




Contributed Papers Session / Communications libres (Gail Wolkowicz, Organizer)
 ANNE BOURLIOUX, Université de Montréal, Montréal, Quebec
The mathematics behind the computation of turbulent flames

Premixed and diffusion flames propagate according to very different
physical mechanisms, yet both types can be described using the same
advectiondiffusionreaction equation. In the case of a turbulent
advection flow field with a wide range of length scales, brute force
computation is hopeless and practitioners have developed a number of
adhoc strategies to account for the effects of the small advection
scales on the flame propagation. On the other hand, there exists a
number of theoretical asymptotic results regarding the homogenization
of small scales that could be directly relevant to the practical
computational strategies: I will show how the large eddy
simulations framework is a good setup to bridge the gap between
theory and computations, and how idealized asymptotic testcases can be
used to validate popular adhoc models.
 CHRISTINA CHRISTARA, Department of Computer Science, University of Toronto, Toronto,
Ontario M5G 2A1
Fast solvers for quadratic spline collocation equations

Optimal spline collocation methods for elliptic Boundary Value Problems
(BVPs) have been relatively recently developed, offering an alternative
to Galerkin finite element methods as well as to Hermite spline
collocation methods. Fast solvers, though, for spline collocation
equations are still in the making. A variety of solvers has been
studied, including acceleration techniques with various preconditioners
and domain decomposition, but the analysis has been carried out for a
few only solvers and for restricted classes of PDE operators.
In this talk, we focus on a class of solvers for quadratic spline
collocation (QSC) equations based on Fast Fourier Transforms (FFTs).
We first develop FFT solvers for QSC equations arising from the
discretization of Helmholtz problems with constant coefficients or
problems with variable coefficients in one variable. The types of
boundary conditions that are handled include Dirichlet, Neumann,
periodic, as well as combinations of these. We then use the FFT
solvers as preconditioners to handle general PDE operators. Onestep
methods and acceleration techniques are studied. The Fourier methods
for QSC equations are compared to multigrid techniques recently
developed for the same equations and are shown to be very efficient
solvers and suitable for parallel computation. The FFT methods for QSC
equations have been extended to systems of elliptic PDEs and to
threedimensional PDEs.
 JOHN CLEMENTS, Department of Mathematics and Statistics and School of Biomedical
Engineering, Dalhousie University, Halifax, Nova Scotia B3H 3J5
Investigation of the equal anisotropy ratio assumption in the
bidomain model for a human ventricular myocardium

Ventricular tachyarrhythmias (VTs) are excessively fast aberrant
heart rhythms that interfere with the heart's ability to pump blood.
Precise analysis of the activation mechanism involved is essential for
successful treatment. To accomplish this, comprehensive, anatomically
accurate bidomain mathematical models of the propagated activation in
the human heart have been developed based on the equal anisotropy ratio
assumption that the conductivity tensors D_{i} and D_{e} which model
the intracellular and extracellular currents satisfy D_{e} = kD_{i} for
some constant k. This assumption simplifies the governing partial
differential equations and the numerical solution procedures required.
However, there is now evidence to suggest that this assumption may in
some cases generate inaccurate activation sequences and lead to
incorrect conclusions regarding the origin of a given VT. This talk
will explore the results obtained todate in analysing the propagation
properties of the myocardium when realistic (experimentally estimated)
unequal conductivity ratios are incorporated into the bidomain models.
 BORIS DEKSTER, Mount Allison University, Sackville, New Brunswick E4L 1E8
Each convex body in E^{3} symmetric about a plane can be illuminated
by 8 directions

Let C be a convex body in E^{d}, d ³ 2. Let x be a point on
¶C and v be a direction (nonzero vector). Consider the
axis l having the direction v and passing through x. The
direction v is said to illuminate x if l contains a point
y Î int C which succeeds x. If each point of a part of
¶C is illuminated by at least one of a few directions, the
body C is said to be illuminated by these directions. We prove the
following.
Theorem. Each convex body in E^{3} symmetric about a plane can be illuminated by 8 directions.
For polyhedral bodies, this result was established by K. Bezdek in
1991. Our method however is quite different. Both results are partial
proofs of the Hadwiger Conjecture according to which each convex body
in E^{d} can be illuminated by 2^{d} directions.
 DAVID EARN, Department of Mathematics and Statistics, McMaster University, Hamilton
Ontario L8S 4K1
A simple model for complex dynamical transitions in epidemics

Dramatic changes in patterns of epidemics have been observed
throughout the last century. For childhood infectious diseases such
as measles, the major transitions are between regular cycles and
irregular, possibly chaotic epidemics, and from regionally
synchronized oscillations to complex, spatially incoherent epidemics.
A simple model can explain both kinds of transitions as the
consequences of changes in birth and vaccination rates. Measles is a
natural ecological system that exhibits different dynamical
transitions at different times and places, yet all of these
transitions can be predicted as bifurcations of a single nonlinear
model.
 A.B. GUMEL, BENI SAHAI, AND P.N. SHIVAKUMAR, Department of Mathematics, University of Manitoba, Winnipeg,
Manitoba R3T 2N2
Numerical model for the dynamics between HIV and CD4+ T
cells in vivo

CD4+ T cells are the principal target and site of replication of HIV in
vivo. Such a replication cycle, however, can only be completed in
activatednot restingCD4+ T cells. A novel deterministic model
which predicts the population of CD4+ T cells with integrated HIV and
HIV particles in a typical untreated HIV patient will be presented and
analysed. The model also enables the populations of the uninfected CD4
T cells and the viral load to be monitored with time.
 RANIS IBRAGIMOV, Department of Mathematics, University of New Brunswick
Fredericton, New Brunswick E3B 5A3
On the tidal motion around the earth complicated by the
circular geometry of the ocean's shape

The CauchyPoisson free boundary problem on the stationary motion of a
perfect fluid around the Earth is considered in this paper. Since we
consider strictly longitudinal flow, such problem can be associated
with the 2D model to the tide which take the form of long gravity waves,
but they are complicated by the circular geometry of the ocean's
shape.
The main concern is to develop the method of the inverse conformal
mapping of the unknown free boundary in the hodograph plane onto some
fixed one in the physical domain.
Approximate solution to the problem is derived as the application of a
such method.
It is shown that one of the features of the positively curved bottom is
the fact that the problem admits two different higher order systems of
shallow water equations while the classical problem for the flat bottom
admits only system.
 DAVID IRON, University of British Columbia, Department of Mathematics,
Vancouver, British Columbia V6T 1Z2
The stability of spike solutions to the 1dimensional
GiererMeinhardt model

The GiererMeinhardt model uses a system of reaction diffusion equations
of activator inhibitor type to model cellular differentiation. It speculated
that spike type solutions to this sysem could explain the formation of locally
specilized cells in a group of identical cells.
The stability properties for an Nspike equilibrium solution to the
GiererMeinhardt activatorinhibitor model in a onedimensional
domain is studied asymptotically in the limit of small activator
diffusivity e. For a certain range of parameters in the model
and for N ³ 2 and e® 0, it is shown that there is a
critical value D_{N} of the inhibitor diffusivity D such that an
Nspike equilibrium solution is stable when D < D_{N} and is unstable
when D > D_{N}. A formula for the critical value D_{N}, which depends
on N but is independent of e, is given when N ³ 2.
 JUN LI AND S. DUBUC, Université de Montreal, Montreal, Quebec
Le pavage du plan par la coube de Lévy

En utilisant la courbe de Lévy, on peut paver le plan. Si l'on fait
tourner la courbe de Lévy autour de son point milieu selon un angle
multiple entier de 90 degrés, on obtient quatre courbes. Le
déplacement de ces quatre courbes par des vecteurs à composantes
entières donne un pavage du plan. Ce surprenant pavage du plan permet
de justifier le fait que la courbe de Lévy recouvre une aire positive
dont la valeur est 1/4 et qu'elle admet un intérieur non vide. Enfin,
nous verrons comment l'aire de la courbe de Lévy se distribue sur les
triangles de la triangulation obtenue par les droites x = m, y = n,
x±y = k, m,n,k Î Z.
 X. LI, C. ESSEX, AND M. DAVIDSON, Department of Applied Mathematics,
The University of Western Ontario, London, Ontario N6A 5B7
Oscillation and stability of age structured population

We are concerned with the relation between the stability of a
population and its age structure. The model we study is


¶u ¶a

+ 
¶u ¶t

= m(a,p)u(a,t)+f(a) 
ó õ

¥
0

b(a,p)u(a,t) da, 
  p = 
ó õ

+¥
¥

w(a)u(a,t) da, u ( a,0) = u_{0}(a). 

 

u(a,t) is the population density of individuals aged a at time
t. m and b are agespecific death and birth rates. f(a) is a density function which describes the variable gestation
period of the species. p is a linear functional of population
density in terms of a weight function w.
If we denote u(·,t) by u(t), the nonlinear agestructured
population equation can be written as the following abstract evolution
equation in Banach space:
where for any t, u(t) Î L^{1}(R), p is a measure on
L^{1}(R) and for fixed p, A(p) is a linear operator.
The nonlinear structured population dynamics model is characterized by
two aspects:
THE DAMPING MECHANISM. The dependence of A on p.
THE POPULATION MEASURE. The functional p. This arises from
the introduction of age structure.
An interesting question is: which of the two factors is the cause of
oscillations and instability? Our studies suggest the choice of
population measure rather than the nonlinear damping mechanism causes
the instability.
We found that, for any damping mechanism, there is always a possible
choice of population measure E, called the ``newborn equivalent
quantity'', which results in a global asymptotically stable
equilibrium. Moreover, when p is chosen to be E the size of the
population is a monotone function of time, showing no oscillations.
When p is chosen to be other than E, there will be oscillations,
whose extent is proportional to the difference between p and E.
Our conclusions are:
 The choice of population measure can significantly affect the
oscillating behavior of the population.
 When p is chosen to be the special measure E the population
goes to a steady state without any oscillations.
 The oscillation's extent is decided by the magnitude of the
difference between p and the special population measure.
These ideas may help us understand the intrinsic cause of population
oscillations, and to control population oscillations where, as in the
protection of endangered animals, human intervention is necessary.
 DONG LIANG, HUAXIONG HUANG, AND BRIAN WETTON, Department of Mathematics and Statistics,
York University, Toronto, Ontario M3J 1P3; Department of Mathematics and
Statistics, York University, Toronto, Ontario M3J 1P3; and Department of
Mathematics, University of British Columbia, Vancouver, British Columbia V6T
1Z2
Motion of a liquid drop on a partially wetting surface under shear

In this paper we study the motion of a liquid drop on a partially
wetting surface exposed to a shear flow. The objective of the study is
to investigate the behavior of the drop under the influence of flow
conditions of the external fluid and the wetting property of the flat
solid surface. In particular, whether the drop will stay attached to
the solid surface, or it will undergo sliding, rolling, or lifting
motion when the shear rate/velocity of the external fluid increases and
the solid surface is considered to be wetting or nonwetting. The
freesurface between the liquid and gas phases is handled by a
fronttracking method similar to the approach used by Unverdi and
Traggvason. The moving contact line is modeled by a slip velocity.
Instead of using a constant slip coefficient b as in Hockings'
original model, an implicit relation between b and the dynamic
contact angle q_{d} is established, based a local force balance.
Numerical examples are also given.
 ROLAND MALLIER, Department of Applied Mathematics, University of Western Ontario,
London, Ontario N6A 5B7
Modal interactions in a bickley jet by direct numerical simulation

The plane (Bickley) jet, which has a sech^{2}y velocity profile has
been used by numerous authors to provide a good approximation to the
wake behind a bluff body. It is wellknown that this flow possesses
both varicose and sinuous instability modes,
and in two recent papers (Mallier, 1996; Mallier & Haslam, 1999),
we used aysmptotic (nonlinear critical layer) techniques to
study analytically how a threedimensional disturbance to a Bickley jet
would develop if the disturbance were comprised of both the
varicose and sinuous modes, with the end result of both papers being
a set of highly nonlinear coupled integrodifferential equations,
the solutions to which had a finitetime singularity. Unfortunately,
at the time those studies were performed, there was little if any
experimental or numerical evidence to corroborate our analyis.
To address this, in this study, we use a threedimensional
spectral (Fourier) method to perform direct numerical simulations
of the Bickley jet.
In our simulations, we will examine how small threedimensional
disturbances consisting of both varicose and sinuous modes develop
with time, and we will discuss the connection with our earlier asymptotic
analysis.
References
[Mallier R., 1996] Fully coupled resonant triad interactions
in a Bickley jet. European J. Mech. B Fluids 15, 507526.
[Mallier R. and Haslam M., 1999] Interactions between pairs
of oblique waves in a Bickley jet. European J. Mech. B Fluids 18,
227243.
 SERGEY SADOV, Department of Mathematics, University of Manitoba, Winnipeg,
Manitoba R3T 2N2
A mathematical model of ice melting on transmission lines

During winters and ice storms, ice forms on high voltage electrical
transmission lines. This ice formation often results in downed lines
and has been responsible for considerable damage to life and property.
The model concerns melting of ice due to a higher current applied to
the transmission line. We consider a two dimensional crosssection
which contains four material layers: (i) transmission line, where the
Joulean heat is generated, (ii) water due to melting of ice, (iii) ice,
and (iv) atmosphere. Heat propagation and ice melting are put as
a Stefan like problem. The model takes into account gravity. This leads
to downward motion of ice and to forced convection in the water layer,
in addition to natural buoyancy driven convection. The convection is
described by the NavierStokes equation. The most intensive melting
occurs in a region near the top of the electrical wire. A very thin
layer of water carries weight of the ice shell due to big pressure
gradients. Big temperature gradients are also present there. In order
to make the model computationally tractable, we single out simplified
submodels and demonstrate estimations of melting time using values
obtained for those submodels treated quasistatically. The main
submodels are: (i) heat transfer and melting, assuming known velocity
field in the liquid, and (ii) boundary layer equations assuming known
melting rate and a geometry of ice/water frontier. We also discuss the
validity of physical assumptions, sensitivity to external boundary
conditions, and present numerical results.
This is a joint work with P. N. Shivakumar (Department of Mathematics,
University of Manitoba) and J. F. Peters (Department of Electrical and
Computer Engineering, University of Manitoba). Support by a grant from
Manitoba Hydro is acknowledged.
 PAUL J. SULLIVAN AND TOM SCHOPFLOCHER, University of Western Ontario, London, Ontario N6A 5B7 and Division
of Clinical Epidemiology, Department of Medicine, Montreal General
Hospital, Montreal, Quebec H3G 1A4
The PDF of scalar concentration in turbulent flows

The probability density function of concentration is important in
problems of combustion, toxicity and malador for example. In turbulent
flows the PDF equation is intractable and it is difficult to
accurately measure especially in environmental flows. The objective is
to devise a parametric form of the PDF and determine the parameters
from some few low ordered moments. The moment equations are less
intractable and moments are easier to measure than the PDF. Using
general physical constraints, and application of extreme value theory
in statistics and observations on the finescale texture of the
concentration field a fourparameter mixture PDF consisting of two
Beta PDFs is derived. The numerical solution of nonlinear algebraic
equations to determine the parameters from measured moments provides
satisfactory results. The double Beta PDF represents the
experimental PDFs well and somewhat better than a selection of
competing mixture PDFs that do not quite satisfy the constraints.
 W. YAO, C. ESSEX, AND P. YU, Department of Applied Mathematics, University of Western Ontario,
London, Ontario N6A 5B7
Analysis on a simple quiet standing model

A simple delayed, stochastic differential equation model is proposed to
describe human postural control during quiet standing. The model
includes only three parts: (1) white noise which destabilizes the
equilibrium state, (2) the inertial effects which accelerate the
destabilization process, and (3) delayed negative feedback controlling
the process. Some part of the nervous system participates in the last
part, and takes some time to detect, transmit and process the postural
information. Our analytical and numerical results obtained from the
model show that the model exhibits Hopf bifurcation. Under suitable
parameters, the result from the model approximately agrees with that
from the experimental data.
 PEI YU, Department of Applied Mathematics, University of Western Ontario,
London, Ontario N6A 5B7
Computation of simplest normal forms using a perturbation technique

We present a simple perturbation technique for computing the simplest
normal forms of vector fields, associated with certain singularities at
an equilibrium. The method can uniformly deal with a general
ndimensional system which is not necessarily described on a center
manifold. It can be used to systematically compute the explicit
expressions of unique normal forms as well as the corresponding
nonlinear transformations. The simple recursive procedure can be
easily implemented on a computer algebra system such as Maple to
develop userfriendly computer software. Examples chosen from
engineering problems are given to show the applicability of the
methodology and the efficiency of computer software.
 YUAN YUAN AND PEI YU, Department of Applied Mathematics, University of Western Ontario,
London, Ontario N6A 5B7
Computation of simplest normal forms of BodganovTakens
singularities

Normal form theory plays an important role in the study of dynamical
systems. The basic idea of normal forms is employing successive
coordinate transformations to construct a simpler form of the original
differential equations. Recently, more attention has been paid to the
simplest normal forms. This presentation is focused on the computation
of simplest normal form of BodganovTakens singularity. In 1992, Baider
and Sanders developed grading functions on the basis of Lie Algebra to
study the unique BodganovTakens normal form and classified three
cases. They solved two of the cases and the remaining one was solved
recently by Wang et al. using the same approach. However, their method
is not easy to be used for a symbolic computation in order to find
explicit expressions of the simplest normal form, though the approach
is elegant in theoretical proofs. It is also noticed that Algebra et
al. have used Lie Algebra to consider the computation of so called
``hypernormal form'' for the BodganovTakens singularity. Their results
provide a very detailed formula derivations, but it seems that their
approach cannot be straightforwardly implemented on a computer algebra
system such as Maple in a recursive manner. In this work we present an
efficient method, with the aid of Maple, which can be used to
systematically compute the simplest normal form of the BodganovTakens
singularity. The key step of our method is to find the relation between
the original differential equation and the simplest normal form through
an appropriate pattern of nonlinear transformations so that the
resulting normal form is the simplest. Userfriendly computer software
is being developed which can be applied to ``automatically'' compute
the simplest normal form for a given dynamical system associated with
the BodganovTakens singularity.
 HUAIPING ZHU, Department of Applied Mathematics, University of Waterloo,
Waterloo, Ontario N2L 3G1
Stability and bifurcation of a multiple delayed system of coupled
excitatory and inhibitory neurons

Consider a multipledelayed system of coupled neurons, one is excitatory and
the other is inhibitory. Conditions for the linear stability of the trivial
solution are given, and various of bifurcations, especially the Hopf and
BogdanovTakens bifurcations for this system are discussed using the center
manifold analysis.

