Symmetry analysis, conservation laws and exact solutions of ill-posed Boussinesq equation

This thesis presents a comprehensive study on the construction of conservation laws and the application of double reduction techniques to obtain exact solutions for the ill-posed boussinesq equation Conservation laws are utilized to perform a double reduction of the original PDE. This reduction process involves two key stages: firstly, the application of the derived conservation laws to convert the PDE to ODE then reduction of the order of the ODE and secondly, the further simplification of the reduced equation by exploiting additional symmetries to obtain the exact solutions. The exact solutions are analyzed and graphically demonstrated to gain insight into the underlying physical and mathematical properties of the original PDE. The dissertation contributes to the field of applied mathematics by providing a rigorous framework for constructing conservation laws and applying reduction techniques to nonlinear PDEs. The exact solutions obtained not only advance the theoretical understanding of the equation but also offer potential applications in areas such as fluid dynamics, nonlinear optics, and other fields where similar equations arise.