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An Elementary Review of the Mathieu-Hill Equation of Real Variable Based on Numerical Solutions

Description: Abstract: description is given of a large number of trajectories of Mathieu's equation made on the ENIAC in 1948 and available at BRL. A large chart describing the behaviour of the solutions is given. The occasion is taken to review the essentials of the Mathieu theory for the benefit of the occasional user-proceeding from the point of view of an inspection of the numerical data.
Date: April 1955
Creator: Zaroodny, Serge J.
Partner: UNT Libraries Government Documents Department

Similar Solutions for the Compressible Laminar Boundary Layer with Heat Transfer and Pressure Gradient

Description: Note applying Stewartson's transformation to the laminar compressible boundary-layer equations and the requirement of similarity, resulting in a set of ordinary nonlinear differential equations previously quoted by Stewartson, but unsolved. The requirements of the system are specified. It is then transformed to an integral system with the velocity ratio as the independent variable.
Date: February 1955
Creator: Cohen, Clarence B. & Reshotko, Eli
Partner: UNT Libraries Government Documents Department

Differential Equations in Airplane Mechanics

Description: In the following report, we will first draw some conclusions of purely theoretical interest, from the general equations of motion. At the end, we will consider the motion of an airplane, with the engine dead and with the assumption that the angle of attack remains constant. Thus we arrive at a simple result, which can be rendered practically utilizable for determining the trajectory of an airplane descending at a constant steering angle.
Date: March 1922
Creator: Carleman, M. T.
Partner: UNT Libraries Government Documents Department

Electronic Analog Computer Study of Effects of Motor Velocity and Driving Voltage Limits upon Servomechanism Performance

Description: The object of this thesis is (1) to demonstrate the value of an electronic analog computer for the solution of non-linear ordinary differential equations particularly when a large family of solutions is required; and (2) to obtain as a by-product results of practical applicability to servomechanism selection and analysis.
Date: 1956
Creator: Haynes, Joe Preston
Partner: UNT Libraries

On the Existence and Uniqueness of Solutions of Two Differential Equations

Description: The purpose of this paper is to study two differential equations. A method of approximation by iteration is used to define sequences of functions which converge to solutions of these equations. Some properties of the solutions are proved for general boundary conditions and certain special solutions are studied in detail.
Date: August 1965
Creator: Keath, Mary Katherine
Partner: UNT Libraries

Mathematical and numerical studies of nonstandard difference equation models of differential equations. Final technical report

Description: This report summarizes the complete research findings of the PI. Included are titles and places of publication of all journal, book, and conference papers, and abstracts. A listing of major conferences and meetings where these research results were reported is also provided.
Date: October 3, 2001
Creator: Mickens, Ronald E.
Partner: UNT Libraries Government Documents Department

Seismic imaging of reservoir flow properties: Time-lapse pressurechanges

Description: Time-lapse fluid pressure and saturation estimates are sensitive to reservoir flow properties such as permeability. In fact, given time-lapse estimates of pressure and saturation changes, one may define a linear partial differential equation for permeability variations within the reservoir. The resulting linear inverse problem can be solved quite efficiently using sparse matrix techniques. An application to a set of crosswell saturation and pressure estimates from a CO{sub 2} flood at the Lost Hills field in California demonstrates the utility of this approach. From the crosswell estimates detailed estimates of reservoir permeability are produced. The resulting permeability estimates agree with a permeability log in an adjacent well and are in accordance with water and CO{sub 2} saturation changes in the interwell region.
Date: April 8, 2003
Creator: Vasco, Don W.
Partner: UNT Libraries Government Documents Department

Time-periodic solutions of the Benjamin-Ono equation

Description: We present a spectrally accurate numerical method for finding non-trivial time-periodic solutions of non-linear partial differential equations. The method is based on minimizing a functional (of the initial condition and the period) that is positive unless the solution is periodic, in which case it is zero. We solve an adjoint PDE to compute the gradient of this functional with respect to the initial condition. We include additional terms in the functional to specify the free parameters, which, in the case of the Benjamin-Ono equation, are the mean, a spatial phase, a temporal phase and the real part of one of the Fourier modes at t = 0. We use our method to study global paths of non-trivial time-periodic solutions connecting stationary and traveling waves of the Benjamin-Ono equation. As a starting guess for each path, we compute periodic solutions of the linearized problem by solving an infinite dimensional eigenvalue problem in closed form. We then use our numerical method to continue these solutions beyond the realm of linear theory until another traveling wave is reached (or until the solution blows up). By experimentation with data fitting, we identify the analytical form of the solutions on the path connecting the one-hump stationary solution to the two-hump traveling wave. We then derive exact formulas for these solutions by explicitly solving the system of ODE's governing the evolution of solitons using the ansatz suggested by the numerical simulations.
Date: April 1, 2008
Creator: Ambrose , D.M. & Wilkening, Jon
Partner: UNT Libraries Government Documents Department

Multiscale integration schemes for jump-diffusion systems

Description: We study a two-time-scale system of jump-diffusion stochastic differential equations. We analyze a class of multiscale integration methods for these systems, which, in the spirit of [1], consist of a hybridization between a standard solver for the slow components and short runs for the fast dynamics, which are used to estimate the effect that the fast components have on the slow ones. We obtain explicit bounds for the discrepancy between the results of the multiscale integration method and the slow components of the original system.
Date: December 9, 2008
Creator: Givon, D. & Kevrekidis, I.G.
Partner: UNT Libraries Government Documents Department

ROTATION OF MERCURY: THEORETICAL ANALYSIS OF THE DYNAMICS OF A RIGID ELLIPSOIDAL PLANET

Description: The second-order nonlinear differential equation for the rotation of Mercury is shown to imply locked-in motion when the period is within the range (2T/3) [1-{lambda} cos 2{pi}t/T {+-} 2/3 (21{lambda}e/2){sup 1/2}], where e is the eccentricity and T the period of Mercury's orbit, the time t is measured from perihelion, and {lambda} = (B-A)/C measures the planet's distortion. For values near 2T/3, the instantaneous period oscillates about 2T/3 with period (21{lambda}e/2){sup -1/2}T.
Date: January 1, 1966
Creator: Laslett, L. Jackson & Sessler, Andrew M.
Partner: UNT Libraries Government Documents Department

Mathematical and Numerical Studies of Nonstandard Difference Equation Models of Differential Equations

Description: This research examined the following items/issues: the NSFD methodology, technical achievements and applications, dissemination efforts and research related professional activities. Also a list of unresolved issues were identified that could form the basis for future research in the area of constructing and analyzing NSFD schemes for both ODE's and PDE's.
Date: December 22, 2008
Creator: Mickens, Ronald E.
Partner: UNT Libraries Government Documents Department

The Evaluation of Definite Integrals, and a Quasi-Monte-Carlo Method Based on the Properties of Algebraic Numbers

Description: A formula is given for the approximate evaluation of multiple definite integrals using the ergodic property of a certain transformation of the unit cube into itself. Estimates of the rate of convergence are made for sufficiently smooth integrand. The work was motivated by a belief, that appeared at one time justified, that a substantial improvement of the accuracy of Monte Carlo method would result from use of the principles described herein. Although that belief proved groundless in numerical tests, it is deemed worthwhile to give this report of the work. Part of the theoretical development, the work of L.G. Peck, will be reported independently. (auth)
Date: October 13, 1951
Creator: Richtmyer, R. D.
Partner: UNT Libraries Government Documents Department

Performance and scaling of locally-structured grid methods forpartial differential equations

Description: In this paper, we discuss some of the issues in obtaining high performance for block-structured adaptive mesh refinement software for partial differential equations. We show examples in which AMR scales to thousands of processors. We also discuss a number of metrics for performance and scalability that can provide a basis for understanding the advantages and disadvantages of this approach.
Date: July 19, 2007
Creator: Colella, Phillip; Bell, John; Keen, Noel; Ligocki, Terry; Lijewski, Michael & Van Straalen, Brian
Partner: UNT Libraries Government Documents Department

AN IBM-7090 SUBROUTINE PACKAGE FOR LAGRANGIAN INTERPOLATION

Description: A FORTRAN subroutine was writteni for the purpose of generating a table of data by Lagrangian interpolation in a smaller table of data. There are three options for the interpolation scheme: linear, semilog, and log-log. The number of points to be used in the Lagrangian interpolation is optional. This subroutine provides information for a table lookup, which is usually much less costly than direct computation. (auth)
Date: May 16, 1963
Creator: Penny, S.K. & Emmett, M.B.
Partner: UNT Libraries Government Documents Department

Variable Metric Method for Minimization

Description: A method for determining numerically local minima of differentiable functions of several variables is presented. In the process of locating each minimum, a matrix which characterizes the behavior of the iunction about the minimum is determined. For a region in which the function depends quadratically on the variables, no more than N iterations are required, where N is the number of variables. By suitable choice of starting values and without modification of the procedure, linear constraints can be imposed upon the variables. (auth)
Date: November 1, 1959
Creator: Davidon, W. C.
Partner: UNT Libraries Government Documents Department