Lie Symmetries Of Differential Equations

This thesis studies the geometry of the one parameter transformations which leave invariant the differential equations and investigates the mathematical properties of 2nd order differential equations of Lie symmetries,as well as Noether symmetries.Moreover,we found an advanced geometric approach which describes the symmetries of differential equations.According to this advanced approach,we resolved symmetries of mathematical models.This advanced geometric approach is used to establish the Newton Kepler Ermakov scheme about Riemannian space,in order to find the Newtonian dynamical system of all two and three dimensional which introduce Noether and Lie symmetries,to investigate the point symmetries among quantum and classical schemes and study Type-II Hidden symmetries for Laplace equation as well as for a heat equation within Riemannian space.Furthermore,by means of Painlevé analysis,(2+1)dimensional Nonlinear Schr?dinger Equation(NLSE)is investigated with the help of Weiss et al approach and Kruskal's simplification method.Finally,many generalized Lie symmetries for(2+1)dimensional NLSE are obtained by formal series symmetry approach.