Perturbed ODEs in cosmology and invariance analysis of ordinary diﬀerence equations - Mnguni, Nkosingiphile (My University, 2017)
This 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.
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 2017
Keywords: Difference equations; Symmetry (Mathematics)