Perturbed ODEs in Cosmology and Invariance Analysis of Ordinary Difference Equations

A dissertation submitted to the faculty of science, University of the Witwatersrand, in partial fulfilment of the requirements for the degree of Master of Science, Johannesburg December 2017This dissertation contains two parts ,the first part involves the novel study of the use of symmetries to find exact solutions of ordinary difference equations. Non-trivial symmetries of third-order difference equations of the form (1) are obtained. These symmetries are used to investigate their solutions for some random sequences (An) and (Bn). We extend the results obtained in [15] and conditions for well-defined solutions are obtained. Furthermore, a full Lie point symmetry analysis of the fifthorder rational difference equations of the form (5), where λ and µ are real numbers, is considered and exact solutions are found. In the second part we find new geometric conditions that can be specialized to obtain the approximate Noether symmetries of a large class of variational equations. We will also look at the ordinary differential equations that model cosmological and relativistic phenomena, which we will analyse using an approximate symmetry method.XL201