Use this URL to cite or link to this record in EThOS:
Title: Some exact singularities of Burger's and heat equations
Author: Reynolds, B. T.
Awarding Body: University of Wales Swansea
Current Institution: Swansea University
Date of Award: 2000
Availability of Full Text:
Access from EThOS:
Our main aim in the present work is to study explicit examples of shock waves for Burger's equation and the corresponding level surfaces for the heat equation. This thesis is in two parts: In Part I (Chapters 1 and 2) we introduce many of the concepts required for our study. Particularly we give a brief introduction to the heat equation, hydrodynamics and the underlying dynamical theory. Towards the end of Chapter 1 we discuss the stochastic Hamilton-Jacobi theory and begin formulating the notion of caustics and wavefronts in both the classical and stochastic cases. Chapter 2 contains an account of the results presented in [1] and [2] in which the exact Greens functions are calculated for the zero, linear and harmonic oscillator potentials, for both the classical and stochastic cases. We employ these results in the derivation of exact formulae for the corresponding caustics and wavefronts. In Part II (Chapter 3 and 4) the results obtained from Chapter 2 are applied for several explicit chosen examples. We consider the initial conditions that lead us to the Thom catastrophes of the Cusp and the Butterfly, see [3], and their corresponding wavefronts, the Tricorn and Fish. Some time is taken to consider the meeting points of the caustics and wavefronts and the connection with meeting points of the pre-curves. In several instances we provide conditions for the existence of the pre-wavefronts with respect to the time. A large portion of the work in these chapters relied heavily on Mathematica and required an enhanced understanding of the handling of graphics within the package. In examples where we have been unable to calculate the wavefronts directly we have developed numerical methods within Mathematica to provide graphical images of them.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available