In this paper results concerning a semilinear elliptic boundary value problem are proven. This problem has five solutions when the range of the derivative of the nonlinearity ƒ includes the first two eigenvalues. The existence and multiplicity or radially symmetric solutions under suitable conditions on the nonlinearity when Ω is a ball in R^N.

Results are presented for towing tank experiments of a surface-piercing flat plate with superimposed Stokes wave in order to examine free surface and wave effects on the boundary layer and wake. Measurements with servo wave gauges are made to characterize the Stokes-wave wave field in terms of its two-dimensionality, amplitude, and wavelength. Flow field measurements using stereo particle image velocimetry are used to identify the boundary layer and wake velocities. Particular attention is drawn to the juncture region to resolve the complex and poorly understood secondary flow patterns. Four test cases are presented (1) flat free surface without plate, (2) Stokes-wave without plate, (3) flat free surface with plate, and (4) Stokes-wave with plate; the cases were chosen in order to isolate and identify the performance of the velocimeter system, Stokes-wave flow field, free-surface effects, and combined Stokes-wave and free surface effects, respectively. All cases are conducted at Froude numbers of Fn = 0.4, length-based Reynolds number of Re = 1.64×106, and momentum thickness-based Reynolds number of about Re = 4000. Results show, as expected, that the free surface effects penetrate to a depth slightly greater than the boundary layer thickness and wave effects diminish at roughly one half the wavelength. The juncture region flow was resolved to levels that far exceed previous towing tank experiments, but leave more to be desired. The data and analysis are important, not only from a scientific perspective, but have a practical application with regard to development of turbulence models for computational fluid dynamic techniques.

We study boundary value problems for linear elliptic differential operators of order one. The underlying manifold may be noncompact, but the boundary is assumed to be compact. We require a symmetry property of the principal symbol of the operator along the boundary. This is satisfied by Dirac type operators, for instance. We provide a selfcontained introduction to (nonlocal) elliptic boundary conditions, boundary regularity of solutions, and index theory. In particular, we simplify and generalize the traditional theory of elliptic boundary value problems for Dirac type operators. We also prove a related decomposition theorem, a general version of Gromov and Lawson's relative index theorem and a generalization of the cobordism theorem.

In the ¡Krst chapter, separation of variables is used to derive the explicit particular solutions for a class of singularly perturbed di¤erential equations with constant coe¢ cients on a rectangular domain. Although only the Dirichlet boundary condition is taken into account; it can be similarly extended to other boundary conditions. Based on these results, the behavior of the solutions and their derivatives can be easily illustrated. Moreover, we have proposed a model with exact solution, which can be used to explore the behavior of layer and to test numerical methods. Hence, these analytic solutions are very important to the study in this ¡Keld. In the second chapter, we study the model of Shi¤ et al. [20]. It is a biharmonic equation on the rectangular domain [¡ a; a]£ [0; b] with clamped boundary condition. We compute its most accurate numerical solution by boundary approximation method (BAM), which is a special version of spectral method or collocation method. Its convergence unfortunately is not as good as the usual spectral method with exponential decay rate. We discover that the slowdown is due to the very mild singularity at two corners not considered by BAM. We further simplify the basis functions and their partial derivatives. Using these functions we can construct several models useful for testing numerical methods. We also explore how the stress intensity factor depends on the sizes of domain a and b, and the load ¸ by reducing the original problem with three parameters lambda, a, b to that with only one parameter t.

No description available.

No description available.

No description available.

No description available.

Thesis (M.App.Sc.) - Carleton University, 2002. / Includes bibliographical references (p. 221-225). Also available in electronic format on the Internet.

Thesis (Ph. D.)--University of Texas at Austin, 2004. / Supervisor: David B. Goldstein. Vita. Includes bibliographical references. Also available from UMI.