Thesis (Ph. D.)--University of Washington, 2002. / Vita. Includes bibliographical references (p. 97-104).

Thesis (M.S)--Ohio University, June, 2005. / Includes bibliographical references (p. 55-56)

Thesis (M.S)--Ohio University, June, 2005. / Title from PDF t.p. Includes bibliographical references (p. 55-56)

A recent method for solving singular perturbation problems is examined. It is designed for the applied mathematician or engineer who needs a convenient, useful tool that requires little preparation and can be readily implemented using little more than an industry-standard software package for spreadsheets. In this paper, we shall examine singularly perturbed two point boundary value problems with the boundary layer at one end point. An initial-value technique is used for its solution by replacing the problem with an asymptotically equivalent first order problem, which is, in turn, solved as an initial value problem by using cubic splines. Numerical examples are provided to show that the method presented provides a fine approximation of the exact solution. The first chapter provides some background material to the cubic spline and boundary value problems. The works of several authors and a comparison of different solution methods are also discussed. Finally, some background into the specific singularly perturbed boundary value problems is introduced. The second chapter contains calculations and derivations necessary for the cubic spline and the initial value technique which are used in the solutions to the boundary value problems. The third chapter contains some worked numerical examples and the numerical data obtained along with most of the tables and figures that describe the solutions. The thesis concludes with some reflections on the results obtained and some discussion of the error bounds on the calculated approximations to the exact solutions for the numeric examples discussed

We prove that the solution of the Boussinesq equation with relatively small initial data exists globally and decays exponentially under some boundary conditions. / Ph. D.

El objetivo de esta tesis doctoral es desarrollar nuevos métodos basados en splines para la resolución de sistemas de ecuaciones diferenciales del tipo Y'(x)=f(x,Y(x)) , a<x<b Y(a)=Y_a (1) donde Y_a, Y(x) son matrices rxq, comenzando con splines de tipo cúbico y con un algoritmo similar al propuesto por Loscalzo y Talbot en el caso escalar [20], intentando poder aumentar el orden del spline, lo que con el método dado en [20] no puede hacerse de forma convergente. Trataremos también de aplicar dicho método al problema Y''(x)=f(x,Y(x),Y'(x)) , a<x<b Y(a)=Y_a Y'(a)=Y_b (2) sin aumentar la dimensión del problema para evitar el sobrecoste computacional. Los métodos presentados se compararán con los existentes en la literatura y serán implementados en algoritmos para ponerlos, debidamente documentados, a disposición de la comunidad científica. / Tung, MM. (2013). Spline approximations for systems of ordinary differential equations [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/31658 / Premios Extraordinarios de tesis doctorales

Wir betrachten ein System nichtlinearer parabolischer partieller Differentialgleichungen zur Modellierung des biologischen Phänomens Chemotaxis, das unter anderem in Aggregationsprozessen in Lebenszyklen bestimmter Einzeller eine wichtige Rolle spielt. Unser Chemotaxismodell benutzt Sensitivitäts funktionen, die die vorkommenden biologischen Prozesse genauer spezifizieren. Trotz der durch die Sensitivitätsfunktionen eingebrachten, zusätzlichen Nichtlinearitäten in den Gleichungen erhalten wir zeitlich globale Existenz von Lösungen für verschiedene biologisch realistische Klassen von Sensitivitätsfunktionen und können unter unterschiedlichen Bedingungen an die Systemdaten Konvergenz der Lösungen zu trivialen und nicht-trivialen stationären Punkten beweisen. / We consider a system of non-linear parabolic partial differential equations modeling chemotaxis, a biological phenomenon which plays a crucial role in aggregation processes in the life cycle of certain unicellular organisms. Our chemotaxis model introduces sensitivity functions which help describe the biological processes more accurately. In spite of the additional non-linearities introduced by the sensitivity functions into the equations, we obtain global existence of solutions for different classes of biologically realistic sensitivity functions and can prove convergence of the solutions to trivial and non-trivial steady states.

by Tam Yue Hung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2002. / Includes bibliographical references (leaves 49-52). / Abstracts in English and Chinese. / List of Tables --- p.vi / List of Figures --- p.vii / Chapter 1 --- Solving Systems of ODEs with BVMs --- p.1 / Chapter 1.1 --- Introduction --- p.1 / Chapter 1.2 --- Background --- p.4 / Chapter 1.2.1 --- Linear Multistep Formulae --- p.4 / Chapter 1.2.2 --- Preconditioned GMRES Method --- p.6 / Chapter 1.3 --- Strang-Type Preconditioners with BVMs --- p.7 / Chapter 1.3.1 --- Block-BVMs and Their Matrix Forms --- p.8 / Chapter 1.3.2 --- Construction of the Strang-type Preconditioner --- p.10 / Chapter 1.3.3 --- Convergence Rate and Operation Cost --- p.12 / Chapter 1.3.4 --- Numerical Result --- p.13 / Chapter 1.4 --- Strang-Type BCCB Preconditioner --- p.15 / Chapter 1.4.1 --- Construction of BCCB Preconditioners --- p.15 / Chapter 1.4.2 --- Convergence Rate and Operation Cost --- p.17 / Chapter 1.4.3 --- Numerical Result --- p.19 / Chapter 1.5 --- Preconditioned Waveform Relaxation --- p.20 / Chapter 1.5.1 --- Waveform Relaxation --- p.20 / Chapter 1.5.2 --- Invertibility of the Strang-type preconditioners --- p.23 / Chapter 1.5.3 --- Convergence rate and operation cost --- p.24 / Chapter 1.5.4 --- Numerical Result --- p.25 / Chapter 1.6 --- Multigrid Waveform Relaxation --- p.27 / Chapter 1.6.1 --- Multigrid Method --- p.27 / Chapter 1.6.2 --- Numerical Result --- p.28 / Chapter 1.6.3 --- Concluding Remark --- p.30 / Chapter 2 --- Restoration of Chopped and Nodded Images --- p.31 / Chapter 2.1 --- Introduction --- p.31 / Chapter 2.2 --- The Projected Landweber Method --- p.35 / Chapter 2.3 --- Other Numerical Methods --- p.37 / Chapter 2.3.1 --- Tikhonov Regularization --- p.38 / Chapter 2.3.2 --- MRNSD --- p.41 / Chapter 2.3.3 --- Piecewise Polynomial TSVD --- p.43 / Chapter 2.4 --- Numerical Result --- p.46 / Chapter 2.5 --- Concluding Remark --- p.47 / Bibliography --- p.49

This thesis is devoted to studies of initial-boundary value problems (IBVPs) for systems of partial differential equations (PDEs) arising from fluid mechanics modeling, especially for the compressible Euler equations with frictional damping, the Boussinesq equations, the Cahn-Hilliard equations and the incompressible density-dependent Navier-Stokes equations. The emphasis of this thesis is to understand the influences to the qualitative behavior of solutions caused by boundary effects and various dissipative mechanisms including damping, viscosity and heat diffusion.

This study seeks to develop a method that generalises the use of Adams-Bashforth to solve or treat partial differential equations with local and non-local differentiation by deriving a two-step Adams-Bashforth numerical scheme in Laplace space. The resulting solution is then transformed back into the real space by using the inverse Laplace transform. This is a powerful numerical algorithm for fractional order derivative. The error analysis for the method is studied and presented. The numerical simulations of the method as applied to the four-dimensional model, Caputo-Lu-Chen model and the wave equation are presented. In the analysis, the bifurcation dynamics are discussed and the periodic doubling processes that eventually caused chaotic behaviour (butterfly attractor) are shown. The related graphical simulations that show the existence of fractal structure that is characterised by chaos and usually called strange attractors are provided. For the Caputo-Lu-Chen model, graphical simulations have been realised in both integer and fractional derivative orders. / Mathematical Sciences / M. Sc. (Applied Mathematics)