Browsing by Subject "Reaction-diffusion equations"
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Item Dynamics of boundary-controlled convective reaction-diffusion equations(Texas Tech University, 1995-05) Okasha, Nahed AThis research is concerned with an initial value boundary problem for a class of convective reaction-diffusion equations for which a feedback control law is implemented through the boundary conditions. This class contains, as a special case the well-known Burgers' equation which has been studied rather extensively. Using methods based on Functional Analysis, in particular, the energy method and Galerkin Approximations, solvability for the above class is established. In addition, we prove the global in time existence and the regularity of solutions of the controlled problem for sufficiently small L^2-initial data. To do this, additional explicit restrictions on the nonlinear terms are imposed. Then we prove the local Lyapunov stability of the system, the existence of an absorbing ball, and the existence of a compact local attractor in this ball. Similar results for the same equation with Dirichlet boundary conditions are obtained for arbitrary L^2-initial data. The solutions of the boundary-controlled problem are shown to depend continuously on the boundary control parameters. As these parameters tend to infinity, we prove that the trajectories of the boundary-controlled problem converge, uniformly on any finite interval, to the trajectories of the corresponding problem with Dirichlet boundary conditions.Item Reaction-diffusion fronts in inhomogeneous media(2006) Nolen, James Hilton; Xin, Jack; Souganidis, PanagiotisIn this thesis, we study the asymptotic behavior of solutions to the reaction-advection-diffusion equation ut = ∆zu + B(z, t) · ∇zu + f(u), z ∈ R n , t > 0 under various conditions on the prescribed flow B. Our goal is to characterize, bound, and compute the speed of propagating fronts that develop in the solution u and to describe their dependence on the flow B. We focus mainly on the case when f is the KPP nonlinearity f(u) = u(1 − u). In the first section, we consider the case that B is a temporally random field having a spatial shear structure and Gaussian statistics. We show that the solution to the initial value problem develops traveling fronts, almost surely, which are characterized by a deterministic variational principle. In the second section, we use this and other variational principles to derive analytical estimates on the speed of propagating fronts. In the final section, we use the variational principle to compute the front speed numerically. The mathematical analysis involves perturbation expansions, ergodic theorems, and techniques from the theory of large deviations. We use numerical methods for computing the principal Lyapunov exponents of parabolic operators, which appear in the variational characterization of the front speed.Item Remote quantitative transport and imaging investigations of small fluorescent molecular probes in an interstitial tissue model(Texas Tech University, 1998-08) Houlne, Michael PatrickThe diffusive transport characteristics of a unique class of small fluorescent molecular probes in an interstitial tissue model are investigated using micro-endoscopy. The probes employed in the present work are organo-metallic complexes of polyazamacrocycles chelated to Terbium. These particular molecules have large Stoke's shifts, making them amendable to tissue analysis. The delocalized electronic structure of the organic chelate absorbs ultra-violate light (~270 nm) and, after inter-molecular transfer, the lanthanide cation fluoresces in the visible region (550 nm). The diffusive transport properties of the probe molecules are related to their chemical structure, which governs their affinity toward the components of the interstitial model. The basic polyazamacrocyde is functionalized with three phosphate groups. Presently, methyl, ethyl, propyl and butyl alkyl chains are added to the phosphate groups on the polyazamacrocyde to modify the affinity of the probes toward the components of the interstitial model. The interstitial tissue model is constructed by preparing a Type I collagen gel in phosphate buffer solution. Known quantities of the probe are injected into the gel and the resulting diffusive transport of the probe is digitally imaged through a micro-endoscope as a function of time. Microendoscopy coupled with digital imaging allows remote, quantitative analysis of the transport process in near real time. Cross sectional analysis of the images yields the concentration profile of the probe as it diffuses through the gel. The concentration profile is fit to Fick's second law of diffusion to determine the diffusion coefficient for each of the probe molecules.Item Singular limits of reaction diffusion equations of KPP type in an infinite cylinder(2007) Carreón, Fernando; Souganidis, TakisIn this thesis, we establish the asymptotic analysis of the singularly perturbed reaction diffusion equation [cataloger unable to transcribe mathematical equations].... Our results establish the specific dependency on the coefficients of this equation and the size of the parameter [delta] with respect to [epsilon]. The analyses include equation subject to Dirichlet and Neumann boundary conditions. In both cases, the solutions u[superscript epsilon] converge locally uniformally to the equilibria of the reaction term f. We characterize the limiting behavior of the solutions through the viscosity solution of a variational inequality. To construct the coefficients defining the variational inequality, we apply concepts developed for the homogenization of elliptic operators. In chapter two, we derive the convergence results in the Neumann case. The third chapter is dedicated to the analysis of the Dirichlet case.Item Superconvergence of convection-diffusion equations in two dimensions(Texas Tech University, 1999-12) Moran, Daniel L.This thesis studies the convergence of a singularly perturbed two-dimensional problem of the convection-diflfusion type. The problem is solved using the bilinear finite element method on a Shishkin Mesh. This thesis will consider the results of two separate types of Shishkin Meshes, as well as a quick consideration of the uniform mesh and its shortcomings. Results will show a superconvergence rate close to 0 using a discrete energy norm. Results will also consider stability of the method by examining the condition number of the element stiffness matrix.