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.

Fluid power technology has been widely used in industrial practice; however, its energy efficiency became a big concern in the recent years. Much progress has been made to improve fluid power energy efficiency from many aspects. Among these approaches, using a valve-less system to replace a traditional valve-controlled system showed eminent energy reduction. This thesis studies the valve-less solution-pump displacement controlled actuators- from the view of controls background. Singular perturbations have been applied to the fluid power to account for fluid stiffness; and a novel hydraulic circuit for single rod cylinder has been presented to increase the hydraulic circuit stabilities. Recursive Least Squares has been applied to account for measurement noise thus the parameters have fast convergence rate, square root algorithm has further applied to increase the controller's numerical stability and efficiency. It was showed that this technique is consistent with other techniques to increase controller's robustness. The developed algorithm is further extended to a hybrid adaptive control scheme to achieve desired trajectory tracking for general cases. A hardware test-bed using the invented hydraulic circuit was built up. The experimental results are presents and validated the proposed algorithms and the circuit itself. The end goal of this project is to develop control algorithms and hydraulic circuit suitable for industrial practice.

We consider a collocation method for Cauchy singular integral equations on the interval based on weighted Chebyshev polynomials , where the coefficients of the operator are piecewise continuous. Stability conditions are derived using Banach algebra methods, and numerical results are given.

This paper is devoted to asymptotic estimates for the condition numbers $\kappa(T_n(a))=||T_n(a)|| ||T_n^(-1)(a)||$ of large $n\cross n$ Toeplitz matrices $T_N(a)$ in the case where $\alpha \element L^\infinity$ and $Re \alpha \ge 0$ . We describe several classes of symbols $\alpha$ for which $\kappa(T_n(a))$ increases like $(log n)^\alpha, n^\alpha$ , or even $e^(\alpha n)$ . The consequences of the results for singular values, eigenvalues, and the finite section method are discussed. We also consider Wiener-Hopf integral operators and multidimensional Toeplitz operators.

We consider a finite section (Galerkin) and a collocation method for Cauchy singular integral equations on the interval based on weighted Chebyshev polymoninals, where the coefficients of the operator are piecewise continuous. Stability conditions are derived using Banach algebra techniques, where also the system case is mentioned. With the help of appropriate Sobolev spaces a result on convergence rates is proved. Computational aspects are discussed in order to develop an effective algorithm. Numerical results, also for a class of nonlinear singular integral equations, are presented.

Die Arbeit beschäftigt sich mit Faltungsoperatoren auf Kegeln, die in Lebesgueräumen L^p(R^2) (1<p<\infty) von Funktionen auf der Ebene wirken. Es werden asymptotische Spektraleigenschaften der zugehörigen Finite Sections studiert. Im Falle p=2 (Hilbertraum) wird das Invertierbarkeitsproblem von Operatoren vom Faltungstyp auf Kegeln mit Hilfe der Methode der Standard-Modell-Algebren untersucht.

Singularly perturbed reaction-diffusion problems exhibit in general solutions with anisotropic features, e.g. strong boundary and/or interior layers. This anisotropy is reflected in the discretization by using meshes with anisotropic elements. The quality of the numerical solution rests on the robustness of the a posteriori error estimator with respect to both the perturbation parameters of the problem and the anisotropy of the mesh. The simplest local error estimator from the implementation point of view is the so-called hierarchical error estimator. The reliability proof is usually based on two prerequisites: the saturation assumption and the strengthened Cauchy-Schwarz inequality. The proofs of these facts are extended in the present work for the case of the singularly perturbed reaction-diffusion equation and of the meshes with anisotropic elements. It is shown that the constants in the corresponding estimates do neither depend on the aspect ratio of the elements, nor on the perturbation parameters. Utilizing the above arguments the concluding reliability proof is provided as well as the efficiency proof of the estimator, both independent of the aspect ratio and perturbation parameters.

This paper is the last part of a three-fold article aimed at some efficient numerical methods for solving the Poisson problem in three-dimensional prismatic and axisymmetric domains. In the first and second parts, the Fourier singular complement method (FSCM) was introduced and analysed for prismatic and axisymmetric domains with reentrant edges, as well as for the axisymmetric domains with sharp conical vertices. In this paper we shall mainly conduct numerical experiments to check and compare the accuracies and efficiencies of FSCM and some other related numerical methods for solving the Poisson problem in the aforementioned domains. In the case of prismatic domains with a reentrant edge, we shall compare the convergence rates of three numerical methods: 3D finite element method using prismatic elements, FSCM, and the 3D finite element method combined with the FSCM. For axisymmetric domains with a non-convex edge or a sharp conical vertex we investigate the convergence rates of the Fourier finite element method (FFEM) and the FSCM, where the FFEM will be implemented on both quasi-uniform meshes and locally graded meshes. The complexities of the considered algorithms are also analysed.

We extend our results on the degree of ill-posedness for linear integration opera- tors A with weights mapping in the Hilbert space L^2(0,1), which were published in the journal 'Inverse Problems' in 2005 ([5]). Now we can prove that the degree one also holds for a family of exponential weight functions. In this context, we empha- size that for integration operators with outer weights the use of the operator AA^* is more appropriate for the analysis of eigenvalue problems and the corresponding asymptotics of singular values than the former use of A^*A.

Thesis (Ph. D.)--University of Missouri-Columbia, 2001. / Typescript. Vita. Includes bibliographical references (leaves 136-138). Also available on the Internet.