Lie Symmetry Analysis of Nonlinear Fourth Order Evolution Equation - Wambura, Kevin Odundo (JOOUST, 2020)
Problems involving nonlinear partial differential equations arise in various fields of science for instance in propagation of shallow water waves, fluid physics. quantum mechanics. chemical kinematics and other related areas. Therefore, the task of obtaining the exact solutions to a nonlinear partial differential equation is of great importance and has attracted high attention among researchers. Nikolai and Maria made an attempt. in obtaining the exact solution of a special nonlinear fourth order evolution equation expressed as using Newton polygon method. The method allowed them to express the exact. solution of the equation studied through the solution of another equation using the properties of the basic equation itself. However, they were only able to obtain one exact. solution of the equation. In the recent years, lie symmetry analysis has played a significant. role in the construction of exact solutions to nonlinear evolution equations. The history of Lie symmetry group classification method goes back to Sophus Lie at the end of the 19th century. A variety of methods have been developed in the past few years by Ovsyarrnikov. Ibragimov and others. Therefore, by setting the coefficient of the nonlinear term uu; to zero, we have attempted in this study to determine more than one exact explicit solution of the nonlinear fourth order evolution equation using Lie symmetry analysis approach. The equation is a Korteweg-De Vries (KdV) type of equation this {at has been applied to describe a wide range of physical phenomena of the evolution and interaction to nonlinear waves. such as fluid dynamics, aerodynamics, continuum mechanics, solitons and turbulence. The exact solutions to the nonlinear fourth order evolution equation has been obtained through the prolongation of infinitesimal generators, determination of geometric vector fields, similarity reduction and use of power series method. We believe this will be a great contribution to the knowledge and further research.