TITLES AND ABSTRACTS

A Comparison of Three Different Stochastic Population Models With Regard to Persistence Time
L. J. S. Allen and E. J. Allen
Texas Tech University

Results are summarized from the literature on three commonly-used stochastic population models with regard to persistence time. In addition, several new results are introduced to clearly illustrate similarities between the models. Specifically, the relations between the mean persistence time and higher-order moments for discrete-time Markov chain models, continuous-time Markov chain models, and stochastic differential equation models are compared for populations experiencing demographic variability. Similarities between the models are demonstrated analytically, and computational results are provided to show that estimated persistence times for the three stochastic models are generally in good agreement when the models are consistently formulated. As an example, the three stochastic models are applied to a population satisfying logistic growth. Logistic growth is interesting as different birth and death rates can yield the same logistic differential equation. However, the persistence behavior of the population is strongly dependent on the explicit forms for the birth and death rates. Computational results demonstrate how dramatically the mean persistence time can vary for different populations that experience the same logistic growth.

Estimates and local existence of solutions of a two-dimensional angiogenesis model
Jeffrey R. Anderson
Winona State University

Recently, a model for the growth of new capillaries (angiogenesis) in the vicinity of a developing solid tumor has been introduced and studied numerically. The model is set in a two-dimensional space, which is meant to represent the space between an existing capillary and the tumor, with a set of one-dimensional differential equations at one side of the boundary in order to simulate cell kinetics in and transfers through the capillary wall. It incorporates degenerate diffusion of growth factors, mean curvature evolution for collagen (or fibronectin), and a reinforced random walk for endothelial cell growth and migration. Numerical investigations have shown the model predicts behavior in remarkable agreement with known laboratory results. In this presentation, we introduce the model, discuss a notion of weak solvability, and derive the primary estimates used to establish a local existence theory.

Spectral Hermite Approximations and Transient Growth for the Actively Mode-Locked Laser
Kelly Black and John B. Geddes
University of New Hampshire

An approximation technique for the governing equations for the mode-locked laser is examined. The technique centers on a transformation of the governing equations in which the resulting equations closely resemble the Hermite equation. The approximation of the system is constructed through a linear combination of Hermite polynomials resulting in a Hermite-spectral method. The rate of decay of the resulting modes is examined for a simplified problem and difficulties in showing the stability of the method are also discussed. Numerical comparisons with a finite difference scheme are also presented.

A Matrix Decomposition Method for Solving Large Scale Partial Differential Equations Using Radial Basis Functions.
C.S. Chen
University of Nevada, Las Vegas
and
A. Karageorghis and Y.-S. Smyrlis
University of Cyprus

Radial basis functions (RBFs) are effective tools in multivariate function approximation. In recent years, RBFs have been extended to solve partial differential equations (PDEs). Despite many attractive features of RBFs, the resultant matrix is full and dense and thus costly to solve, in particular, for large scale problems which often require ten of thousand of interpolation points. In this talk, we propose an efficient algorithm using matrix decomposition for the evaluation of the particular solutions of elliptic PDEs. The collocation points are placed on concentric circles and the resulting matrix has a block circulant structure. We exploit this circulant structure to develop an efficient algorithm for the solution of the resulting system.

Some Recent Studies of Wave Phenomena in Continuum Mechanics
P. M. Jordan
Naval Research Laboratory
Stennis Space Center

In this talk, we explore some recent topics of interest in wave propagation. We do so in the context of physical problems from continuum mechanics that involve shear (or transverse) waves, compressional (or longitudinal) waves, and kinematic waves. Specifically, the following three topics will be considered: (1) Instant steady-state and other remarkable features of dipolar fluids, (2) Diffusive solitons and bifurcations in nonlinear acoustics, and (3) Shock and acceleration waves in a traffic flow model with relaxation. Employing both analytical and numerical methods, we carry out this investigation with the purpose of gaining a better understanding of, and deeper insight into, the physical phenomena represented in the mathematical models. (Work supported by ONR/NRL funding.)

Probabilities, intervals, what next? Extension of interval computations to situations with partial information about probabilities.
Vladik Kreinovich
University of Texas at El Paso

When we have only interval ranges [xi-,xi+] of sample values x1,...,xn, what is the interval [V-,V+] of possible values for the variance V of these values? We prove that the problem of computing the upper bound V+ is NP-hard. We provide a feasible (quadratic time) algorithm for computing the lower bound V- on the variance of interval data. We also provide a feasible algorithm that computes V+ under reasonable easily verifiable conditions.
We also extend the main formulas of interval arithmetic for different arithmetic operations x1*x2 to the case when, for each input xi, in addition to the interval [xi]=[xi-,xi+] of possible values, we also know its mean Ei (or an interval [Ei] of possible values of the mean), and we want to find the corresponding bounds for x1*x2 and its mean.

Stochastic stability of inflation-unemployment process
G. S. Ladde
University of Texas at Arlington

In this work, by assuming a speed of adjustment of the rate of unemployment to be a Gaussian White noise as well as unanticipated changes to be Markov chain, stochastic mathematical models of inflation-unemployment processes are outlined. Furthermore, sufficient conditions are given to insure stability of inflation-employment processes in the quadratic mean as well as in the sense of almost sure. The presented study sheds a light on effects of random structural perturbations in the system. The study also exhibits robust stability with respect to both deterministic as well stochastic parameters of the system. This is achieved by the usage of comparison theorems in the context energy functions.

Various Aspects of the Shear Band Problem
W. E. Olmstead
Northwestern University

The mathematical model governing the formation of shear bands in high strength materials is a nonlinearly coupled parabolic-hyperbolic system of PDE's. Various aspects of this problem will be presented including: (i) treating the shear band as a boundary layer with discontinuous gradients (ii) numerical simulations using a boundary element approach, (iii) a singular perturbation analysis of a thermal inhomogeneity, (iv) multiple steady-states associated with the Arrhenius law of plastic deformation and (v) the use of maximum principles to establish energy localization. Also, some still open questions regarding the shear band problem will be mentioned.

A Nonstandrd Discretization Method that Preserves Stability
Lih-Ing Wu Roeger
Texas Tech University

Differences and similarities between differential equations and difference equations interest many people. Difference equations can be derived from differential equations through discretization methods. Sometimes, the stability is totally lost through the discretization process. The most standard discretization method is the Euler method. In my talk, I will discuss a nonstandard method by W. Kahan that discretizes the differential equation dx/dt=f(x) where f(x) is at most quadratic. I will explain why Kahan's method preserves local stability including Hopf bifurcations. I will also present some examples by applying Kahan's method to the classical Lotka-Volterra competition differential equations, especially the famous May-Leonard "rock-paper-scissors" competition model.

Hybrid Fuzzy Differential Equations on Time Scale
M. Sambandham
Morehouse College

Hybrid fuzzy dynamic systems on time scales are developed and practical stability of such systems is discussed by the application of Lyapunov-like functions and comparison principle.

Global Behavior of Solutions of a Nonautonomous Delay Logistic Difference Equation
D. Stutson, V.L. Kocic, and G. Arora
Xavier University

Our aim in this paper is to investigate the asymptopic behavior of solutions of the following difference equation x_{n+1}=(a_{n}x_{n})/(1+x_{n-k}), n=0,1,... where {a_{n}} is a positive bounded sequence and k is a positive integer. Sufficient conditions for boundedness and attractivity are obtained. The results are applied to the special case when {a_{n}} is a periodic sequence with prime period p.

Modeling Cerebral Blood Flow Control During Posture Change from Sitting to Standing
Hien T. Tran
N.C. State University

Hypertension, decreased cerebral blood flow, and diminished cerebral blood flow control, are among the first signs indicating the presence of cerebral vascular disease. In this talk, we will present our work on developing mathematical models for systemic blood pressure and cerebral blood flow control (auto- and baroreceptor regulation) during posture change from sitting to standing. The mathematical model is based on compartmental modeling describing the pulsatile blood flow and pressure in a number of compartment of the systemic arteries. These compartments include the upper body, the legs, and the brain. Physiologically based control mechanisms will be added to explore how arterial and cerebral blood pressure drop as a consequence of posture change from sitting to standing. The effect of time delays involving a delay for the onset of control as well as the duration of the control will also be presented. Finally, to justify the fidelity of our mathematical model and control mechanisms development, we will show validation results of our model against clinical data. This is a joint work with Mette Olufsen (North Carolina State University), Johnny Ottesen (Roskilde University), and Lewis Lipsitz (Harvard Medical School)

On a nonlinear heat equation
Xuefeng Wang
Tulane University

Of concern is a heat equation with a power source term. The spatial domain is the whole space. The equation has a simple appearance but rich mathematical structure. I shall talk about recent results concerning stability of steady states, sufficient conditions for blow-up and asymptotic behavior of solutions as time goes to infinity.

Computational Theory and Methods for Solving Multiple Saddle Point Problems---Application to Eigenpair Problems of the p-Laplacian Operator
Jianxin Zhou
Texas A&M University

Many nonlinear boundary value problems can be reduced to solve an Euler-Lagrange equation for critical points. When multiple saddle points exist, the case becomes very difficult to handle numerically. Most results in the literature focus on the existence issue and are not for computational purpose.
In this lecture, the speaker will first introduce the computational theory and methods for solving multiple saddle point problems in Banach spaces. Then the speaker will study how to numerically compute eigenpairs of the nonlinear p-Laplacian operator. Several important theoretical and practical issues will be addressed.
Numerical eigenpairs of the p-Laplacian operator will be presented through their graphics. Several interesting phenomena have been observed.