Symmetries and conservation laws of some classes of partial differential equations

Malatsi, Tebogo Doctor
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Using underlying invariance/symmetry properties and related/associated conservation laws, we investigate some ’high’ order nonlinear equations. The multiplier method is mainly used to construct conserved vectors for these equations. This method results in conservation laws that are useful in a wide range of ways. When the partial differential equations are reduced to nonlinear ordinary differential equation (NLODE), exact solutions for the ODEs are constructed and graphical representations of the resulting solutions are provided. In some cases, the solutions obtained are the Jacobi elliptic cosine function and the solitary wave solutions. Firstly, we study the third-order ’equal width equation’ followed by a new fourth-order nonlinear partial differential equation (NLPDE), which was recently established in the literature. We employ Lie symmetry methods and various other techniques to construct closed-form solutions of the NLPDE, which we then visualize in 3D and 2D plots using Mathematica. Secondly, we establish conservation laws using the multiplier method. Finally, we study the Korteweg-de Vries (KdV) equation having three dispersion sources. The exact solutions and conservation laws for this equation are discovered. The multipliers technique is used to derive conservation laws for this equation. The KdV equation is further reduced into a first-order NLODE using the travelling wave reduction method with conservation laws. The Jacobi ellliptic cosine function and solitary wave solutions are established.
A dissertation submitted in fulfillment of the requirements for the degree of Master of Science to the Faculty of Science, School of Mathematics, University of the Witwatersrand, Johannesburg, 2022