Mathematical and Computational Applications,Vol. 15, No. 4, pp. 709-719, 2010. c ⃝Association for Scientific Research SYMMETRY REDUCTION AND NUMERICAL SOLUTION OF A THIRD-ORDER ODE FROM THIN FILM FLOW E. Momoniat 1 and F.M. Mahomed 2 Centre for Differential Equations, Continuum Mechanics and Applications School of Computational and Applied Mathematics, University of the Witwatersrand, Johannesburg, Private Bag 3, Wits 2050, 1 Ebrahim.Momoniat@wits.ac.za, 2 Fazal.Mahomed@wits.ac.za Abstract- A new approach to solving high-order ordinary differential equations numerically is presented. Instead of the usual approach of writing a high-order ordi- nary differential equation as a system of first-order ordinary differential equations, we write the high-order ordinary differential equation in terms of its differential in- variants. The third-order ordinary differential equation y ′′′ = y −k for constant k is used to illustrate this approach for the cases k = 2 and k = 3. Keywords- Thin film; third-order ODE; symmetry reduction. 1. INTRODUCTION We consider the ordinary differential equation (ODE) y ′′′ = y −k , k = constant (1) solved subject to the initial conditions (α, β , γ and λ are constants) y(λ)= α, y ′ (λ)= β, y ′′ (λ)= γ. (2) Analytical solutions to the third-order ODE (1) are given in general in the handbook of Polyanin and Zaitsev [17] for the cases k = 0, 7/2, 5/2, 4/3, 7/6, 2, 1/2 and 5/4. The ordinary differential equation for the case k = 2 has been derived by Tanner [18] to investigate the motion of the contact line for a thin oil drop spreading on a horizontal surface. A review of thin film flow has been presented by Myers [14] in which some third-order ODEs occuring in thin film flow are discussed. The interested reader is also referred to Tuck and Schwartz [19] and the references therein. Analytical solutions admitted by the case k = 2 has also recently been investigated by Duffy and Wilson [4]. The analytical solution for the case k = 2 was initially presented by Ford [5]. The case k = 2, amongst others, has been the subject of intense numerical investigation by Tuck and Schwartz [19]. Existence and uniqueness