<p>This thesis is concerned with singularly perturbed two-parameter problems. We study a tted nite difference method as applied on two different meshes namely a piecewise mesh (of Shishkin type) and a graded mesh (of Bakhvalov type) as well as a tted operator nite di erence method. We notice that results on Bakhvalov mesh are better than those on Shishkin mesh. However, piecewise uniform meshes provide a simpler platform for analysis and computations. Fitted operator methods are even simpler in these regards due to the ease of operating on uniform meshes. Richardson extrapolation is applied on one of the tted mesh nite di erence method (those based on Shishkin mesh) as well as on the tted operator nite di erence method in order to improve the accuracy and/or the order of convergence. This is our main contribution to this eld and in fact we have achieved very good results after extrapolation on the tted operator finitete difference method. Extensive numerical computations are carried out on to confirm the theoretical results.</p>

<p>This thesis is concerned with singularly perturbed two-parameter problems. We study a tted nite difference method as applied on two different meshes namely a piecewise mesh (of Shishkin type) and a graded mesh (of Bakhvalov type) as well as a tted operator nite di erence method. We notice that results on Bakhvalov mesh are better than those on Shishkin mesh. However, piecewise uniform meshes provide a simpler platform for analysis and computations. Fitted operator methods are even simpler in these regards due to the ease of operating on uniform meshes. Richardson extrapolation is applied on one of the tted mesh nite di erence method (those based on Shishkin mesh) as well as on the tted operator nite di erence method in order to improve the accuracy and/or the order of convergence. This is our main contribution to this eld and in fact we have achieved very good results after extrapolation on the tted operator finitete difference method. Extensive numerical computations are carried out on to confirm the theoretical results.</p>

Thesis (M.S.)--Ohio University, November, 2008. / Title from PDF t.p. Includes bibliographical references.

Thesis (M.Sc. (Dept. of Mathematics, Faculty of Natural Sciences))--University of the Western Cape, 2009. / Bibliography: leaves 180-195.

Magister Scientiae - MSc / This thesis is concerned with singularly perturbed two-parameter problems. We study a tted nite difference method as applied on two different meshes namely a piecewise mesh (of Shishkin type) and a graded mesh (of Bakhvalov type) as well as a tted operator nite di erence method. We notice that results on Bakhvalov mesh are better than those on Shishkin mesh. However, piecewise uniform meshes provide a simpler platform for analysis and computations. Fitted operator methods are even simpler in these regards due to the ease of operating on uniform meshes. Richardson extrapolation is applied on one of the tted mesh nite di erence method (those based on Shishkin mesh) as well as on the tted operator nite di erence method in order to improve the accuracy and/or the order of convergence. This is our main contribution to this eld and in fact we have achieved very good results after extrapolation on the tted operator finitete difference method. Extensive numerical computations are carried out on to confirm the theoretical results. / South Africa

The method of singular-perturbation analysis is applied to the determination of range-fuel-time optimal aircraft trajectories. The problem is shown to break down into three sub-problems which are studied separately. In particular, the inner layer containing the altitude path-angle dynamics is analyzed in detail. The outer solutions are discussed in an earlier work. As a step forward in solving the ensuing nonlinear two-point boundary-value problem, linearization of the equations is suggested. Conditions for the stability of the linearized boundary-layer equations are discussed. Also, the question of parameter selection to fit the solution to the split boundary conditions is resolved. Generation of feedback laws for the angle-of-attack from the linear analysis is discussed. Finally, the techniques discussed are applied to a numerical example of a missile. The linearized feedback solution is compared to the exact solution obtained using a multiple shooting method. / Master of Science

The primary purpose of this study is to develop a nonlinear feedback control law for the F / A-I8 fighter aircraft that performs a fast half-loop maneuver. This feedback law is developed using a singular perturbation approach. A secondary purpose of this study is to establish a baseline for time optimal half-loop maneuvers. The singular perturbation approach makes it possible to develop a state feedback control law which rotates the velocity vector through one hundred and eighty degrees at a maximum equilibrium pitch rate with a nearly constant angle of attack. The response of the aircraft to the control law is compared to simulations of half-loop maneuvers generated at NASA Langley Research Center. / Master of Science

In this thesis, we study the normally elliptic singular perturbation problems including both finite and infinite dimensional cases, which could also be non-autonomous. In particular, we establish the existence and smoothness of O(1) local invariant manifolds and provide various estimates which are independent of small singular parameters. We also use our results on local invariant manifolds to study the persistence of homoclinic solutions under weakly dissipative and conservative per- turbations. We apply Semi-group Theory and Lyapunov-Perron Integral Equations with some careful estimates to handle the O(1) driving force in the system so that we can approximate the full system through some simpler limiting system. In the investigation of homoclinics, a diagonalization procedure and some normal form transformation should be first carried out. Such diagonalization procedure is not trivial at all. We discuss this issue in the appendix. We use Melnikov type analysis to study the weakly dissipative case, while the conservative case is based on some energy methods. As a concrete example, we have shown rigrously the persistence of homoclinic solutions of an elastic pendulum model which may be affected by damping, external forcing and other potential fields.

In this thesis, a reversible Lotka-Volterra model was proposed to study the nonlinear oscillation of the Belousov-Zhabotinsky(BZ) reaction in a closed isothermal chemical system. The reaction zone can be divided into two zones, oscillation zone and transition zone, where the oscillation time and the transition time and the number of the complete oscillations are estimated. By applying the geometric singular perturbation method, it was proved that there exist an oscillation axis in the oscillation zone, a strongly stable two-dimensional invariant manifold in transition zone, on which there is also a one-dimensional stable invariant manifold, which is the part of the central axis. There is no oscillation in the vicinity of the equilibrium, as indicated by Onsager's reciprocal symmetry relation. Furthermore, the damped oscillation is studied in terms of the action-action-angle variables. In the end, the model reference control technique is employed to control the oscillation amplitude in the BZ reaction.

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