In this dissertation, we employ variational methods to obtain a new existence result for solutions of a Ginzburg-Landau type equation on a Riemannian manifold. We prove that if $N$ is a compact, orientable 3-dimensional 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 Ginzburg-Landau functional \bd\ds E^\ep(u):=\frac{1}{2\pi |\ln\ep|}\int_N |\nabla u|^2+\frac{(|u|^2-1)^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 Ginzburg-Landau 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 almost-minimal currents that $\gamma$ is a saddle point of $E$ in an appropriate sense.

There are three distinct processes that are predominant in models of flowing media with interacting components: advection, reaction, and diffusion. Collectively, these processes are typically modelled with partial differential equations (PDEs) known as advection-reaction-diffusion (ARD) equations.<p> To solve most PDEs in practice, approximation methods known as numerical methods are used. The method of lines is used to approximate PDEs with systems of ordinary differential equations (ODEs) by a process known as semi-discretization. ODEs are more readily analysed and benefit from well-developed numerical methods and software. Each term of an ODE that corresponds to one of the processes of an ARD equation benefits from particular mathematical properties in a numerical method. These properties are often mutually exclusive for many basic numerical methods.<p> A limitation to the widespread use of more complex numerical methods is that the development of the appropriate software to provide comparisons to existing numerical methods is not straightforward. Scientific and numerical software is often inflexible, motivating the development of a class of software known as problem-solving environments (PSEs). Many existing PSEs such as Matlab have solvers for ODEs and PDEs but lack specific features, beyond a scripting language, to readily experiment with novel or existing solution methods. The PSE developed during the course of this thesis solves ODEs known as initial-value problems, where only the initial state is fully known. The PSE is used to assess the performance of new numerical methods for ODEs that integrate each term of a semi-discretized ARD equation. This PSE is part of the PSE pythODE that uses object-oriented and software-engineering techniques to allow implementations of many existing and novel solution methods for ODEs with minimal effort spent on code modification and integration.<p> The new numerical methods use a commutator-free exponential Runge-Kutta (CFERK) method to solve the advection term of an ARD equation. A matrix exponential is used as the exponential function, but CFERK methods can use other numerical methods that model the flowing medium. The reaction term is solved separately using an explicit Runge-Kutta method because solving it along with the diffusion term can result in stepsize restrictions and hence inefficiency. The diffusion term is solved using a Runge-Kutta-Chebyshev method that takes advantage of the spatially symmetric nature of the diffusion process to avoid stepsize restrictions from a property known as stiffness. The resulting methods, known as Integrating-factor-based 2-additive Runge-Kutta methods, are shown to be able to find higher-accuracy solutions in less computational time than competing methods for certain challenging semi-discretized ARD equations. This demonstrates the practical viability both of using CFERK methods for advection and a 3-splitting in general.

The qualitative behavior of solutions of differential equations mainly addresses the various questions arising in the study of the long run behavior of solutions. The contents of this thesis are related to three of the major problems of the qualitative theory, namely the stability, the boundedness and the periodicity of the solution. Learning the qualitative behavior of such solutions is crucial part of the theory of differential equations. It is important to know if a solution is bounded or unbounded or if a solution is stable. Moreover, the periodicity of a solution is also of great significance for practical purposes.

Ball State University LibrariesLibrary services and resources for knowledge buildingMasters ThesesThere is no abstract available for this thesis.

Ball State University LibrariesLibrary services and resources for knowledge buildingMasters ThesesThere is no abstract available for this thesis.

Ball State University LibrariesLibrary services and resources for knowledge buildingMasters ThesesThere is no abstract available for this thesis.

In this thesis, it is contemplated to study the asymptotic behaviour of the solutions of the differential equationz2d2y + z(bo +b1zm ) dy +(co + c1zm + c2 z2m)y =0 (1) dz2 dzHere m is a positive integer, the variable z is regarded complex as are the constants bi(i=0,1) and ci(i=0,1,2) with c2 ≠ 0. Then in the language of Fuch's theory the differential equation (1) will have a regular singular point at z=0 and an irregular singular point at z=θ. The indicial equation about z=0 is found to beh(h-1) + boh + co = 0(2)It is also assumed that the difference of the roots of (2) are incongruent to zero modulo m.Ball State UniversityMuncie, IN 47306

In this dissertation, we employ variational methods to obtain a new existence result for solutions of a Ginzburg-Landau type equation on a Riemannian manifold. We prove that if $N$ is a compact, orientable 3-dimensional 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 Ginzburg-Landau functional \bd\ds E^\ep(u):=\frac{1}{2\pi |\ln\ep|}\int_N |\nabla u|^2+\frac{(|u|^2-1)^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 Ginzburg-Landau 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 almost-minimal currents that $\gamma$ is a saddle point of $E$ in an appropriate sense.

Under the assumption of mass-action kinetics, systems of chemical reactions can give rise to a wide variety of dynamical behaviour, including stability of a unique equilibrium concentration, multistability, periodic behaviour, chaotic behaviour, switching behaviour, and many others. In their canonical papers, M. Feinberg, F. Horn and R. Jackson developed so-called Chemical Reaction Network theory which drew a strong connection between the topological structure of the reaction graph and the dynamical behaviour of mass-action systems. A significant amount of work since that time has been conducted expanding upon this connection and fleshing out the theoretical underpinnings of the theory. In this thesis, I focus on three topics within the scope of Chemical Reaction Network theory. 1. Linearization: It is known that complex balanced systems possess within each invariant space of the system a unique positive equilibrium concentration and that that concentration is locally asymptotically stable. F. Horn and R. Jackson determined this through the use of an entropy-like Lyapunov function. In Chapter 4, I approach this problem through the alternative approach of linearizing the mass-action system about its equilibrium points. I show that this approach reproduces the results of F. Horn and R. Jackson and has the advantage of being able to give explicit exponential bounds on the convergence near equilibria. 2. Persistence: A well-known limitation of the theory is that the stabilities of the positive equilibrium concentrations guaranteed are locally limited. The conjecture that the equilibrium concentrations of complex balanced systems are global attractors of their respective invariant spaces has become known as the Global Attractor Conjecture and has received significant attention recently. This theory has been significantly aided by the realization that trajectories not tending toward the set of positive equilibria must tend toward the boundary of the positive orthant; consequently, persistence is a sufficient condition to affirm the conjecture. In Chapter 5, I present my contributions to this problem. 3. Linear Conjugacy: It is known that under the mass-action assumption two reaction networks with disparate topological structure may give rise to the same set of differential equations and therefore exhibit the same qualitative dynamical behaviour. In Chapter 6, I expand the scope of networks which exhibit the same behaviour to include ones which are related by a non-trivial linear mapping. I have called this theory Linear Conjugacy theory. I also show how networks exhibiting a linear conjugacy can be found using the mixed integer linear programming (MILP) framework introduced by G. Szederkenyi.

In this thesis, we consider the two-space homogenization method, which produces macroscopic expressions out of descriptions of the behaviour of the microstructure. Specifically, we focus on its application to poroelastic media. After describing the method, we provide examples to demonstrate that the resultant expressions are equivalent to an explicit derivation, which might not always be possible, and to outline the method for proving that the expressions converge to their macroscopic equivalents. Upon providing the basis for this method, we follow Burridge and Keller’s work for using this to prove the existence of Biot’s consolidation equations for poroelastic media and to provide expressions for the derivation of the parameters of these equations from the microstructure [5]. We then discuss the benefits and challenges that arise from this formulation of Biot’s consolidation equations.