Lie Symmetry Analysis of Third Order NonLinear Ordinary Differential Equation - Odongo, Peter Miginjo (JOOUST, 2018)
A few third order nonlinear Ordinary Differential Equations can be solved explicitly and one must typically rely on numerical scheme to approximate the solution. However to overcome this problem Lie symmetry reduction of Ordinary Differential Equation to lower order can be used to obtain analytic solution of the equation of the form F(x, y, y', y", ... yn) = O .The most interesting properties of a differential equation is their symmetry. Symmetry analysis was developed in nineteenth century by Sophus Lie. Symmetry analysis plays an important role in this study. Attempt was made by Ahmad et al, to develop a new systematic method to find relative invariant differential operators . They incorporated Lie's symmetry anal¬ysis to study the general class, y"' = F(x, y, y', y") under general point equivalence transformations in the generic case. As a result all third order relative and absolute invariant differential equation operators were deter¬mined for third order Ordinary Differential Equation, y111 = F(x, y, y', y") which are not quadratic second order derivative. Therefore in our study we have used Lie Symmetry analysis approach to transform the equation by subjecting it to an extension generator and obtained determining equa¬tions, to reduce the order of the equation to lower order and to find the general solution of the third order nonlinear heat conduction from the in¬variance related potential system under scaling . We exploited the use of prolongations ( extended transformations), infinitesimal generators, vari¬ation of symmetries, invariant transformation problems and integrating factors. We expect that the research will be useful in image processing, variation of variable will help in modeling options.
Keywords: Lie Symmetry Analysis; Third Order NonLinear Ordinary Differential Equation