# York University Applied and Industrial Mathematics Seminar

## Past Conferences

DYNAMICS day, on March 27, 2003, York University

MINI-SYMPOSIUM ON
Evolution Equations: Theory, Numerics and Applications,
on March 4-5, 2002, York University

## Past Seminars

Friday, Nov. 21, 2003

TITLE: Asset Allocation and Annuitization

SPEAKER: Virginia R. Young
Department of Mathematics, University of Michigan
(Joint work with Moshe Milevsky, Schulich School of Business, York University)

ABSTRACT:

In this talk, we find the optimal investment, consumption, and annuitization policies of a utility-maximizing retiree facing a random time of death. We use techniques from optimal control to discover that the individual will buy annuities in order to keep wealth to one side of a barrier in wealth-annuity space--a type of barrier-control. In the region of no-annuity purchasing, we obtain a partial differential equation that determines the maximum utility. In the time-homogeneous case, we obtain an explicit solution to the problem and present a numerical example.

TIME: 3:00p.m.-4:000p.m., Friday, Nov. 21.

PLACE: N638 Ross

Friday, Nov. 7, 2003

TITLE: On a Two-Point Boundary Value Problem With Spurious Solutions

SPEAKER: Dr. C. H. Ou, York University

ABSTRACT:

The singularly perturbed problem $\varepsilon \frac{d^{2}u}{dx^{2}}+Q(u)=0$ with various boundary conditions at $x=\pm 1$ is studied from a rigorous point of view where $\varepsilon$ is a small parameter. Known solutions obtained from the method of matched asymptotics are shown to approximate true solutions with an exponentially small error estimate. The so-called spurious solutions turn out to be approximations of true solutions, when the locations of their internal layers, spikes'' or shocks'', are properly assigned. An estimate is also given for the maximum number of spikes'' or shocks'' that these solution can have.

TIME: 3:00p.m.-4:000p.m., Friday, Nov. 7.

PLACE: N638 Ross

Friday, Oct. 24, 2003

TITLE: PATTERN FORMATION IN NONLOCAL PHASE TRANSITIONS MODELS

ABSTRACT:

A classical way of modeling phase transition phenomena (e.g., in spin systems, binary materials, superconductors, etc.) is to consider an appropriate free energy Helmholtz functional and its local minimizers. This type of approach dates back to van der Waals, and its popularity is growing, as manifested by this years' Nobel prize in physics, 1/3 awarded to V.L. Ginzburg. In this talk, I will discuss the original van der Waals' fully nonlocal functional, and show that it admits a variety of periodic L^2 local minimizers. Depending on the absolute temperature of the system, these minimizers can have discontinuous interfaces. This is in sharp contrast to the local approximations of the fully nonlocal functional, which are often known as Ginzburg-Landau or Cahn-Hilliard models. In other words, this result apparently establishes a middle ground between the ambiguous local phase-field and free boundary models. Finally, which might sound unbelievable, it is not clear how to recover the above result numerically.

TIME: 3:00p.m.-4:000p.m., Friday, Oct. 24.

PLACE: N638 Ross

Thursday, Oct. 9, 2003

TITLE: A Recent Multi-scale Time-frequency Analysis and Its Biomedical Applications

SPEAKER: Dr. Hongmei Zhu
Seaman Family Magnetic Resonance Research Centre
University of Calgary

ABSTRACT:

Biomedical signals are typically finite duration, dynamic and non-stationary processes whose frequency characteristics vary over time or space. This often requires algorithms capable of locally analyzing and processing signals. The recently developed S-transform (ST) combines the time-frequency representation of the windowed Fourier transform with the multi-scale analysis of the wavelet transforms. Applying this transform to a temporal signal reveals information on what and when frequency events occur. In addition, its multi-scale analysis allows more accurate detection of subtle signal changes while interpretation in a time-frequency domain is easy to understand. Based on the ST, a series of adaptive time-frequency analysis techniques can be derived, which may provide valuable information for disease diagnosis and treatment. In this talk, we overview the theory of the ST and illustrate its effectiveness in de-noising and analyzing magnetic resonance imaging data.

TIME: 2:00p.m.-3:00p.m., Thursday, Oct. 9.

PLACE: N638 Ross

Tuesday, Oct. 7, 2003

TITLE: The bifurcation of Aero-elastic Airfoils with Structural Non-linearities

SPEAKER: Dr. Liping Liu
Duke University

ABSTRACT:

Classical works on aero-elasticity assume linear models for dynamics, aerodynamics, and structures. However, structural non-linearities arise from worn hinges of control surfaces, loose control linkages, material behavior and various other sources. Aging aircraft and combat aircraft that carry heavy external stores are more likely to be influenced by effects associated with nonlinear structures. An understanding of the nonlinear behavior of the system is crucial to the efficient and safe design of aircraft wings and control surfaces. There are three types of structural non-linearities: cubic spring, free-play and hysteresis. The principle interest for the aero-elastician is the asymptotic motion behavior (convergence, divergence, limit cycle oscillation) and the amplitude and frequency of the limit cycle oscillations. For a two-degree-of-freedom aero-elastic airfoil motion placed in an incompressible flow, by using the analytical techniques: the center manifold theory, the principle of normal form, the perturbation method, and the point transformation method, we accurately predict the nonlinear response. Various types of nonlinear response, damped, period-one, period-one with harmonics, period-two, period-two with harmonics, and chaotic motions are detected and the amplitudes and frequencies of limit cycle oscillations are predicted for the velocities beyond the linear flutter speed. In particular, a secondary bifurcation after the primary Hopf (flutter) bifurcation is detected for a cubic hard spring in the pitch degree of freedom. Furthermore, there is a hysteresis in the secondary bifurcation: starting from different initial conditions the motion may jump from one limit cycle to another at different flow velocities. Higher order harmonic balance method is employed to investigate the possible bifurcation branches. A more up to date model, a three-degree-of-freedom aero-elastic airfoil with control surface/free-play, will be introduced. The recent theoretical/experimental study will be briefly discussed, and the future topics will be presented.

TIME: 1:30p.m.-2:30p.m., Tuesday, Oct. 7.

PLACE: N638 Ross

Friday, Oct. 3, 2003

TITLE: High-Reynolds Number Solutions of Navier-Stokes Equations

SPEAKER: Dr. Rossitza S. Marinova
Enabled Simulation and Optimization Software

ABSTRACT:

The Navier-Stokes equations are a system of nonlinear, partial differential equations that describe the motion of a viscous, incompressible fluid. It is of importance to understand the long time behavior of solutions of these equations. The numerical treatment of high-Reynolds number viscous flows is of considerable interest for the applications because of the fact that the predominant part of the practically important flows take place either in large scales and high speeds or with small viscosity. My research has primarily focused on the problem of identifying the long-time behavior of solutions, as well as their asymptotic when the coefficient 1/Re of the highest-order derivatives approaches zero. In this talk I am going to present a new method for solving the incompressible Navier-Stokes equations and advection-diffusion equations. The main advantage of the method is that due to the economy of the computer time and memory required it is very efficient for solving multidimensional problems. Finally, I shall discuss my future research plans.

TIME: 1:00p.m.-2:00p.m., Friday, Oct. 3.

PLACE: N638 Ross

Friday, Sept. 26, 2003

TITLE: Affine stochastic differential equations with finite and infinite delay

SPEAKER: Markus Riedle, Humboldt University of Berlin

ABSTRACT:

Stochastic differential equations with finite delay have been intensively studied in the last years and fundamental results on the behaviour of their solutions were derived. But albeit deterministic equations with infinite delay are often encountered in applications, e.q. viscoelasticity and population dynamics, only a few work has so far been devoted to stochastic differential equations with infinite delay.

In this talk we introduce affine stochastic differential equations with both finite and infinite delay. After we have explained some differences between the solutions of the underlying deterministic differential equations with finite and infinite delay we present consequences of these differences for stochastic equations with infinite delay.

Treating equations with infinite delay often requires more sophisticated methods and techniques as the finite delay case. But on the other hand there exists a subclass of equations with infinite delay which can be reduced to ordinary differential equations without delay. We consider in detail the stochastic equations in this subclass. Moreover we establish that various linear hereditary models can be described by equations in this subclass.

TIME: 3:00p.m.-4:00p.m., Friday, Sept. 26.

PLACE: N638 Ross

Friday, Sept. 19, 2003

TITLE: Critical Role of Nosocomial Transmission in the Toronto SARS Outbreak

SPEAKER: Huaiping Zhu, York University

ABSTRACT:

We develop a compartmental mathematical model to address the role of hospitals in SARS transmission dynamics, which partially explains the heterogeneity of the epidemic. Comparison of the effects of two major policies, strict hospital infection control procedures and community-wide quarantine measures, implemented in Toronto two weeks into the initial outbreak, shows that their combination is the key to short-term containment and that quarantine is the key to long-term containment.
This is a joint wotk with Glenn Web, Matin Blaser, Sten Ardal, Jianhong Wu.

TIME: 3:00p.m.-4:00p.m., Friday, Sept. 19.

PLACE: N638 Ross

Friday, Sept. 12, 2003

TITLE: Fast algorithms for the electromagnetic scattering from a large cavity

SPEAKER: Weiwei Sun
Department of Mathematics City University of Hong Kong

ABSTRACT:

We present a fast algorithm for solving electromagnetic scattering from a rectangular open cavity embedded in an infinite ground plane. The medium inside the cavity is assumed to be (vertically) layered. By introducing a transparent (artificial) boundary condition, the problem in the open cavity is reduced to a bounded domain problem. A simple finite difference method is then applied to solve the model Helmholtz equation. The fast algorithm is designed for solving the resulting discrete system in terms of the discrete {\it Fourier transform} and a preconditioning conjugate gradient (PCG) method with a complex diagonal preconditioner for the indefinite interface system. The existence and uniqueness of the finite difference solution is proved for arbitrary wave numbers. Our numerical experiments for large numbers of mesh points, up to 16 million unknowns, and for large wave numbers, {\em e.g.}, between 100 and 200 wavelengths, show that the algorithm is extremely efficient. The cost for calculating the Radar Cross Section, which is of significant interest in practice, is $O(M^2)$ for an $M \times M$ mesh. The proposed algorithm may be extended easily to solve discrete systems from other discretization methods of the model problem.

TIME: 3:00p.m.-4:00p.m., Friday, Sept. 12.

PLACE: N638 Ross

Wednesday, July 16, 2003

TITLE: Crystal Growth from 3D to 1D

SPEAKER: C. Sean Bohun, Penn State University

ABSTRACT:

A semi-analytical approach for computing the temperature distribution and thermal stress inside an InSb crystal grown with the Czochralski technique is described. An analysis of the growing conditions indicates that the crystal growth occurs on the conductive time scale. A perturbation method for the temperature field is developed using the Biot number as a (small) expansion parameter whose zeroth order solution is one-dimensional (in the axial direction) and is obtained for a cylindrical and a conical crystal. For the growth conditions of InSb a parabolic temperature profile in the radial direction is shown to arise naturally as the first order correction. Both the quasi-steady and unsteady cases are solved for the crystal/melt system and various crucible profiles. Some issues relevant to growth conditions are also discussed. This is joint work with I. Frigaard, H. Huang, and S. Liang.

TIME: 2:30p.m.-3:30p.m., Wednesday, July 16.

PLACE: N638 Ross

Thursday, May 1, 2003

TITLE: Flexibility and the Molecular Conjecture: from combinatorics to proteins

SPEAKER: Walter Whiteley, York University

ABSTRACT:

We present the background for some algorithms for fast, combinatorial analysis of which sections of proteins (or other molecules) are rigid and which are flexible. We will also briefly present some illustrations from work on proteins, docking of ligands, etc. The algorithms are implemented in the program FIRST for protein analysis, available free (with an academic license agreement) on the web from Michigan State University.

The algorithms are based on ideas of 'generic rigidity of graphs' developed over several decades of mathematical work on the geometry and combinatorics of rigid structures. We will describe some of this mathematical background and a pair of key unsolved mathematical problem which we call The Molecular Conjectures.

Details of the algorithms and current work will be presented in a second seminar in the near future.

The work is part of a joint NIH funded grant with Leslie Kuhn (Biochemistry) and Michael Thorpe (BioPhysics) at Michigan State University.

TIME: 11:00a.m.-12:00p.m., Thursday, May 1st.

PLACE: 3009 Vari Hall

Friday, April 11th, 2003

TITLE: Study of Surface Roughness Evolution for Metal Electrodeposits Using Dynamic Scaling Analysis

SPEAKER: T.Jiang and S. Morin, Department of Chemistry, York University

ABSTRACT:

Tin electroplated coatings are widely used as corrosion protection layers in consumer packaging applications. The desirable smoothness and brightness of the coatings are a function of the grain-size and their micro- and nano-structures. Earlier studies demonstrated that analysis of the electrodeposited metal surface on a micrometer scale offers the potential to provide new insights into the metal deposition mechanism and provides a way of improving the quality of the metal deposits. In this work, atomic force microscopy (AFM) is used to characterize the morphology of the tin electrodeposits grown under various plating conditions. We are interested in the effects of plating solution composition, current density and plating temperature on the morphology and structure of these films. The surface roughness of the tin films grown using different conditions is evaluated by applying scaling analysis from the AFM images. The results show that the tin film surface displays self-affinity, and normal scaling behaviour is observed only when the brightener and stabilizer are both present in the plating electrolyte. This presentation will outline the methodology employed to perform the dynamic scaling analysis and how chemist can use fractal concepts to understand metal growth on surfaces. This is a joint work of T.Jiang, N.Hall and S. Morin.

TIME: 1:00p.m.-2:00p.m., Friday, April 11th.

PLACE: N638 Ross

Friday, March 28th, 2003

TITLE: Robust Multigrid Methods with Application to Graphics Simulation

SPEAKER: Justin W.L. Wan, University of Waterloo

ABSTRACT:

To achieve more realistic image synthesis, sophisticated physically-based models have been recently used in computer graphics for simulating physical phenomena. For instance, in water animation, the two-phase incompressible Navier-Stokes model was used to simulate water motion. Consequently, one needs to solve Poisson equations with many complex interfaces, which are often known as Stefan problems. Moreover, the PDE coefficients often have large jumps in discontinuities. The large number and highly irregular shape of the interface pose great challenge to accurate and efficient numerical solution to the problems. In this talk, we present a fast multigrid preconditioning technique for solving highly irregular interface problems. Our approach takes advantage of the knowledge of the interface location and the jump conditions, which one often knows in practice. Specifically, our interpolation captures the boundary conditions at the interface. Numerical results in 2D and 3D show that the resulting multigrid method is more efficient than other robust multigrid methods, and is independent of both the mesh size and the size of the jump.

TIME: 1:00p.m.-2:00p.m., Friday, Mar. 28th.

PLACE: N638 Ross

Friday, March 21th, 2003

TITLE: Splitting of separatrices in singularly perturbed nonlinear syste ms

SPEAKER: Alexander Tovbis, University of Central Florida

TIME: 12:30p.m.-1:30p.m., Friday, Mar. 21th.

PLACE: N627 Ross

Wednesday, March 19th, 2003

TITLE: Exponential dichotomies at work

SPEAKER: Marion Weedermann, University of Wisconsin-Green Bay

ABSTRACT:

Exponential dichotomies have played and continue to play a significant role in the study of the asymptotic behavior of various types of differentia l equations. After reviewing the basic ideas behind exponential dichotomy we will focus on its application in showing the existence of periodic solutions of ordinary differential equations, parabolic partial differential equations and functional differential equations. We will establish the criteria necess ary to further extend this method to other types of equations.

TIME: 1:00p.m.-2:00p.m., Wednesday, March 19th.

PLACE: N638 Ross

Friday, March 14th, 2003

TITLE: Building better molecules in an imperfect world

SPEAKER: Logan Donaldson, York University

ABSTRACT:

A macromolecular structure is calculated from a combined set of common chemical topologies and experimental data. In this seminar, I will guide the listener through a structure calculation and discuss the problems one encounters along the way.

TIME: 1:00p.m.-2:00p.m., Friday, Mar. 14th.

PLACE: 3009 Vari Hall

Wednesday, March 5th, 2003

TITLE: Variations on a Theme of Morawetz

SPEAKER: Jim Colliander, University of Toronto

ABSTRACT:

The identification of monotone-in-time quantities underpins some of the basic insights into the long-time behavior on nonlinear Schrodinger evolutions. For example, in the focusing setting, the variance identity reveals a monotone behavior implying the existence of blow-up solutions. In the defocusing case on R^3,the Morawetz identity of Lin-Strauss provides space-time norm bounds implying scattering behavior. This talk describes a unified approach to obtaining monotone-in-time quantities for NLS evolutions, generalizing these two classic examples. A scattering result for the R^3 cubic defocusing case will also be discussed. This talk describes joint work with M. Keel, G. Staffilani, H. Takaoka and T. Tao.

TIME: 1:00p.m.-2:00p.m., Wednesday, Mar. 5th.

PLACE: N638 Ross Building

Friday, Feb. 28th, 2003

TITLE: Gaussian Plume Modeling of Contaminant Transport

SPEAKER: Chris A. Kennedy, University of Toronto

ABSTRACT:

A technique for modeling contaminant transport based on Markov Process theory is developed. Transport is quantified by summing the first two moments of independent random displacements and applying the Central Limit Theorem (CLT) to obtain solute distributions of a Gaussian nature. For non-uniform flow fields the CLT is applied in a streamfunction / equi-travel time space and transforms are used to give concentrations in Cartesian co-ordinates. Simulations in uniform, radially converging and circular flow fields show the method to be two to three orders of magnitude faster than modelling with the advection-dispersion equation, using a control volume technique.

TIME: 1:00p.m.-2:00p.m., Friday, Feb. 28th.

PLACE: N638 Ross Building

Friday, Feb. 21th, 2003

TITLE: Nonlinear spectra and applications to boundary value problems

SPEAKER: Wenying Feng, Trent University

ABSTRACT:

As a fascinating field of growing interest on the borderline between functional analysis, operator theory, differential equations and mathematical physics, nonlinear spectral theory has been extensively studied by many authors. Three European mathematicians, Furi, Martelli and Vignoli, made a major contribution by introducing a notion of spectrum for continuous maps. Later, a new spectrum that contains the eigenvalues of the operator as the linear case was introduced by the speaker. Both theories have wide applications.

We shall discuss the application of the new theory to the study of some nonlinear integral operators such as Hammerstein integral operators. The results then will be used to prove existence of a solution for a second order differential equation under three-point boundary conditions. A generalization of the Borsuk-Ulam theorem will also be given.

TIME: 1:30p.m.-2:30p.m., Friday, Feb. 21th.

PLACE: N638 Ross Building

Friday, Feb. 14th, 2003

TITLE: From clone to PDB: protein structure determination by NMR spectroscopy

SPEAKER: Logan Donaldson, York University

ABSTRACT:

Advances in protein expression, instrumentation and computation have made protein structure determination a process that can be measured in weeks/months rather than years. As a result, researchers can now focus more on biological problems than the act of structure determination itself. Increases in throughput are also crucial to drug screening and proteomics initiatives. My seminar will discuss how material (protein/DNA) is manufactured, how data is collected, and how structural calculations are performed.

TIME: 1:00p.m.-2:00p.m., Friday, Feb. 14th.

PLACE: 3009 Vari Hall

Friday, Feb. 7th, 2003

TITLE: Numerical and mathematical challenges related to hydrogen fuel cells (HFC)

SPEAKER: Arian Novruzi, University of Ottawa

ABSTRACT:

HFC make a very intensive research area due to their capability to produce free-pollution electromotive energy from a chemical reaction.

I will make an overview of HFC model equations that give the HFC fluid dynamics. I will present a FEM for 3D HFC fluid dynamics computations, some numerical results and other issues to HFC computations. I will discuss the boundary conditions on liquid-solid interfaces and free air-porous domain. In terms of geometry optimization of HFC performance, I will present some shape optimization techniques to use for porous domain optimal design.

TIME: 1:00p.m.-2:00p.m., Friday, Feb 7th.

PLACE: N638 Ross Building

Friday, Jan. 24th, 2003

TITLE: Inhibitory Rhythms in Hippocampus: Linking Levels with Mathematical Modelling

SPEAKER: Frances K. Skinner, University Health Network & University of Toronto

ABSTRACT:

An essential goal in neuroscience is to develop links between the multiple levels of neural organization that include the molecular, cellular, multi-cellular, circuit and system levels. The complexity of brain networks makes their output impossible to understand using purely experimental techniques. Mathematical models with clear links to experimental data are uniquely poised to forge links between different neural levels. However, the development and the computational and mathematical analyses of appropriate physiological models are challenging. In large part, this is due to the specificity and the interdisciplinarity of the work.

Oscillatory output produced by networks of inhibitory neurons play critical roles in brain function, including hippocampus. In this talk, I will present some of our modelling work which shows that it is possible to link changes in inhibitory kinetics associated with anesthetics to specific alterations in inhibitory network patterns. This suggests that the effects of different anesthetic drugs might lie in the different ways in which these drugs modulate inhibitory network patterns. Future and ongoing work will also be discussed.

TIME: 1:00p.m.-2:00p.m., Friday, Jan. 24th.

PLACE: N638 Ross Building

Friday, Nov. 22th, 2002

TITLE: Semigroups of Operators and Applications to Differential Equations

SPEAKER: Jizhou Zhang, Shanghai Normal University

ABSTRACT:

In this talk, We investigate whether abstract differential operator with the symbol polynomial generates integrated semigroup and regularized semigroup by making different conditions. The application to partial differential operators with constant coefficients can be obtained immediately on several different function spaces. In particular, more exact results are obtained for pseudodifferential operator on certain function space. Finally, the results are applied to partial differential equations and compared by the size of their initial value spaces. It turns out that the regularized semigroup is an appropriate tool for non-elliptic partial differential operators and is far superior to the integrated semigroup approach. Similarly, corresponding results are obtained for integrated cosine function and regularized cosine function.

TIME: 1:30p.m.-2:30p.m., Friday, Nov. 22th.

PLACE: N638 Ross Building

Friday, Nov. 15th, 2002

TITLE: The Pricing of Options for Security Markets with Delayed Response

SPEAKER: Yuriy Kazmerchuk, York University

ABSTRACT:

We consider (B,S)-securities market with a standard riskless asset (bond) and a risky asset (stock) with stochastic volatility depending on time and the history of stock price.

We state some results on option pricing in such market and its completeness. A continuous-time analogue of GARCH model for our stochastic volatility is proposed.

We then show that the equation for the expected squared volatility under risk-neutral measure is a deterministic delay differential equation, and we construct the solutions for such an equation. We also derive the partial integro-differential equation for the evaluation function with boundary conditions defined by the option final payoff function.

And, finally, we propose numerical and estimation procedures for above model and show the comparison of numerical results.

TIME: 1:30p.m.-2:30p.m., Friday, Nov. 15th.

PLACE: N638 Ross Building

Friday, Nov. 8th, 2002

TITLE: Towards mechanistic models of biological systems: predicting stem cell responses

SPEAKER: Peter Zandstra, University of Toronto

ABSTRACT:

Our inability to quantitatively predict stem cell fate limits the use of these cells in a variety of therapeutic applications. While significant inroads have been made in investigating the native or plastic potential of stem cells from adult, embryonic and fetal origins, strategies to control or direct the differentiation of these cells to produce large numbers of purified stem or differentiated cells is at a relatively early stage. A systematic and computational approach would assist in the identification and characterization of parameters that regulate stem cell responses; these parameters can then be further engineered to produce desired cells or cell products. Most computational models of stem cell determination events describe observed data without detailing the underlying regulatory mechanisms, thus limiting the effective and controlled exploitation of stem cells for clinical applications. Our integrated experimental and computational approach allows us to evaluate different mechanisms of stem cell fate regulation, and is useful in the design and development of experiments to improve stem cell culture systems.

TIME: 1:30p.m.-2:30p.m., Friday, Nov. 8th.

PLACE: N638 Ross Building

Friday, Oct. 25th, 2002

TITLE: Stochastic Dynamical Systems in Biology and Finance

SPEAKER: Anatoliy Swishchuk, York University

ABSTRACT:

The talk is devoted to stochastic dynamical systems arising in biology and finance, and consists of two parts:

(1) biological systems in random media;
(2) stochastic financial systems with delayed volatility.

The first part deals with the limit theorems and stability for biological systems in random media. The second part is devoted to the study of (B,S)-securities markets with delayed volatility: completeness, equation for expectation of volatility, option pricing formula.

TIME: 1:30p.m.-2:30p.m., Friday, Oct. 25th.

PLACE: N638 Ross Building

Friday, Oct. 18th, 2002

TITLE: Multiphysics Modeling

SPEAKER: Bjorn Sjodin, COMSOL, Inc.

ABSTRACT:

This seminar will present a study of the applicability of mathematical modeling in engineering. Models will be built interactively from scratch to allow the audience to give their input and to alter the modeling process itself. The idea will be to show the benefits of modeling in an education environment, as a compliment to theoretical and experimental studies.

The seminar will specifically look at:

(1) Coupled momentum and heat transfer in flow through a heat exchanger
(2) Solving fundamental physics on complex geometries, using weak formulations
(3) Specifying two-phase flow in the general mode of FEMLAB
(4) Coupling 2D and 3D geometries in a structural mechanics/acoustics multiphysics problem
(5) Command line programming from the MATLAB environment

Based on MATLAB, FEMLAB is a modeling tool that solves any arbitrary nonlinear coupled Partial Differential Equation. It also consists of specialized application modules for Chemical Engineering/Transport Phenomena, Structural Mechanics and Electromagnetics. Further information can be found at www.comsol.com

Physics can be applied in FEMLAB as model equations in tailored, ready-to-use forms, or specified freely to suit any arbitrary type of physical phenomenon (linear, non-linear or time dependent). Several problems can be combined and coupled in a single model - multiphysics modeling - meaning that your simulations can encompass all fields of physics and engineering.

TIME: 1:30p.m.-3:30p.m., Friday, Oct. 18th.

PLACE: N638 Ross Building

Friday, Oct. 11th, 2002

TITLE: Optimal Quadratic and Cubic Spline Collocation on Non-Uniform Partitions

SPEAKER: Christina C. Christara, University of Toronto

ABSTRACT:

Collocation with piecewise polynomials is a simple-to-implement and integration-free discretization method for one- or multi-dimensional Boundary Value Problems (BVPs). Collocation using splines (i.e. piecewise polynomials with maximum continuity) gives rise to small linear systems, usually nicely behaved, and with small bandwidth. Therefore, spline collocation is a reasonable alternative to Galerkin. However, spline collocation has not yet been extensively used for the solution of BVPs. It is known that the standard formulation of the method gives rise to suboptimal approximations with respect to convergence order. Relatively recently several optimal spline collocation methods have been derived based on splines of degree 2, 3, 4 and 5. However, the success of the methods is hindered by the fact that all these methods are derived on uniform grids.

We will first review the development of optimal spline collocation methods on uniform partitions. Next, we will describe the extension of optimal Quadratic and Cubic Spline Collocation (QSC and CSC) methods for the solution of linear second-order two-point Boundary Value Problems (BVPs) discretized on non-uniform partitions. To derive the methods, we use a mapping function between uniform and non-uniform partitions and develop expansions of the error at the non-uniform collocation points of some appropriately defined spline interpolants. The existence and uniqueness of the QSC and CSC approximations are shown, under some conditions. Optimal global and local orders of convergence of the spline approximations and derivatives are derived, similar to those of the respective methods for uniform partitions. The jth derivative of the QSC approximation, for j>=0, is O(h^{3-j}) globally, and O(h^{4-j}) locally on certain points. The jth derivative of the CSC approximation, is O(h^{4-j}) globally, for j>=0, and O(h^{5-j}) locally on certain points, for j>0. The non-uniform partition QSC and CSC methods are integrated with adaptive grid techniques, and grid size and error estimators. Numerical results on a variety of problems, including problems with boundary or interior layers, verify the theoretically expected behaviour of the methods.

TIME: 1:30p.m.-2:30p.m., Friday, Oct. 11th.

PLACE: N638 Ross Building

Friday, Oct. 4th, 2002

TITLE: Motion of a liquid drop: mathematical modelling and computation

SPEAKER: Huaxiong Huang, York University

ABSTRACT:

Inside a proton-exchange-membrane fuel cell, condensation of water vapor may occur and subsequent removal of liquid drops is of primary interest. In this talk, we will discuss the motion of a liquid drop on a solid surface driven by gas flows. In particular, we will discuss the behaviour of the liquid-gas-solid three-phase contact point and related modeling issues. We will then outline a front-tracking approach for computing the motion of a two-dimensional drop, based on Peskin's immersed boundary method. We will also present a simple analytical approach to handle the singularity at the contact point and to model the dynamical contact angle. Numerical results will be presented.

TIME: 1:30p.m.-2:30p.m., Friday, Oct. 4th.

PLACE: N638 Ross Building

Friday, Sept. 20th, 2002

TITLE: Time lag and spatial diffusion together: modeling, dynamics, biological invasion and numerics

SPEAKER: Jianhong Wu, York University

ABSTRACT:

A short survey will be provided for the recent development in the theory and applications of reaction diffusion equations with both retarded arguments and non-local spatial interactions. Models arising from structured populations with spatial dispersal will be discussed, and some new approaches and results regarding traveling waves will be reported together with their implication for biological invasion and range expansion.

TIME: 1:30p.m.-2:30p.m., Friday, Sept.20.

PLACE: N638 Ross Building

Friday, Sept. 13th, 2002

TITLE: Nonlinear Dynamic Prediction Using Data Mining Approach

SPEAKER: Yau Shu Wong, Department of Mathematical & Statistical Sciences, University of Alberta

ABSTRACT:

Prediction of the nonlinear response of a dynamical system is a crucial step in many science and engineering applications. For example, in the study of nonlinear aeroelasticity, understanding the nonlinear behavior of aircraft structures will lead to more efficient and safe design of aircraft wings and control surfaces. Traditionally, mathematical theory and numerical simulation have been successfully applied to study the response of nonlinear dynamical systems. In this approach, a mathematical model is developed and the system parameters must be known. In some practical applications, only the dynamic response due to a given excitation is available. The recorded nonlinear response is usually noisy, nonstationary, and may have high dimensional dynamics. Consequently, the traditional approach may be difficult to deal with these practical problems. In this talk, we propose to analyze the dynamics from data instead of using mathematical equations and numerical simulations. An expert data mining system (EDMS) is developed, in which a short term data is taken as input to EDMS. The output of EDMS provides a prediction of the long term dynamic behavior and it can also extract important features of the corresponding nonlinear response. The key modules in the proposed EDMS include artificial neural networks, nonlinear time series models and filtering techniques. Applications of the proposed EDMS to simulated data and real experimental data from nonlinear aeroelastic systems modeling a two degree of freedom airfoil oscillating in pitch and plunge will be reported.

TIME: 11:30a.m.-12:30p.m., Friday, Sept.13.

PLACE: N638 Ross Building

Friday, June 14, 2002

TITLE: MANUFACTURING, ROBOTICS AND COMPUTATIONAL GEOMETRY

SPEAKER: David Field, General Motors Research

ABSTRACT:

This SIAM Visiting Lecture features examples of geometry's dominating influence in the automotive manufacturing process. The lecture begins with the design and manufacture of sheetmetal components that motivated advances in mathematical applications for Computer Aided Design. After a discussing the mathematics developed for the geometric aspects of this manufacturing process, the lecture examines an application of the same mathematics to robotics. The next topic relates the previous geometric constructions with the analysis of automotive components for fatigue, stress and strain. The lecture ends with the award winning video tape "Ballet Robotique".

TIME: 11:00a.m.-12:00p.m., Friday, June 14.

PLACE: N638 Ross Building

Tuesday, June 11, 2002

TITLE: Nonlinear Dispersive Waves in a Circular Rod Composed of a Mooney-Rivlin Material

SPEAKER: H.-H. Dai, City University of Hong Kong

ABSTRACT:

Nonlinear dispersive waves in rods have been the subject of many studies. In particular, nonlinear axisymmetric waves that propagate axial-radial deformation in circular cylindrical rods composed of a homogeneous isotropic material have been considered by many authors. Here, we mention in particular three related pieces of work by Wright (1982, 1985) and Coleman and Newman (1990). Wright (1982) seems to have been the first to take into account the full nonlinearity for such a problem. In a sequel paper, Wright (1985) considered traveling waves in a rod composed of an incompressible hyperelastic material. He pointed out the existence of a variety of types of waves. In particular, he conjectured that sharp crested solitary waves can arise. This is one type of solitary waves with a first-order derivative discontinuity at the wave peak. Here, we shall call them solitary shock waves. Coleman and Newman (1990) derived the one-dimensional rod equation for a general incompressible hyperelastic material. Their work was concerned with smooth solutions only. Here, we shall resolve Wright's conjecture by showing that solitary shock waves can indeed arise in a Mooney-Rivlin rod. The explicit solution expressions for these waves and the physical existence conditions are also obtained.

References

Coleman, B.D. and Newman, D.C. 1990 On waves in slender elastic rods. Arch. Rational Mech. Anal. 109, 39-61.
Wright, T. 1982 Nonlinear waves in rods. In Proc. IUTAM Symp. on Finite Elasticity (ed. D.E. Clarkson and R.T. Shields). The Hague: Martinus Nijhoff.
Wright, T. 1985 Nonlinear waves in rods: results for incompressible elastic material. Stud. Appl. Math. 72, 149-160.

TIME: 1:30p.m.-2:30p.m., Tuesday, June 11.

PLACE: N638 Ross Building

Thursday, June 6, 2002

TITLE: Solving inverse problems with radial basis function

SPEAKER: Yongji Tan, Fudan University

ABSTRACT:

In this talk some inverse problems of reconstruction the boundary, initial condition or lower odder term for the boundary value problem of heat equation are investigated. The radial basis functions are used to solve corresponding direct problems. Some numerical results show the efficiency of this method.

TIME: 1:30p.m.-2:30p.m., Thursday, June 6.

PLACE: N638 Ross Building

Thursday, April 18, 2002

TITLE: Operator splitting using SSP Runge-Kutta methods for taxis-diffusion-reaction systems

SPEAKER: Alf Gerisch, The Fields Institute and University of Guelph

ABSTRACT:

We describe a method of lines (MOL) technique for the simulation of taxis-diffusion-reaction (TDR) systems. These time-dependent PDE systems arise when modelling the spatio-temporal evolution of a population of organisms which migrate in direct response to e.g. concentration differences of a diffusible chemical in their surrounding (chemotaxis). Examples include pattern formation and different processes in cancer development. The effect of taxis is modelled by a nonlinear advection term in the TDR system (the taxis term).

The MOL-ODE is obtained by replacing the spatial derivatives in the TDR system by finite volume approximations. These respect the conservation of mass property of the TDR system, and are constructed such that the MOL-ODE has a nonnegative analytic solution (positivity). The latter property is natural (because densities/concentrations are modelled).

The MOL-ODE is stiff and of large dimension. We develop integration schemes which treat the discretization of taxis and diffusion/reaction differently (splitting). This is achieved through operator (Strang-)splitting. To solve the resulting non-stiff subproblem we employ strong-stability preserving (SSP) Runge-Kutta methods and for the stiff subproblems a linearly implicit W-method with approximate matrix factorization is applied. Optimal SSP Runge-Kutta methods with number of stages larger than their order are used because of their favourable positivity preserving properties.

Numerical experiments confirm the broad applicability of the splitting schemes for the solution of TDR systems and show the effect of using SSP Runge-Kutta methods with many stages.

TIME: 1:30p.m.-2:30p.m., Thursday, April 18.

PLACE: N638 Ross Building

Thursday, March 14, 2002

TITLE: Approximation on Differentiable Manifolds

SPEAKER: Marshall Walker, York University

ABSTRACT:

Given a sequence of points which lie over a differentiable 2-manifold M embedded in R^3, we propose a method which allows the construction of approximating or interpolating curves which respect intrinsic geometry of the manifold. In particular we desire exact representation of geodesic arcs and of a class of spiral-like curves which orthogonally project to geodesic arcs on the manifold. In the particular case when M is a sphere applications exist in the domain of geological and geographical mapping, for instance the creation of topographical contour lines or isotherms, and in the field of video production where it is desirable to have smooth camera trajectories interpolating fixed camera positions.

For a differentiable Riemannian manifold M and a point x\in M, it is well known that there is a neighborhood V