An artificial neural network (ANN) method was developed for solving boundary value problems (BVPs) on an arbitrary irregular domain in such a manner that all Dirichlet and/or Neuman boundary conditions (BCs) are automatically satisfied. Exact satisfaction of BCs is not available with traditional numerical solution techniques such as the finite element method (FEM). The ANN is trained by reducing error in the given differential equation (DE) at certain points within the domain. Selection of these points is significantly simpler than the often difficult definition of meshes for the FEM. The approximate solution is continuous and differentiable, and can be evaluated at any location in the domain independent of the set of points used for training. The continuous solution eliminates interpolation required of discrete solutions produced by the FEM. Reducing error in the DE at a particular location in the domain does not necessarily imply improvement in the approximate solution there. A theorem was developed, proving that the solution will improve whenever error in the DE is reduced at all locations in the domain during training. The actual training of ANNs reasonably approximates the assumptions required by the proof. This dissertation offers a significant contribution to the field by developing a method for solving BVPs where all BCs are automatically satisfied. It had already been established in the literature that such automatic BC satisfaction is beneficial when solving problems on rectangular domains, but this dissertation presents the first method applying the technique to irregular domain shapes. This was accomplished by developing an innovative length factor. Length factors ensure BC satisfaction extrapolate the values at Dirichlet boundaries into the domain, providing a solid starting point for ANN training to begin. The resulting method has been successful at solving even nonlinear and non-homogenous BVPs to accuracy sufficient for typical engineering applications.

I show that a class of semilinear Laplace-Beltrami equations has infinitely many solutions on the unit sphere which are symmetric with respect to rotations around some axis. This equation corresponds to a singular ordinary differential equation, which we solve using energy analysis. We obtain a Pohozaev-type identity to prove that the energy is continuously increasing with the initial condition and then use phase plane analysis to prove the existence of infinitely many solutions.

Call number: LD2668 .T4 1978 K34 / Master of Science

No description available.

Classically, fluid dynamics have been dealt with analytically because of the lack of numerical resources (Yeung, 1982). With the development of computational ability, many formulations have been developed which typically use the traditional Navier-Stokes equations along with an Eulerian grid. Today, there exists the possibility of using a moving grid (Lagrangian) along with a meshless discretization. The first issue in meshless fluid dynamics is the equations of motion. There are currently two types of Lagrangian formulations. Spherical Particle Hydrodynamics (SPH) is a method which calculates all equations of motion explicitly. The Moving Particle Semi-implicit (MPS) method uses a mathematical foundation based on SPH. However, instead of calculating all laws of motion explicitly, a fractional time step is performed to calculate pressure. A proposed method, Lagrange Implicit Fraction Step (LIFS), has been created which improves the mathematical formulations on the fluid domain. The LIFS method returns to Continuum mechanics to construct the laws of motion based on decomposing all forces of a volume. It is assumed that all forces on this volume can be linearly superposed to calculate the accelerations of each mass. The LIFS method calculates pressure from a boundary value problem with the inclusion of proper flux boundary conditions. The second issue in meshless Lagrangian dynamics is the calculation of derivatives across a domain. The Monte Carlo Integration (MCI) method uses weighted averages to calculate operators. However, the MCI method can be very inaccurate, and is not suitable for sparse grids. The Radial Basis Function (RBF) method is introduced and studied as a possibility to calculate meshless operators. The RBF method involves a solution of a system of equations to calculate interpolants. Machine expenses are shown to limit the viability of the RBF method for large domains. A new method of calculation has been created called Multi-dimensional Lagrange Interpolating Polynomials (MLIP). While Lagrange Interpolating Polynomials are essentially a one-dimensional interpolation, the use of "dimensional-cuts" and Gaussian quadratures can provide multi-dimensional interpolation. This paper is divided into three sections. The first section specifies the equations of motion. The second section provides the mathematical basis for meshless calculations. The third section evaluates the effectiveness of the meshless calculations and compares two fluiddynamic codes. / Fund number: N62271-97-G-0041. / US Navy (USN) author.

Approved for public release; distribution in unlimited. / Numerical ocean modeling usually composes various initial- and boundary-value problems. It integrates hydrodynamic and thermodynamic equations numerically with atmospheric forcing and boundary conditions (lateral and vertical) from initial states of temperature, salinity and velocity. Past observations, historical datasets and climatological datasets of the ocean have contributed greatly to the knowledge of the data fields of initial condition, atmospheric forcing and boundary conditions. Change in either initial or boundary condition leads to a variety of model solutions. It is necessary to specify realistic initial and boundary conditions to achieve better understanding and prediction of the ocean behavior. However, uncertainty often exists in both initial and boundary conditions. Up to now, most studies on ocean predictability have usually been for one particular type of model input uncertainty within the three types of uncertainty (initial conditions, open boundary conditions, atmospheric forcing function). This thesis investigates the response of ocean model to the three types of model input uncertainty simultaneously using Princeton Ocean Model (POM) implemented for the Japan/East Sea (JES). / Lieutenant Commander, Taiwan, R.O.C. Navy

Thesis. 1979. Ph.D.--Massachusetts Institute of Technology. Dept. of Ocean Engineering. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND ENGINEERING. / Bibliography: leaves 154-158. / Ph.D.

Thesis. 1977. M.S.--Massachusetts Institute of Technology. Dept. of Aeronautics and Astronautics. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND AERO. / Bibliography : leaves 51-53. / M.S.

Thesis. 1976. Ph.D.--Massachusetts Institute of Technology. Dept. of Ocean Engineering. / Microfiche copy available in Archives and Engineering. / Bibliography: leaves 83-84. / by Robert J. Van Houten. / Ph.D.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1980. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND ENGINEERING. / Vita. / Includes bibliographical references. / by Weng Cho Chew. / Ph.D.