The application of symmetry methods and conservation laws to ordinary differential equations and a linear wave equation

Symmetry analysis and conservation laws are widely used in analyzing and solving differential equations. Conservation laws are also called first integrals when dealing with ordinary differential equations (ODEs). In this thesis, the complementary nature of these two approaches is explored; specifically, the use of symmetries to find integrating factors and, conversely, the use of conservation laws to seek new symmetries. In Part 1, building upon results in [3] and [10], it is shown that a higher-order symmetries of an ODE induces a point symmetry of the corresponding integrating factor determining equations (IFDE), and an explicit expression for this induced symmetry is obtained. Secondly, it is shown that the converse also holds for a special class of Lie point symmetries of the IFDE; namely, all Lie point symmetries of the IFDE which are of this form project onto point symmetries of the original scalar ODE. In Part 2, the use of conservation laws to find non-local symmetries is shown for a linear one-dimensional wave equation in a two-layered medium with a smooth transition layer. The resulting analytic solutions are then studied in order to investigate the effect of the transmission and reflection of energy between the two media. It is found that the reflection and transmission coefficients depend on the ratio of the wave speeds in the two media as well as the ratio of the characteristic length of the incoming signal to the width of the transition layer. Approximations of the dependence of the reflection and transmission coefficients on these two parameters are also presented, obtained via numerical experiments performed using both the analytic solution and a finite element method.Science, Faculty ofMathematics, Department ofGraduat