In this thesis, we examine the role of accretion disks in the formation of stellar systems, focusing on young massive disks which regulate the flow of material from the parent molecular core down to the star. We study the evolution of disks with high infall rates that develop strong gravitational instabilities. We begin in chapter 1 with a review of the observations and theory which underpin models for the earliest phases of star formation and provide a brief review of basic accretion disk physics, and the numerical methods which we employ. In chapter 2 we outline the current models of binary and multiple star formation, and review their successes and shortcomings from a theoretical and observational perspective. In chapter 3 we begin with a relatively simple analytic model for disks around young, very massive stars, showing that instability in these disks may be responsible for the higher multiplicity fraction of massive stars, and perhaps the upper mass to which they grow. We extend these models in chapter 4 to explore the properties of disks and the formation of binary companions across a broad range of stellar masses. In particular, we model the role of global and local mechanisms for angular momentum transport in regulating the relative masses of disks and stars. We follow the evolution of these disks throughout the main accretion phase of the system, and predict the trajectory of disks through parameter space. We follow up on the predictions made in our analytic models with a series of high resolution, global numerical experiments in chapter 5. Here we propose and test a new parameterization for describing rapidly accreting, gravitationally unstable disks. We find that disk properties and system multiplicity can be mapped out well in this parameter space. Finally, in chapter 6, we address whether our studies of unstable disks are relevant to recently detected massive planets on wide orbits around their central stars.

Thesis (M.S.)--University of Missouri-Columbia, 2007. / The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file (viewed on March 19, 2008) Includes bibliographical references.

Thesis (Ph.D.)--University of Delaware, 2008. / Principal faculty advisor: Tobin A. Driscoll, Dept. of Mathematical Sciences. Includes bibliographical references.

Thesis (Ph. D.)--Technische Universiteit Delft, 1989. / Summary also in Dutch. "Stellingen" (3 p.) inserted. Vita. Includes bibliographical references.

Thesis (M.S.)--Nuclear and Radiological Engineering, Georgia Institute of Technology, 2008. / Committee Chair: Stacey, Weston M.; Committee Co-Chair: de Oliveira, Cassiano R.E.; Committee Member: Hertel, Nolan; Committee Member: van Rooijen, Wilfred F.G.

Thesis (Ph.D.)--University of Western Australia, 2007.

Thesis (Ph.D.)--Memorial University of Newfoundland, 2004. / Bibliography: leaves 170-183.

Thesis (Ph. D.)--University of Texas at Austin, 2007. / Vita. Includes bibliographical references.

Thesis (MTech (Mechanical Engineering))--Cape Peninsula University of Technology, 2009. / Includes bibliographical references (leaves 94-98). Also available online.

Kernel-based meshless methods for approximating functions and solutions of partial differential equations have many applications in engineering fields. As only scattered data are used, meshless methods using radial basis functions can be extended to complicated geometry and high-dimensional problems. In this thesis, kernel-based least-squares methods will be used to solve several direct and inverse problems. In chapter 2, we consider discrete least-squares methods using radial basis functions. A general l^2-Tikhonov regularization with W_2^m-penalty is considered. We provide error estimates that are comparable to kernel-based interpolation in cases in which the function being approximated is within and is outside of the native space of the kernel. These results are extended to the case of noisy data. Numerical demonstrations are provided to verify the theoretical results. In chapter 3, we apply kernel-based collocation methods to elliptic problems with mixed boundary conditions. We propose some weighted least-squares formulations with different weights for the Dirichlet and Neumann boundary collocation terms. Besides fill distance of discrete sets, our weights also depend on three other factors: proportion of the measures of the Dirichlet and Neumann boundaries, dimensionless volume ratios of the boundary and domain, and kernel smoothness. We determine the dependencies of these terms in weights by different numerical tests. Our least-squares formulations can be proved to be convergent at the H^2 (Ω) norm. Numerical experiments in two and three dimensions show that we can obtain desired convergent results under different boundary conditions and different domain shapes. In chapter 4, we use a kernel-based least-squares method to solve ill-posed Cauchy problems for elliptic partial differential equations. We construct stable methods for these inverse problems. Numerical approximations to solutions of elliptic Cauchy problems are formulated as solutions of nonlinear least-squares problems with quadratic inequality constraints. A convergence analysis with respect to noise levels and fill distances of data points is provided, from which a Tikhonov regularization strategy is obtained. A nonlinear algorithm is proposed to obtain stable solutions of the resulting nonlinear problems. Numerical experiments are provided to verify our convergence results. In the final chapter, we apply meshless methods to the Gierer-Meinhardt activator-inhibitor model. Pattern transitions in irregular domains of the Gierer-Meinhardt model are shown. We propose various parameter settings for different patterns appearing in nature and test these settings on some irregular domains. To further simulate patterns in reality, we construct different kinds of domains and apply proposed parameter settings on different patches of domains found in nature.