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Moving mesh methods for convection-dominated equations and nonlinear conservation lawsZhang, Zhengru 01 January 2003 (has links)
No description available.
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Some topics on hyperbolic conservation laws.January 2008 (has links)
Xiao, Jingjing. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (p. 46-50). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgement --- p.ii / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Backgrounds and Our Main Results --- p.4 / Chapter 2.1 --- Backgrounds --- p.4 / Chapter 2.1.1 --- The Scalar Case --- p.4 / Chapter 2.1.2 --- 2x2 Systems --- p.5 / Chapter 2.1.3 --- General n x n(n ≥ 3) Systems --- p.9 / Chapter 2.2 --- Our Main Results --- p.18 / Chapter 3 --- Lifespan of Periodic Solutions to Gas Dynamics Systems --- p.21 / Chapter 3.1 --- Riemann Invariant Formulation --- p.21 / Chapter 3.2 --- Calculation along Characteristics --- p.26 / Chapter 3.3 --- Estimate of the Global Wave Interaction --- p.35 / Chapter 3.4 --- Proof of Theorem 2.2.1 --- p.38 / Chapter 4 --- Proof of Theorem 2.2.2 and a Special Case --- p.40 / Chapter 4.1 --- Proof of Theorem 2.2.2 --- p.40 / Chapter 4.2 --- A Special Case --- p.43 / Chapter 5 --- Appendix --- p.45
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Nonclassical symmetry reductions and conservation laws for reaction-diffusion equations with application to population dynamicsLouw, Kirsten 29 May 2015 (has links)
A research report submitted to the Faculty of Science, in partial fulfilment of the requirements for the degree of Master of Science, University of the Witwatersrand, Johannesburg, 2015. / This dissertation analyses the reaction-di usion equations, in particular the modi ed
Huxley model, arising in population dynamics. The focus is on determining the classical
Lie point symmetries, and the construction of the conservation laws and group-invariant
solutions for reaction-di usion equations. The invariance criterion for determination
of classical Lie point symmetries results in a system of linear determining equations
which can be solved analytically. Furthermore, the Lie point symmetries associated
with the conservation laws are determined. Reductions by associated Lie point symmetries
are carried out. Nonclassical symmetry techniques are also employed. Here
the invariance criterion for symmetry determination results in a system of nonlinear
determining equations which may be solved albeit di cult. Nonclassical symmetries
results in exact solutions which may not be constructed by classical Lie point symmetries.
The highlight in construction of exact solution using nonclassical symmetries
is the introduction of the modi ed Hopf-Cole transformation. In this dissertation, the
di usion term and the coe cient of the source term are given as quadratic functions
of space variable in one case, and the coe cient as the generalised power law in the
other. These equations admit a number of classical Lie point symmetries. The genuine
nonclassical symmetries are admitted when the source term of the reaction-di usion
equation is a cubic.
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Some problems on conservation laws and Vlasov-Poisson-Boltzmann equation /Zhang, Mei. January 2009 (has links) (PDF)
Thesis (Ph.D.)--City University of Hong Kong, 2009. / "Submitted to Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy." Includes bibliographical references (leaves [90]-94)
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Phase transition problems of conservation lawsChen, Chunguang, January 1900 (has links)
Thesis (Ph. D.)--West Virginia University, 2010. / Title from document title page. Document formatted into pages; contains vii, 48 p. : ill. (some col.). Includes abstract. Includes bibliographical references (p. 48).
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Moving mesh methods for singular problems in two dimensionsLee, Wan Lung 01 January 2004 (has links)
No description available.
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Symmetries and conservation laws of difference and iterative equationsFolly-Gbetoula, Mensah Kekeli 22 January 2016 (has links)
A thesis submitted to the Faculty of Science, University of the Witwatersrand,
Johannesburg, in ful lment of the requirements for the degree of Doctor of Philosophy.
Johannesburg, August 2015. / We construct, using rst principles, a number of non-trivial conservation
laws of some partial di erence equations, viz, the discrete Liouville equation
and the discrete Sine-Gordon equation. Symmetries and the more recent
ideas and notions of characteristics (multipliers) for di erence equations are
also discussed.
We then determine the symmetry generators of some ordinary di erence
equations and proceed to nd the rst integral and reduce the order of the
di erence equations. We show that, in some cases, the symmetry generator
and rst integral are associated via the `invariance condition'. That is,
the rst integral may be invariant under the symmetry of the original di erence
equation. We proceed to carry out double reduction of the di erence
equation in these cases.
We then consider discrete versions of the Painlev e equations. We assume
that the characteristics depend on n and un only and we obtain a number
of symmetries. These symmetries are used to construct exact solutions in
some cases.
Finally, we discuss symmetries of linear iterative equations and their transformation
properties. We characterize coe cients of linear iterative equations
for order less than or equal to ten, although our approach of characterization
is valid for any order. Furthermore, a list of coe cients of linear iterative
equations of order up to 10, in normal reduced form is given.
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Symmetry reductions of systems of partial differential equations using conservation lawsMorris, R. M. 07 February 2014 (has links)
There is a well established connection between one parameter Lie groups of transformations and conservation laws for differential equations. In this thesis, we construct conservation laws via the invariance and multiplier approach based on the wellknown result that the Euler-Lagrange operator annihilates total divergences. This
technique will be applied to some plasma physics models. We show that the recently
developed notion of the association between Lie point symmetry generators and conservation laws lead to double reductions of the underlying equation and ultimately
to exact/invariant solutions for higher-order nonlinear partial di erential equations
viz., some classes of Schr odinger and KdV equations.
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Symmetries and conservation laws of certain classes of nonlinear Schrödinger partial differential equationsMasemola, Phetego 08 May 2013 (has links)
A dissertation submitted to the Faculty of Science, University of the Witwatersrand, in fulfilment of the requirements for the degree of Master of Science. Johannesburg, 2012. / Unable to load abstract.
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Symmetries and conservation laws of higher-order PDEsNarain, R. B. 19 January 2012 (has links)
PhD., Faculty of Science, University of the Witwatersrand, 2011 / 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.
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