Ordinary Differential Equations and Applied Dynamics Commons^™

Steady Incompressible Magnetohydrodynamic Flow Near A Point Of Reattachment, J. M. Dorrepaal, S. Moosavizadeh

Mathematics & Statistics Faculty Publications

The oblique stagnation-point flow of an electrically conducting fluid in the presence of a magnetic field is a highly nonlinear problem whose solution is of interest even in the simplest of geometries. The problem models the flow of a viscous conducting fluid near a point where a separation vortex reattaches itself to a rigid boundary. A similarity solution exists which reduces the problem to a coupled system of four ordinary differential equations which can be integrated numerically. The problem has two independent parameters, the conductivity of the fluid and the strength of the magnetic field. Solutions are tabulated for a …

Lyapunov Exponents Of Linear Stochastic Functional-Differential Equations. Ii. Examples And Case Studies, Salah-Eldin A. Mohammed, Michael K. R. Scheutzow

Articles and Preprints

We give several examples and examine case studies of linear stochastic functional differential equations. The examples fall into two broad classes: regular and singular, according to whether an underlying stochastic semi-flow exists or not. In the singular case, we obtain upper and lower bounds on the maximal exponential growth rate $\overlineλ₁$(σ) of the trajectories expressed in terms of the noise variance σ . Roughly speaking we show that for small σ, $\overlineλ₁$(σ) behaves like -σ² /2, while for large σ, it grows like logσ. In the regular case, it is shown that a discrete Oseledec …

Droplet Evaporation And Deformations In An Amplitude Modulated Ultrasonic Field, Nihad E. Daidzic, Rene Stadler, Adrian Melling

Aviation Department Publications

The aim of the report presented is the measurements of droplet oscillations.

Series Solutions, Factorials, And The Gamma Function, Steven J. Wilson

Topics in Mathematics

Using a power series to solve an ordinary differential equation will often result in a solution whose general term involves a product of terms from an arithmetic sequence. This paper explores how factorials and the gamma function can be used to rewrite such general terms in a closed form.

Robot Space Coordinate Representation Of Objects In Euclidean Space, Justin Gallagher

Mathematical Sciences Technical Reports (MSTR)

Robot motion control strategies generally center around trajectory planning schemes which are point-to-point. This paper explores the problem of planning robot trajectories which sweep an area in a two-link robot's work space. A diffeomorphism which transforms the linear coordinates of Euclidean space to the non-linear angular coordinates which represent the displacements of the joint motors is developed. It is used to determine the distortion of an object's area at different locations in the robot's work space and for different robot link length geometries. Study of such distortions may lead to an optimization scheme by which the placement of the object …

Electrostatic Positioning Of Droplets In Turbulent Flows (Lstm 375/Te/93), Nihad E. Daidzic, Adrian Melling

Aviation Department Publications

Report LSTM 375/TE/93, Lehrstuhl fuer Stroemungsmechanik Universitaet Erlangen-Nuernberg Cauerstr. 4, 8520 Erlangen Germany.

A Bifurcation Theorem And Applications, Alfonso Castro, Jorge Cossio

All HMC Faculty Publications and Research

In this paper we give a sufficient condition on the nonlinear operator N for a point (λ, u) to be a local bifurcation point of equations of the form u + λL^-1(N(u)) = 0, where L is a linear operator in a real Hilbert space, L has compact inverse, and λ ∈ R is a parameter. Our result does not depend on the variational structure of the equation or the multiplicity of the eigenvalue of the linear operator L. Applications are made to systems of differential equations and to the existence of periodic solutions of nonlinear second order …

Some Stability Problems In Droplet Formation And Breakup (Report Lstm 351/T/92)., Nihad E. Daidzic

Aviation Department Publications

In this study the instability of droplets and cylindrical jets is investigated. The understanding of these processes has both academic and practical value. Instability of cylindrical jets is theoretically investigated for infinitesimal and finite, but small initial amplitudes (linear and nonlinear stability). For droplets, only linear theory is presented. It is assumed that the capillary force play a dominant role. It is determined that the viscosity exerts a damping effect. In the first section we give an introduction, after which the linear stability theory of cylindrical liquid jets is presented. In the third section the nonlinear jet stability theory is …

A Mathematical Model And Simulation Of Natural Circulation Loop Thermalhydraulics, Nihad E. Daidzic

Aviation Department Publications

In this work, a simple, nonlinear, seventh-order mathematical model with lumped parameter analysis of the thermalhydraulic processes in a natural circulation steam boiler loop is presented. Digital simulation has been performed in order to predict open loop transients in severe accidental situations. Parts of the circulation loop such as the steam drum, downcomer, and riser were modelled separately on the basis of the integral balance equations. The two-phase flow in the steam drum and in the riser was modelled with the aid of a homogeneous-equilibrium, two-phase flow model. Special care was taken to ensure the stability of the numerical scheme, …

A Generalization Of Linear Multistep Methods, Leon Arriola

Mathematics & Statistics Theses & Dissertations

A generalization of the methods that are currently available to solve systems of ordinary differential equations is made. This generalization is made by constructing linear multistep methods from an arbitrary set of monotone interpolating and approximating functions. Local truncation error estimates as well as stability analysis is given. Specifically, the class of linear multistep methods of the Adams and BDF type are discussed.

Nonnegative Solutions For A Class Of Radially Symmetric Nonpositone Problems, Alfonso Castro, Ratnasingham Shivaji

All HMC Faculty Publications and Research

We consider the existence of radially symmetric non-negative solutions for the boundary value problem

\begin{displaymath}\begin{array}{*{20}{c}} { - \Delta u(x) = \lambda f(u(x))\qua... ...\\ {u(x) = 0\quad \left\Vert x \right\Vert = 1} \\ \end{array} \end{displaymath}

where $ \lambda > 0,f(0) < 0$ (non-positone), $ f' \geq 0$ and $ f$ is superlinear. We establish existence of non-negative solutions for $ \lambda $ small which extends some work of our previous paper on non-positone problems, where we considered the case $ N = 1$. Our work also proves a recent conjecture by Joel Smoller and Arthur Wasserman.

Infinitely Many Radially Symmetric Solutions To A Superlinear Dirichlet Problem In A Ball, Alfonso Castro, Alexandra Kurepa

All HMC Faculty Publications and Research

In this paper we show that a radially symmetric superlinear Dirichlet problem in a ball has infinitely many solutions. This result is obtained even in cases of rapidly growing nonlinearities, that is, when the growth of the nonlinearity surpasses the critical exponent of the Sobolev embedding theorem. Our methods rely on the energy analysis and the phase-plane angle analysis of the solutions for the associated singular ordinary differential equation.

Existence And Uniqueness For A Variational Hyperbolic System Without Resonance, Peter W. Bates, Alfonso Castro

All HMC Faculty Publications and Research

In this paper, we study the existence of weak solutions of the problem

□u + ∇G(u) = f(t,x) ; (t,x) є Ω ≡ (0,π)x(0,π)

u(t,x) = 0 ; (t,x) є ∂Ω

where □ is the wave operator ∂²/∂t² - ∂²/∂x², G: Rⁿ→R is a function of class C² such that ∇G(0) = 0 and f:Ώ→R^n is a continuous function having first derivative with respect to t in (L₂,(Ω))ⁿ and satisfying

f(0,x) = f(π,x) = 0

for all x є [0,π].

Moment-Generating Functions And Laplace Transforms, James F. Miller

Journal of the Arkansas Academy of Science

No abstract provided.

Graphical Representations Of Singular Solutions Of Differential Equations, Kathryn Lois Pitman

Bachelors’ Theses

The first men to detect singular solutions of differential equations were Leibniz, Brook Taylor and Clairaut. The direct method of attack used by each man in attaining these solutions is not known, but a short history of each man and what he has contributed to mathematics can be given. Gottfried Wilhelm von Leibniz (1646-1716) was born in Leipsic, Germany, and was the son of a professor of law in a nearby university. He was well-educated in law himself, expecting to follow in the profession of his father. But not being under the necessity of earning his living, he enrolled at …