Momoniat, E.; Mahomed, F. M. Symmetry reduction and numerical solution of a third-order ODE from thin film flow. (English) Zbl 1248.76122 Math. Comput. Appl. 15, No. 4, 709-719 (2010). A new approach to solving high-order ordinary differential equations numerically is provided in this work. Instead of the usual approach of writing a high-order ordinary differential equation as a system of first-order ordinary differential equations, the high-order ordinary differential equation is written in terms of its differential invariants. The third-order ODE is reduced, through the successive reduction of order, to a second-order ODE, and then to a first-order ODE. Unfortunately, generally, the first-order reduction cannot be solved analytically. The latter is used, however, to determine an efficient way to solve the equation numerically.Two methods are introduced: Firstly, a second-order ordinary differential equation is solved numerically as a system of three first-order equations using a fourth-order Runge-Kutta method. The second method, which can be employed due to the double reduction of the original equation, is to solve an equivalent first-order ordinary differential equation. For the second-order case, the authors solve a first-order ODE obtained via reduction using the fourth-order Runge-Kutta method. Reviewer: Charis Harley (Johannesburg) Cited in 5 Documents MSC: 76M60 Symmetry analysis, Lie group and Lie algebra methods applied to problems in fluid mechanics 76A20 Thin fluid films 76M20 Finite difference methods applied to problems in fluid mechanics Keywords:differential invariants; fourth-order Runge-Kutta method Software:LIE; DIMSYM PDFBibTeX XMLCite \textit{E. Momoniat} and \textit{F. M. Mahomed}, Math. Comput. Appl. 15, No. 4, 709--719 (2010; Zbl 1248.76122) Full Text: DOI