SCHEDULE OF TALKS

TITLES AND ABSTRACTS

Global Solvability, Blow Up, and Positivity for a Degenerate Diffusion Model with Transmission Boundary Conditions
Jeffrey Anderson
Department of Mathematics
Winona State University

A model proposed for tumor-induced angiogenesis incorporates degenerate diffusion and nonlinear flux conditions, where a portion of the flux is determined according to a differential equation on the boundary. We discuss results on local and global solvability, as well as blow up in finite time. In developing the theory, various conditions on the nonlinear terms have been introduced in previous work which are sufficient for the existence of nonnegative solutions. By considering a related model, with flux determined by a linear differential equation on the boundary, we investigate the necessity of these conditions toward a better understanding of the behavior of solutions in cases when positivity may not be satisfied. Extensions to a situation where the flux is determined according to a diffusion equation on the boundary are also discussed.

Boundary Value Problems for Div-Curl Systems
Giles Auchmuty
Department of Mathematics
University of Houston

Div-curl systems constitute an overdetermined system of first order linear differential equations - so they may be solved subject to a variety of different classes of boundary conditions. In viscous fluid mechanics, they typically have 3 boundary conditions imposed at each point of the boundary, in electrostatics there generally are 2 boundary conditions given while for inviscid fluids and magnetostatics there is just one boundary condition.
 
This talk will describe variational principles and outline existence-uniqueness results for these different problems. In particular, the number of holes and handles in the domain affects the data that must be prescribed to have well-posed electromagnetic boundary value problems.

Flocculating Infectious Bacteria: Smoluchowski and Sepsis
David M. Bortz
Applied Mathematics
University of Colorado

Klebsiella pneumoniae and Staphylococcus epidermidis are the most common causes of intravascular catheter infections, potentially leading to bacteremia. These bloodstream infections dramatically increase the mortality of illnesses and often serve as an engine for sepsis. Our current model for the dynamics of the size-structured population of aggregates in a flowing system is based on the Smoluchowski coagulation equations, which we are using to study properties of these flocculating bacteria.
 
In this talk, I will discuss the progress of several investigations into properties of our model equations as well as the comparison with data. In particular, I will focus on the derivation of an alternative fragmentation kernel in laminar flow as well as the post-fragmentation aggregate size distribution.

The Variational Approach to Fracture
Blaise Bourdin
Department of Mathematics
Louisiana State University

Most widely accepted theories for modeling brittle fracture are based on Griffith's criterion and limited to the propagation of an isolated, pre-existing crack along a given path. The variational formulation, proposed by G. Francfort and J.-J. Marigo extends Griffith's theory into a global minimization principle, while preserving its essence, the concept of energy restitution in between surface and bulk terms. In its current state, it involves successive minimizations of a total energy with respect to any admissible displacement and crack field. The main advantage of this approach is to be capable of predicting the initiation of new cracks, computing their path, and accounting the interactions between several cracks, in two and three space dimensions. Of course, this has a price both theoretically and numerically. In particular, in order to achieve global minimization with respect to any crack set, one has to devise special numerical methods, typically based on the idea of approximation of functionals in the sense of the Gamma-convergence.
 
After a brief presentation of the variational model, I will describe its numerical implementation and highlight the challenges it poses. I will present various large scale numerical experiments. Time permitting, I will present some extensions of the model, and their requirements in terms of numerical implementation.

The Method of Particular Solutions for Solving Berger Plate Equation
C.S. Chen
Coauthor: Goungming Yao
Department of Mathematics
University of Southern Mississippi

In this presentation, we focus on the derivation of closed form particular solution for Berger equation with large deformation in the context of radial basis functions. The Berger equation is given as follows:
$(\Delta^2 - \lambda^2)u = f(x,y)$,
where $\Delta = \partial^2 / \partial x^2 + \partial^2 / \partial y^2$ and $\lambda$ is a constant. Radial basis functions have been used to approximate $f(x,y)$. An approximate particular solution can be obtained using elementary algebra. Berger equation is widely used in the study of the plate bending in engineering. Two numerical examples are given to demonstrate the effectiveness of the method.

Mathematical Models and their Numerical Schemes for Thermal Analysis in an N-Carrier System
Weizhong Dai
Department of Mathematics
Louisiana Tech University

In this study, we extend the concept of the well-known parabolic and hyperbolic two-step models for micro heat transfer to the case of energy exchanges in a generalized N-carrier system with heat sources. We show that the models for thermal analysis in the multi-carrier system satisfy energy estimates and hence they are well-posed. Based on these results, finite difference schemes are then developed for solving the parabolic and hyperbolic models for the multi-carrier system. The schemes are shown to satisfy a discrete analogue of the corresponding energy estimate, implying that they are unconditionally stable. Finally, the schemes are tested by an example. The difference between the hyperbolic model and the parabolic model for a multi-carrier system is also compared.

Nonlinear Acoustic Phenomena in Viscous, Thermally Relaxing Fluids: Shock Bifurcation and the Emergence of Diffusive Solitons
P. M. Jordan
Naval Research Laboratory
Stennis Space Center, Mississippi

In this talk we will consider the propagation of finite-amplitude acoustic waves in fluids that exhibit both viscosity and thermal relaxation. Under the assumption that the thermal flux vector is given by the Maxwell-Cattaneo law, which is a well known generalization of Fourier's law that includes the effects of thermal inertia, we derive the weakly nonlinear equation of motion in terms of the acoustic potential. We then use singular surface theory to determine how an input signal in the form of a shock wave evolves over time, and for different values of the Mach number. Then, numerical methods are used to illustrate our analytical findings. In particular, it is shown that the shock amplitude exhibits a transcritical bifurcation; that a stable, nonzero equilibrium solution is possible; and that a Taylor shock (i.e., a diffusive soliton), in the form of a "tanh" profile, can emerge from the input shock wave. Finally, an application related to the kinematic-wave theory of traffic flow is noted.

Dynamics of a Discontinuous Discrete Model of West Nile-Like Epidemics
Vlajko L. Kocic
Mathematics Department
Xavier University of Louisiana
New Orleans, LA

The system of non-autonomous nonlinear difference equations models the spread of the West Nile Encephalitis. The disease is transmitted by mosquitoes to both birds and humans; mosquitoes can be infected only from birds; infected birds and infected humans can recover, while infected mosquitoes can not recover. The system of difference equations models the effects of the West Nile-Like Virus on populations of birds, humans, and mosquitoes. The model includes the effects of spraying to reduce the population of mosquitoes as a main tool for control of the epidemics. In the case when the spraying function (actually the kill rate of mosquitoes) is a step function of mosquito population size, the epidemics model becomes discontinuous and it exhibits complex dynamics. Some extensions of this model to hybrid systems will also be discussed.

An Inverse Problem for Identification of the Coefficient in Euler-Bernoulli Equation
Tchavdar T. Marinov
Coauthor: Aghalaya S. Vatsala
Department of Natural Sciences
Southern University at New Orleans

In the present work we have displayed the performance of technique called Method of Variational Imbedding for solving the inverse problem of coefficient identification in Euler-Bernoulli equation from over-posed data. The original inverse problem is replaced by a minimization problem. The Euler-Lagrange equations for minimization comprise an eight-order equation for the solution of the original equation and an explicit equation for the unknown coefficient. It is shown that the solution of the original inverse problem is among the solutions of the variational problem, i.e., the inverse problem is imbedded into a higher-order but well posed boundary value problem. The imbedding problem possesses a unique solution which means that when the imbedding functional is zero, the over-posed data is consistent and the solution of the imbedding problem coincides with the sought solution of the inverse problem. Featuring examples are elaborated numerically with different coefficients through solving the direct problem with given coefficient and preparing the over-posed boundary data for the imbedding problem. The numerical results confirm that the solution of the imbedding problem coincides with the exact solution of the original problem within the order of approximation error.

Comparison of Solutions of Model Evolution Equations
Michael Tom
Department of Mathematics
Louisiana State University

The solutions of the pure initial-value problems for some model evolution equations are compared to those of their regularized counterparts. It is shown that the solutions are the same to within the order of accuracy attributable to either model.

An Epidemic Model for Influenza with Fixed Time Post-Contact Prophylaxis Treatment
Abdessamad Tridane
Co-authors: Horst Thieme, and Yang Kuang
School of Applied Arts & Sciences
Arizona State University, Polytechnic Campus

Although some studies have provided information on the effective dosages of antiviral treatment and the appropriate length of treatment during winter epidemic influenza seasons, there is no clear understanding of the correlation between the duration of the antiviral treatment and virulence or the duration of treatment and the emergence of resistant mutants, which is one of the potential problems of antiviral treatment. In order to understand some of these issues, I will consider in this talk a delay differential equation model that incorporates the fixed duration of the antiviral treatment for a population of exposed individuals in contact with an infected individual but not infective with the influenza infection. This model is a continuation of previous studies on the impact of the treatment length distribution on the large-time behavior of the model solutions, namely whether the solutions converge to an equilibrium or whether they are driven into undamped oscillations. A special case of possible the switch of stability is also considered.

Riesz Potentials and Solutions of Korteweg-de Vries Equation
Vladimir Varlamov
Department of Mathematics
University of Texas Pan American

Riesz potentials are defined as fractional powers of Laplacian. They are well known for their role in studying solvability of equations of the Korteweg-de Vries type (KdV henceforth) and obtaining Lp - Lq estimates in fractional order Sobolev spaces. New fractional properties are established for the fundamental solution of Cauchy problem for the linearized KdV. The latter is expressed in terms of the Airy function of the first kind Ai(x). On the basis of this analysis new properties are obtained for solutions of KdV-type equations. As an example Riesz potentials are considered for the well known soliton solution of KdV. Various analytic properties are established for this family of functions, and in particular the zero mean property.

Estimating the Effectiveness of Anisotropic Insulators
Xuefeng Wang
Mathematics Department
Tulane University

Of concern is the thermal insulation ability of an anisotropic material (anisotropy means the thermal conductivity is direction- dependent). We propose to use the Dirichlet eigenvalues and eigenmodes, especially the principal ones, to measure the thermal insulation. More specifically, we propose to use the principal Dirichlet eigenvalue of the elliptic operator on the unit ball (occupied by the anisotropic material) as a simple thermal insulation measurement; numerically we obtain some user-friendly formulas for this Dirichlet eigenvalue in terms of the trace and determinant of the thermal tensor. We also study the scenario of protecting a thermal conductor (e.g., a space shuttle) from overheating by coating it with an insulator. We establish and prove some easy-to-use rules for the optimal thickness of the coating. We achieve this by studying the behavior of Dirichlet and Robin eigenvalues and eigenfunctions, as well as the heat equation itself, in the singular limit as the coating thickness approaches 0.

On Vibration of Some Materially Nonlinear Structures
Dongming Wei
Department of Mathematics
University of New Orleans

A class of nonlinear wave equations of p-Laplacian type is presented for modeling vibration of rods, beams, and plates made of heat treated metals that satisfy a nonlinear stress-strain power-law. These metals, also called Ludwick-type materials or Hencky plastics are commonly used in industrial applications. Numerical solutions are presented and compared with analytical solutions in some cases. Some traveling wave solutions are also presented to illustrate the qualitative feature of these models.

The Generalized Monotone Iterative Method for Impulsive Differential Systems with Applications
Ianna West
Department of Mathematics and Computer Science
Nicholls State University

Systems of differential equations with impulses can occur in the mathematical modeling of science and engineering. In this paper, we will develop the generalized monotone method for impulsive differential systems using upper and lower solutions. In our case, the forcing functions are sums of increasing and decreasing functions. We will use the method to solve an ecological model.