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Contributed Papers Session / Communications libres
(Gail Wolkowicz, Organizer)

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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 advection-diffusion-reaction 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 ad-hoc 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 set-up to bridge the gap between theory and computations, and how idealized asymptotic test-cases can be used to validate popular ad-hoc 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. One-step 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 three-dimensional 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 to-date 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 E³ 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 (non-zero 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 E³ 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 activated-not resting-CD4+ 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 Cauchy-Poisson 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 1-dimensional Gierer-Meinhardt model

The Gierer-Meinhardt 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 N-spike equilibrium solution to the Gierer-Meinhardt activator-inhibitor model in a one-dimensional 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 N-spike 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, t > 0,  -¥ < a < ¥ p = ó õ +¥ -¥  w(a)u(a,t) da,   u ( a,0) = u0(a).

u(a,t) is the population density of individuals aged a at time t. m and b are age-specific 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 age-structured population equation can be written as the following abstract evolution equation in Banach space:

du dt = A(p[u])u

where for any t, u(t) Î L¹(R), p is a measure on L¹(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:

1. The choice of population measure can significantly affect the oscillating behavior of the population.
2. When p is chosen to be the special measure E the population goes to a steady state without any oscillations.
3. 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 non-wetting. The free-surface between the liquid and gas phases is handled by a front-tracking 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 sech²y velocity profile has been used by numerous authors to provide a good approximation to the wake behind a bluff body. It is well-known 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 three-dimensional 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 integro-differential equations, the solutions to which had a finite-time 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 three-dimensional spectral (Fourier) method to perform direct numerical simulations of the Bickley jet. In our simulations, we will examine how small three-dimensional 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, 507-526.

[Mallier R. and Haslam M., 1999]  Interactions between pairs of oblique waves in a Bickley jet. European J. Mech. B Fluids 18, 227-243.

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 cross-section 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 Navier-Stokes 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 quasi-statically. 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 fine-scale texture of the concentration field a four-parameter 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 n-dimensional 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 user-friendly 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 Bodganov-Takens 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 Bodganov-Takens singularity. In 1992, Baider and Sanders developed grading functions on the basis of Lie Algebra to study the unique Bodganov-Takens 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 Bodganov-Takens 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 Bodganov-Takens 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. User-friendly computer software is being developed which can be applied to ``automatically'' compute the simplest normal form for a given dynamical system associated with the Bodganov-Takens 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 multiple-delayed 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 Bogdanov-Takens bifurcations for this system are discussed using the center manifold analysis.