Symmetry group classification of ordinary differential equations: Survey of some results

Abstract

SUMMARYAfter the initial seminal works of Sophus Lie on ordinary differential equations, several important results on point symmetry group analysis of ordinary differential equations have been obtained. In this review, we present the salient features of point symmetry group classification of scalar ordinary differential equations: linear nth-order, second-order equations as well as related results. The main focus here is the contributions of Peter Leach, in this area, in whose honour this paper is written on the occasion of his 65th birthday celebrations.

Symmetry Solutions of a Third-Order Ordinary Differential Equation which Arises from Prandtl Boundary Layer Equations

“…Thus equation (4.10) possesses a non-solvable Lie symmetry algebra L 3 and, therefore, cannot be solved by the Lie approach (Ibragimov and Nucci [11], Mahomed [12] and Olver [13]). Consider the subalgebra…”

“…For special values of α three Lie point symmetry generators exist and the third-order ordinary differential equation is solved by the Lie approach as described, for example, by Ibragimov and Nucci [11], Mahomed [12] and Olver [13]. For B = 0 the third-order ordinary differential equation (1.1) describes radial and two-dimensional liquid jets and admits a solvable Lie algebra.…”

“…For B = 0 the third-order ordinary differential equation (1.1) describes radial and two-dimensional liquid jets and admits a solvable Lie algebra. We solve the equation by the Lie approach ( Ibragimov and Nucci [11], Mahomed [12], Olver [13]). For α = −1 (two-dimensional) and α = −4 (radial), the third-order ordinary differential equation (1.1) admits a non-solvable Lie algebra and can be reduced to the Chazy equation [4,5,6,8].…”

Conditional Linearizability of Fourth-Order Semi-Linear Ordinary Differential Equations

“…First-order ordinary differential equations (ODEs) can always be linearized (i.e., converted to linear form) [13] by point transformations [11]. Lie [12] showed that all second-order ODEs that can be converted to linear form must be cubically semi-linear, i.e.…”

Inequivalence of Classes of Linearizable Systems of Second Order Ordinary Differential Equations Obtained by Real and Complex Symmetry Analysis

“…Numerical schemes also suffer from the same problems. All the first and linear second order ODEs are equivalent under the change of dependent and independent variables [15]. Lie derived a canonical method for obtaining the exact solution of an ODE, or system of ODEs, provided these are invariant under certain transformations [14].…”