Abstract:
The construction of conserved vectors using Noether’s theorem via a knowledge
of a Lagrangian (or via the recently developed concept of partial Lagrangians) is
well known. The formulae to determine these for higher-order flows is somewhat
cumbersome and becomes more so as the order increases. We carry out these for
a class of fourth, fifth and sixth order PDEs. In the latter case, we involve the
fifth-order KdV equation using the concept of ‘weak’ Lagrangians better known for
the third-order KdV case.
We then consider the case of a mixed ‘high-order’ equations working on the Shallow
Water Wave and Regularized Long Wave equations. These mixed type equations
have not been dealt with thus far using this technique. The construction of conserved
vectors using Noether’s theorem via a knowledge of a Lagrangian is well known.
In some of the examples, our focus is that the resultant conserved flows display some
previously unknown interesting ‘divergence properties’ owing to the presence of the
mixed derivatives.
We then analyse the conserved flows of some multi-variable equations that arise
in Relativity. In addition to a larger class of conservation laws than those given
by the isometries or Killing vectors, we may conclude what the isometries are and
that these form a Lie subalgebra of the Noether symmetry algebra. We perform
our analysis on versions of the Vaidya metric yielding some previously unknown
information regarding the corresponding manifold. Lastly, with particular reference
to this metric, we also show the variations that occur for the unknown functions.
We discuss symmetries of classes of wave equations that arise as a consequence
of the Vaidya metric. The objective of this study is to show how the respective
geometry is responsible for giving rise to a nonlinear inhomogeneous wave equation
as an alternative to assuming the existence of nonlinearities in the wave equation
due to physical considerations. We find Lie and Noether point symmetries of the
corresponding wave equations and give some reductions. Some interesting physical
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conclusions relating to conservation laws such as energy, linear and angular momenta
are also determined. We also present some interesting comparisons with the standard
wave equations (on a ‘flat geometry’).
Finally, we pursue the nature of the flow of a third grade fluid with regard to
its underlying conservation laws. In particular, the fluid occupying the space over
a wall is considered. At the surface of the wall, suction or blowing velocity is
applied. By introducing a velocity field, the governing equations are reduced to a
class of PDEs. A complete class of conservation laws for the resulting equations
are constructed and analysed using the invariance properties of the corresponding
multipliers/characteristics.
Description:
PhD., Faculty of Science, University of the Witwatersrand, 2011