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Symmetry reduction and numerical solution of a third-order ODE from thin film flow. (English) Zbl 1248.76122

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.

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

Software:

LIE; DIMSYM
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