Ahmed, Mogahid Mamoon Abkar2024-11-052024-11-052024Ahmed, Mogahid Mamoon Abkar . (2024). The role of invariants in obtaining exact solutions of differential equations [ PhD thesis, University of the Witwatersrand, Johannesburg]. WireDSpace.https://hdl.handle.net/10539/42172A thesis submitted to the Faculty of Science, University of the Witwatersrand, in fulfillment of the requirements for the degree of Doctor of Philosophy. Johannesburg, 2024We show here that variational and gauge symmetries have additional appli- cations to the integrability of differential equations. We present a general method to construct first integrals for some classes. In particular, we present a broad class of diffusion type equations, viz., the Fisher Kolmorov and Fitzhugh Nagumo equations, which satisfy the Painlev´e properties of their respective travelling wave forms and solitons. It is then shown how a study of invari- ance properties and conservation laws is used to ‘twice’ reduce the equations to solutions. We further constructing the first integrals of a large class of the well-known second-order Painlev´e equations. In some cases, variational and gauge symmetries have additional applications following a known Lagrangian in which case the first integral is obtained by Noether’s theorem. Generally, it is more convenient to adopt the ‘multiplier’ approach to find the first integrals. The main chapters of this thesis have either been published or submitted for publication in accredited journals. The contents of Chapters 2, 3 and 5 has been published ([54], [55]). All computations were done either by hand or Mapleen© 2024 University of the Witwatersrand, Johannesburg. All rights reserved. The copyright in this work vests in the University of the Witwatersrand, Johannesburg. No part of this work may be reproduced or transmitted in any form or by any means, without the prior written permission of University of the Witwatersrand, Johannesburg.InvariantsExact solutionUCTDSDG-4: Quality educationThesisUniversity of the Witwatersrand, Johannesburg