The role of invariants in obtaining exact solutions of differential equations
Date
2024
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
University of the Witwatersrand, Johannesburg
Abstract
We 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 Maple
Description
A thesis submitted to the Faculty of Science, University of the Witwatersrand, in fulfillment of the requirements for the degree of Doctor of Philosophy.
Johannesburg, 2024
Keywords
Invariants, Exact solution, UCTD
Citation
Ahmed, Mogahid Mamoon Abkar . (2024). The role of invariants in obtaining exact solutions of differential equations [ PhD thesis, University of the Witwatersrand, Johannesburg]. WireDSpace.