Group analysis of ordinary differential equations and the invariance principle in mathematical physics (for the 150th anniversary of Sophus Lie)

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

“…He also showed that if a second-order equation admits an eight-dimensional algebra, it is linearizable by a point transformation (see also Theorem 8) and equivalent to the simplest equation y = 0. [16,43,44] One-and two-dimensional algebras have identical structures over the reals as well as over the complex numbers. Consequently, symmetry algebra classification of second-order equations over the reals is the same as that over the complex numbers for one-and two-dimensional Lie algebras.…”

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

“…we obtain a new variable w = 1 + 9(t − s) 17) which transforms the generator (4.15) to its semi-canonical form By the Vessiot-Guldberg-Lie theorem [10,11], the generators …”

Laplace type invariants for variable coefficient mKdV equations

“…These invariants are useful in various problems, for example in the group classification of differential equations [2] and the solution of initial value problems for hyperbolic equations by Riemann's method [3]. We recall the following simple but fundamental applications of the Laplace invariants:…”