Group and potential similarity transformation methods for the solution of evolution partial differential equations

The study of differential equations is a wide field in pure and applied mathematics, physics, and engineering applications. The investigation of the exact solutions of differential equations plays an important role in the study of nonlinear physical systems. A wealth of methods has been developed to find these exact physically significant solutions of a PDE though it is rather difficult. Differential equations such as those used to solve real-life problems may not necessarily be directly solvable, i.e. do not have closed form solutions. Instead, solutions can be approximated using numerical methods.The object of this thesis is to obtain the similarity solution of a given problem by applying the Group and Potential similarity transformation methods where the governing partial differential equations are written in a conserved form to obtain new simpler system of partial differential equations. We then applied the group method which reduces the new system with the auxiliary condition to a system of ordinary differential equation with the appropriate corresponding conditions that can be solved analytically or numerically.Because of the absence of this combination between the potential and the group methods so far, this search is considered as an innovation in the field of mathematics.In the present work the similarity solutions of four different problems were obtained using this new combination. We first solved Burgers’ equation and the general Korteweg-de Vries (KdV) equation as a test on the method. The third application was published in an international journal [1] wherein we found new similarity solution of the nonlinear Boussinesq equation which describes the propagation of solitary waves with small amplitude on the surface of shallow water. Finally we investigated the similarity solution of cylindrical KdV equation which describes solitary waves in a channel with slo