1 
Inertial manifoldsRobinson, James Cooper January 1995 (has links)
No description available.

2 
The application of transmissionline modelling implicit and hybrid algorithms to electromagnetic problemsWright, S. J. January 1988 (has links)
No description available.

3 
Homoclinic bifurcationsDrysdale, David January 1994 (has links)
Previously obtained results from the study of homoclinic bifurcations in ordinary differential equations are presented. The standard technique of analysis involves the construction of a Poincaré map on a surface near to the homoclinic point. This map is the composition of an inside map, with behaviour linearized about the homoclinic point, together with an outside map, with behaviour linearized about the homoclinic orbit. The Poincaré map is then reduced to a onedimensional map, involving the return time between successive visits to the Poincaré surface. These standard techniques in the contemplation of homoclinic systems are then extended to a class of partial differential equations, on unbounded domains. This follows a method introduced by Fowler [Stud. Appl. Math. 83 (1990), pp. 329–353]. This extension involves more technicalities than in the case of ordinary differential equations. The method of Fowler is extended to cover the case of vectorvalued partial differential equations, and to consider the consequences of symmetry invariances. A Poincaré map is derived, and then is reduced to a finitedimensional map. This map has dimension equal to the number of symmetry invariances of the system. Some simple examples of this finitedimensional map are studied, in isolation. A number of interesting bifurcation pictures are produced for these simple examples, involving considerable variation with the values of coefficients of the map. Partial differential equations on finite domains are then considered, yielding similar results to the ordinary differential equation case. The limit as domain size tends to infinity is examined, yielding a criterion for distinction between the applicability of finite and infinite domain results. Finally, these methods are applied to the GinzburgLandau system. This involves the numerical calculation of coefficients for the finitedimensional map. The finitedimensional map thus derived supports an interesting interlocked isola structure, and moreover correlates with numerical integration data.

4 
Analysis of a reactiondiffusion system of λw typeGarvie, Marcus Roland January 2003 (has links)
The author studies two coupled reactiondiffusion equations of 'λw' type, on an open, bounded, convex domain Ω C R(^d) (d ≤ 3), with a boundary of class C², and homogeneous Neumann boundary conditions. The equations are close to a supercritical Hopf bifurcation in the reaction kinetics, and are model equations for oscillatory reactiondiffusion equations. Global existence, uniqueness and continuous dependence on initial data of strong and weak solutions are proved using the classical FaedoGalerkin method of Lions and compactness arguments. The work provides a complete case study for the application of this method to systems of nonlinear reactiondiffusion equations. The author also undertook the numerical analysis of the reactiondiffusion system. Results are presented for a fullypractical piecewise linear finite element method by mimicking results in the continuous case. Semidiscrete and fullydiscrete error estimates are proved after establishing a priori bounds for various norms of the approximate solutions. Finally, the theoretical results are illustrated and verified via the numerical simulation of periodic plane waves in one space dimension, and preliminary results representing target patterns and spiral solutions presented in two space dimensions.

5 
Divergence form equations arising in models for inhomogeneous materialsKinkade, Kyle Richard January 1900 (has links)
Master of Science / Department of Mathematics / Ivan Blank / Charles N. Moore / This paper will examine some mathematical properties and models of inhomogeneous
materials. By deriving models for elastic energy and heat flow we are
able to establish equations that arise in the study of divergence form uniformly elliptic
partial differential equations. In the late 1950's DeGiorgi and Nash
showed that weak solutions to our partial differential equation lie in the
Holder class.
After fixing the dimension of the space,
the Holder exponent guaranteed by this work depends only on
the ratio of the eigenvalues.
In this paper we will look at a specific geometry and show
that the Holder exponent
of the actual solutions is bounded away from
zero independent of the eigenvalues.

6 
Chaos in models of double convectionRucklidge, Alastair Michael January 1991 (has links)
No description available.

7 
A class of PetrovGalerkin finite element methods for the numerical solution of the stationary convectiondiffusion equationPerella, Andrew James January 1996 (has links)
A class of PetrovGalerkin finite element methods is proposed for the numerical solution of the n dimensional stationary convectiondiffusion equation. After an initial review of the literature we describe this class of methods and present both asymptotic and nonasymptotic error analyses. Links are made with the classical Galerkin finite element method and the cell vertex finite volume method. We then present numerical results obtained for a selection of these methods applied to some standard test problems. We also describe extensions of these methods which enable us to solve accurately for derivative values of the solution.

8 
Bargmann transform and its applications to partial differential equationsAl Asmer, Nabil Abed Allah Ali Jr January 2021 (has links)
This thesis is devoted to the fundamental properties and applications of the Bargmann
transform and the FockSegalBargmann space. The fundamental properties include unitarity
and invertibility of the transformation in L2 spaces and embeddings of the FockSegalBargmann spaces in Lp for any p>0. Applications include the linear partial differential
equations such as the timedependent Schrödinger equation in harmonic potential,
the diffusion equation in selfsimilar variables, and the linearized Kortewegde Vries equation, and one nonlinear partial differential equation given by the GrossPitaevskii model
for the rotating BoseEinstein condensate. The main question considered in this work in
the context of linear partial differential equation is whether the envelope of the Gaussian
function remains bounded in the time evolution. We show that the answer to this question
is positive for the diffusion equation, negative for the Schrödinger equation, and unknown
for the Kortewegde Vries equation. We also address the local and global wellposedness
of the nonlocal evolution equation derived for the BoseEinstein condensates at the lowest
Landau level. / Thesis / Master of Science (MSc)

9 
Strominger's system on nonKähler hermitian manifoldsLee, Hwasung January 2011 (has links)
In this thesis, we investigate the Strominger system on nonKähler manifolds. We will present a natural generalization of the Strominger system for nonKähler hermitian manifolds M with c₁(M) = 0. These manifolds are more general than balanced hermitian manifolds with holomorphically trivial canonical bundles. We will then consider explicit examples when M can be realized as a principal torus fibration over a Kähler surface S. We will solve the Strominger system on such construction which also includes manifolds of topology (k−1)(S²×S⁴)#k(S³×S³). We will investigate the anomaly cancellation condition on the principal torus fibration M. The anomaly cancellation condition reduces to a complex MongeAmpèretype PDE, and we will prove existence of solution following Yau’s proof of the Calabiconjecture [Yau78], and Fu and Yau’s analysis [FY08]. Finally, we will discuss the physical aspects of our work. We will discuss the Strominger system using α'expansion and present a solution up to (α')¹order. In the α'expansion approach on a principal torus fibration, we will show that solving the anomaly cancellation condition in topology is necessary and sufficient to solving it analytically. We will discuss the potential problems with α'expansion approach and consider the full Strominger system with the Hull connection. We will show that the α'expansion does not correctly capture the behaviour of the solution even up to (α')¹order and should be used with caution.

10 
Existence of Critical Points for the GinzburgLandau Functional on Riemannian ManifoldsMesaric, Jeffrey Alan 19 February 2010 (has links)
In this dissertation, we employ variational methods to obtain a new existence result for solutions of a GinzburgLandau type equation on a Riemannian manifold. We prove that if $N$ is a compact, orientable 3dimensional Riemannian manifold without boundary and $\gamma$ is a simple, smooth, connected, closed geodesic in $N$ satisfying a natural nondegeneracy condition, then for every $\ep>0$ sufficiently small, $\exists$ a
critical point $u^\ep\in H^1(N;\mathbb{C})$ of the GinzburgLandau functional \bd\ds E^\ep(u):=\frac{1}{2\pi \ln\ep}\int_N \nabla u^2+\frac{(u^21)^2}{2\ep^2}\ed
and these critical points have the property that $E^\ep(u^\ep)\rightarrow\tx{length}(\gamma)$ as $\ep\rightarrow 0$.
To accomplish this, we appeal to a recent general asymptotic minmax theorem which basically says that if $E^\ep$ $\Gamma$converges to $E$ (not necessarily defined on the same Banach space as $E^\ep$), $v$ is a saddle point of $E$ and some additional mild hypotheses are met, then there exists $\ep_0>0$ such that for every $\ep\in(0,\ep_0),E^\ep$ possesses a critical point $u^\ep$ and $\lim_{\ep\rightarrow 0}E^\ep(u^\ep)=E(v)$.
Typically, $E$ is only lower semicontinuous, therefore a suitable notion of saddle point is needed.
Using known results on $\mathbb{R}^3$, we show the GinzburgLandau functional $E^\ep$ defined above $\Gamma$converges to a functional $E$ which can be thought of as measuring the arclength of a limiting singular set. Also, we verify using regularity theory for almostminimal currents that $\gamma$ is a saddle point of $E$ in an appropriate sense.

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