الفهرس 
INTRODUCTION AND AIM OF THE WORKOnly 14 pages are availabe for public view
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Abstract
Investigation of physical systems described by the nonlinear differential equations – ordinary and partial - and exploring their underlying dynamics remains the central focus of research for the past few decades. Finding the exact solutions of those nonlinear differential equations is one of the most important tasks and plays an important role in studying the nonlinear physical phenomena and identifying their properties closely. There are many known methods for obtaining explicit solutions to the differential equations, but most of them are just special cases of symmetry methods. The symmetries are mathematical transformations that preserve the invariance of the physical laws of the system. The Lie group method provides precise mathematical formulation of the basic ideas of symmetry and also provides constructive methods for solving all differential equations analytically; it also allows finding symmetries of differential equations that cannot be reached in other ways and the advantages of this method Reduce the number of independent variables in partial differential equations by one, but in ordinary differential equations, they reduce the order of the equation. Convert a nonlinear differential equation (partial or ordinary) to a linear differential equation. Finding a lot of solutions for the differential equation. Constructing a new solution to the differential equation from another solution. Converting the variable boundary conditions to fixed boundary conditions. The purpose of this thesis is to perform the application of Lie group method to study some practical physical problems such as the viscous barotropic non-divergent vorticity equation (BVE ), the generalized Burgers’, Burgers’- KdV, GKdV, and the generalized Keller – Segel model. Focus is first placed on discussing of the fundamentals notions for the theory of Lie group method and its applications in solving differential equations. Topics of discussion include groups, Lie groups, and groups of transformations, invariants, prolongation, and the optimal one – dimension Lie subalgebra. A simple example of using Lie group method is thoroughly presented using heat equation to demonstrate the inner calculations behind this technique. Focus is then changed to solving the practical physical problems mention above, where emphasis is placed on finding the symmetries and classifying the types of invariant solutions that were produced via the Lie group method.