### Abstract:

Nonlinear Partial Differential Equations (PDEs) are one of the key area of the interaction of mathematics and other sciences. Most physical phenomena can be described by (PDEs). Many methods of mathematical physics have been developed to solve differential equations, among which Lie group method is an area of mathematics in which we can see a harmonious interaction between the methods of classical analysis and modern algebra. The objective of the research is to obtain the Lie symmetries and the exact solutions of nonlinear partial differential equations, which represent example of the important physical phenomena. The study of these systems of differential equations is often regarded as a difficult and confusing endeavor due to various limitations. The study deal with the methods of group invariant solutions, through the constructive procedures Lie established that, in the case of (ODEs), invariance under one-parameter symmetry group implies that the order of the equation can be reduced by one. The main tool in this study will be Lie classical method for reduction (PDEs) and other method including hyperbolic secant-tangent functions expansion method for some exact solutions of (ODEs). The study considers the regulaised long wave to apply the method. Among which Lie group method is an efficient approach to derive the exact solution of nonlinear partial differential equations and it's easy method that depends on solution of algebra equations. The study recommends that Lie group can be applied to other nonlinear partial differential equations. Computer program could be used in the method of solution, especially in the solution of (ODEs) in order to simplify the process of a solution.

### Description:

A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of
Master of Science
in Mathematics, Department of Mathematics, Faculty of Mathematical and Computer Sciences , University of Gezira, 23 July 2013