𝜕 𝜕 𝑋1 = 𝑏𝑘𝑥 + 4𝑏 − (2𝑎𝑘 + 2𝑏𝑘𝑣) 𝜕𝑥 𝜕𝑢 𝜕𝑣 Journal of Applied Mathematics 11 Table 4: Continued. −𝑚 𝐻(𝑣) = 𝑎 + 𝑏𝑣 , 𝑎, 𝑚, and 𝑏 constants (𝑏, 𝑚 ≠ 0), 𝐺(𝑢) = 𝑐, 𝑐 a constant 𝜕 𝜕 𝜕 𝑋1 = , 𝑋 = 𝑥−2 , 𝑋3 = (𝑐𝑥2 + 8𝑣) , 𝜕𝑢 2 𝜕𝑢 𝜕𝑢 𝜕 𝜕 𝜕 + (8𝑢 − 𝑎𝑚𝑥2 − 8𝑚𝑢) + 8𝑣 𝑋4 = 4𝑥 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑏𝑣−𝑚 , 𝑚 and 𝑏 constants (𝑏, 𝑚 ≠ 0), 𝐺(𝑢) = 𝛼 + 𝛽𝑢, 𝛼 and 𝛽 constants (𝛽 ≠ 0) 𝜕 𝜕 𝜕 𝑋1 = (𝛽𝑥 + 𝛽𝑚𝑥) + (2𝛼 − 2𝛼𝑚 + 2𝛽𝑢 − 2𝛽𝑚𝑢) + 4𝛽𝑣 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑏𝑣−𝑚 , 𝑚 and 𝑏 constants (𝑏, 𝑚 ≠ 0), 𝐺(𝑢) = 𝛽𝑢−𝑝 , 𝛽 and 𝑝 constants (𝛽, 𝑝 ≠ 0) 𝜕 𝜕 𝜕 𝑋1 = (𝑝𝑚𝑥 − 𝑥) + (2𝑚𝑢 − 2𝑢) + (2𝑝𝑣 − 2𝑣) 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑏𝑣−𝑚 , 𝑚 and 𝑏 constants (𝑏, 𝑚 ≠ 0), 𝐺(𝑢) = 𝛽𝑒−𝑘𝑢 , 𝛽 and 𝑘 constants (𝛽, 𝑘 ≠ 0) 𝜕 𝜕 𝜕 𝑋1 = 𝑘𝑚𝑥 + (2𝑚 − 2) + 2𝑘𝑣 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑎 + 𝑏𝑒−𝑚𝑣 , 𝑎, 𝑚, and 𝑏 constants (𝑏, 𝑚 ≠ 0), 𝐺(𝑢) = 𝑐, 𝑐 constant 𝜕 𝜕 𝜕 𝜕 𝜕 𝑋1 = , 𝑋 = 𝑥−2 , 𝑋3 = (𝑐𝑥2 + 8𝑣) , 𝑋4 = (𝑎𝑚𝑥2 + 8𝑚𝑢) −8 𝜕𝑢 2 𝜕𝑢 𝜕𝑢 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑏𝑒−𝑚𝑣 , 𝑚 and 𝑏 constants (𝑏, 𝑚 ≠ 0), 𝐺(𝑢) = 𝛼 + 𝛽𝑢, 𝛼 and 𝛽 constants (𝛽 ≠ 0) 𝜕 𝜕 𝜕 𝑋1 = 𝛽𝑚𝑥 − (2𝛼𝑚 + 2𝛽𝑚𝑢) + 4𝛽 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑏𝑒−𝑚𝑣 , 𝑚 and 𝑏 constants (𝑏, 𝑚 ≠ 0), 𝐺(𝑢) = 𝛽𝑢−𝑝 , 𝛽 and 𝑝 constants (𝛽, 𝑝 ≠ 0) 𝜕 𝜕 𝜕 𝑋1 = 𝑚𝑝𝑥 + 2𝑚𝑢 + (2𝑝 − 2) 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑏𝑒−𝑚𝑣 , 𝑚 and 𝑏 constants (𝑏, 𝑚 ≠ 0), 𝐺(𝑢) = 𝛽𝑒−𝑘𝑢 , 𝛽 and 𝑘 constants (𝛽, 𝑘 ≠ 0) 𝜕 𝜕 𝜕 𝑋1 = 𝑚𝑘𝑥 + 2𝑚 + 2𝑘 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝐻(𝑣) = 𝑎 + 𝑏 ln 𝑣, 𝑎, 𝑚, and 𝑏 constants (𝑏 ≠ 0), 𝐺(𝑢) = 𝑐, 𝑐 constant 𝜕 𝜕 𝜕 𝜕 𝜕 𝜕 𝑋1 = , 𝑋 = 𝑥−2 , 𝑋3 = (𝑐𝑥2 + 8𝑣) , 𝑋4 = 4𝑥 + (8𝑢 − 𝑏𝑥2 ) + 8𝑣 , 𝜕𝑢 2 𝜕𝑢 𝜕𝑢 𝜕𝑥 𝜕𝑢 𝜕𝑣 𝜕 𝜕 𝜕 𝑋5 = 𝑥−1 − (2𝑥−2 𝑢 − 𝑏 ln 𝑥) − 2𝑥−2 𝑣 𝜕𝑥 𝜕𝑢 𝜕𝑣 These classifying relations are invariant under the equivalence transformation (7) if The classifying relations (20) are also invariant under the equivalence transformation (10) if 𝛼 = 𝛼, 𝛼 = 𝛼, 𝛽 = 𝛼 (𝑎8 𝑥2 + 𝑎6 𝑥1−𝑛 + 𝑎4 ) + 𝛽𝑒−𝑎2 , 𝛽 = 𝛼 (𝑎8 𝑥2 + 𝑎6 𝑥2 ln 𝑥 + 𝑎4 ) + 𝛽𝑒−𝑎2 , 𝛾 = 𝛾, 𝛾 = 𝛾, 𝛿 = 𝛿𝑒2𝑎1 −𝑎3 − 2𝛾 (1 + 𝑛) 𝑎9 , 𝛿 = 𝛿𝑒2𝑎1 −𝑎3 − 2𝛾𝑎7 , 𝜃 = 𝜃, (21) 𝜃 = 𝜃, 𝜆 = 𝜃 (𝑎9 𝑥2 + 𝑎7 𝑥1−𝑛 + 𝑎5 ) + 𝜆𝑒−𝑎3 , 𝜆 = 𝜃 (𝑎9 𝑥2 + 𝑎7 𝑥2 ln 𝑥 + 𝑎5 ) + 𝜆𝑒−𝑎3 , 𝜑 = 𝜑, 𝜑 = 𝜑, 𝜔 = 𝜔𝑒2𝑎1 −𝑎3 − 2𝜑 (1 + 𝑛) 𝑎8 . 𝜔 = 𝜔𝑒2𝑎1 −𝑎2 − 2𝜑𝑎6 . (22) 12 Journal of Applied Mathematics It is also noted that the classifying relations (20) are invariant under the equivalence transformation (13) if 𝛼 = 𝛼, 𝛽 = 𝛼 (𝑎6 ln 𝑥 + 𝑎8 𝑥2 + 𝑎4 ) + 𝛽𝑒−𝑎2 , References 𝛾 = 𝛾, 𝛿 = 𝛿𝑒2𝑎1 −𝑎3 − 4𝛾𝑎9 , 𝜃 = 𝜃, (23) 𝜆 = 𝜃 (𝑎7 ln 𝑥 + 𝑎9 𝑥2 + 𝑎5 ) + 𝜆𝑒−𝑎3 , 𝜑 = 𝜑, 𝜔 = 𝜔𝑒2𝑎1 −𝑎2 − 4𝜑𝑎8 . The classifying relations (20) are also invariant under the equivalence transformation (16) if 𝛼 = 𝛼, 𝛽 = 𝛼 (𝑎9 𝑥−2 + 𝑎7 𝑥2 + 𝑎5 ) + 𝛽 (1 + 2𝑎2 𝑥−2 ) 𝑒−𝑎3 , 𝛾 = 𝛾, 𝛿 = 𝛿𝑒2𝑎1 −𝑎4 − 8𝛾𝑎8 , 𝜃 = 𝜃, Differential Equations, and Applications: Galois Bicentenary,” (SDEA2012) Conference for their kind hospitality during the conference. (24) 𝜆 = 𝜃 (𝑎10 𝑥−2 + 𝑎8 𝑥2 + 𝑎6 ) + 𝜆 (1 + 2𝑎2 𝑥−2 ) 𝑒−𝑎4 , 𝜑 = 𝜑, 𝜔 = 𝜔𝑒2𝑎1 −𝑎3 − 8𝜑𝑎7 . The above relations are now used to find the nonequivalence forms of 𝐻 and 𝐺 and their corresponding Lie point symmetry. Several cases arise and are presented in Tables 1, 2, 3, and 4. The Noether symmetries given in [6] from (25) to (44) always form a proper subalgebra of the Lie algebra that is obtained above. This can be seen from Tables 1, 2, 3, and 4. However, in [6] the first integrals were also presented. 4. Concluding Remarks We have studied a generalized coupled Lane-Emden system from the algebraic viewpoint. A complete group classification of the underlying system was performed. We showed that the generalized coupled Lane-Emden system admits a nine- or ten-dimensional equivalence Lie algebra. The principal Lie algebra, which was found to be trivial, had several possible extensions. We deduced the results for all possible cases of the values of 𝑛. There were in fact four cases that arose. Acknowledgment Two of the authors, B. Muatjetjeja and C. M. Khalique, would like to thank the Organizing Committee of “Symmetries, [1] S. Chandrasekhar, An Introduction to the Study of Stellar Structure, Dover, New York, NY, USA, 1957. [2] H. T. Davis, Introduction to Nonlinear Differential and Integral Equations, Dover, New York, NY, USA, 1962. [3] O. W. Richardson, The Emission of Electricity From Hot Bodies, Longmans, Green & Co., London, UK, 2nd edition, 1921. [4] B. Muatjetjeja and C. M. Khalique, “Exact solutions of the generalized Lane-Emden equations of the first and second kind,” Pramana—Journal of Physics, vol. 77, no. 3, pp. 545–554, 2011. [5] J. S. W. Wong, “On the generalized Emden-Fowler equation,” SIAM Review, vol. 17, pp. 339–360, 1975. [6] B. Muatjetjeja and C. M. Khalique, “Lagrangian approach to a generalized coupled Lane-Emden system: symmetries and first integrals,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 5, pp. 1166–1171, 2010. [7] H. Zou, “A priori estimates for a semilinear elliptic system without variational structure and their applications,” Mathematische Annalen, vol. 323, no. 4, pp. 713–735, 2002. [8] J. Serrin and H. Zou, “Non-existence of positive solutions of Lane-Emden systems,” Differential and Integral Equations, vol. 9, no. 4, pp. 635–653, 1996. [9] J. Serrin and H. Zou, “Existence of positive solutions of the Lane-Emden system,” Atti del Seminario Matematico e Fisico dell’Università di Modena, vol. 46, supplement, pp. 369–380, 1998. [10] Y.-W. Qi, “The existence of ground states to a weakly coupled elliptic system,” Nonlinear Analysis: Theory, Methods & Applications, vol. 48, no. 6, pp. 905–925, 2002. [11] R. Dalmasso, “Existence and uniqueness of solutions for a semilinear elliptic system,” International Journal of Mathematics and Mathematical Sciences, no. 10, pp. 1507–1523, 2005. [12] Q. Dai and C. C. Tisdell, “Nondegeneracy of positive solutions to homogeneous second-order differential systems and its applications,” Acta Mathematica Scientia. Series B, vol. 29, no. 2, pp. 435–446, 2009. [13] N. H. Ibragimov, “Small effects in physics hinted by the Lie group philosophy: are they observable? I. From Galilean principle to heat diffusion,” Lie Groups and Their Applications, vol. 1, no. 1, pp. 113–123, 1994. [14] L. V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, New York, NY, USA, 1982. Advances in Operations Research Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Advances in Decision Sciences Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Mathematical Problems in Engineering Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Journal of Algebra Hindawi Publishing Corporation http://www.hindawi.com Probability and Statistics Volume 2014 The Scientific World Journal Hindawi Publishing Corporation http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 International Journal of Differential Equations Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Volume 2014 Submit your manuscripts at http://www.hindawi.com International Journal of Advances in Combinatorics Hindawi Publishing Corporation http://www.hindawi.com Mathematical Physics Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Journal of Complex Analysis Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 International Journal of Mathematics and Mathematical Sciences Journal of Hindawi Publishing Corporation http://www.hindawi.com Stochastic Analysis Abstract and Applied Analysis Hindawi Publishing Corporation http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com International Journal of Mathematics Volume 2014 Volume 2014 Discrete Dynamics in Nature and Society Volume 2014 Volume 2014 Journal of Journal of Discrete Mathematics Journal of Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Applied Mathematics Journal of Function Spaces Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Optimization Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014

Research Article Group Classification of a Generalized Lane-Emden System Ben Muatjetjeja,

Related documents

Products

Support

Research Article Group Classification of a Generalized Lane-Emden System Ben Muatjetjeja,

Related documents

Add this document to collection(s)

Add this document to saved

Suggest us how to improve StudyLib