Math 451 - Methods of Applied Mathematics II
Spring 2014
H02: Lou Kondic, Cullimore (617; phone: 2996; email: kondic@njit.edu
H04: Horacio G. Rotstein, Cullimore (614; phone 5306; email: horacio@njit.edu)
Textbook: There is no mandatory text for either section. The following books and
materials will be used:
H02:
1. D.J. Acheson, Elementary Fluid Dynamics, Oxford Applied Mathematics and
Computing Science Series, 1990, ISBN-13: 978-0198596790 (available in the library
or from the instructor).
2. K. W. Morton and D. F. Mayers, Numerical Solutions for Partial Differential
Equations: An Introduction, Cambridge, 2005, ISBN-13: 978-0521607933j
(available in the library or from the instructor.)
3. Selected research articles (to be provided by the instructor).
H04:
1. F. Gabbiani, S. J. Cox, Mathematics for Neuroscientists, 2010, Elsevier (ISBN: 9780-12-374882-9)
2. P. Dayan, L. F. Abbott, Theoretical Neuroscience, 2001, MIT Press (ISBN: 0-26254185-8)
3. G. B. Ermentrout, D. H. Terman, Mathematical Foundations of Neuroscience,
2010, Springer (ISBN 978-0-387-87707-5)
4. B. P. Ingalls, Mathematical Modeling in Systems Biology, 2013, MIT Press (ISBN:
978-0-262-01888-3)
5. Selected research articles (to be provided by the instructor).
Grading policy:
Projects and presentations through the semester: 70%
Final report and presentation: 30%
Course Outline:
Kondic (H02)
Theoretical, computational, and experimental research: Instabilities in twophase flow of complex fluids
Overview: Two-phase flow in Hele-Shaw geometry (flow in the gap between two
plates) will be considered. This flow may become unstable leading to pattern
formation. The focus of the class will be exploring the nature of the instabilities
leading to pattern formation and their mathematical description for flows of
Newtonian fluids, liquid crystals, and shear-thinning fluids.
Theoretical component:
(1) Navier-Stokes equations in viscous regime; simplifications in Hele-Shaw
geometry using consistent asymptotic expansion;
(2) Understanding of the concept of instability in a two-phase flow; discussion of
Saffman-Taylor instability;
(3) Introducing and implementing the concept of linear stability analysis;
(4) Discussing modifications of linear stability analysis due to non-Newtonian
(shear thinning) behavior on instability development.
(5) Discuss the formulation of Hele-Shaw limit for liquid crystal flows.
(6) Discuss influence of electric field on instabilities in Hele-Shaw flow of liquid
crystals.
Computational component:
(1) Finite difference based methods for solving linear and nonlinear elliptic
problems;
(2) Boundary integral methods for solving linear elliptic problems;
(3) Applications of the numerical methods (1) and (2) to Saffman-Taylor instability;
(4) Development of diffusion-limited aggregation (Monte-Carlo type of simulations)
approach to simulating Hele-Shaw flow of two fluids; applications to Newtonian,
liquid crystal, and shear-thinning flow configurations.
Experimental component: Carrying out experiments involving an air or water
bubble spreading into (1) viscous fluid (glycerin); (2) nematic liquid crystal with
and without electric field, and (3) shear-thinning fluid such as corn starch. Discuss
the issues involved in carrying out controlled, reproducible experiments and
quantify the error bounds. Interpret the results in the context of the theoretical and
computational results.
Rotstein (H04):
Theoretical, computational, and experimental research: Neuronal dynamics
Overview: This project will focus on the dynamic of single neurons and small
networks of neurons. Electric activity in neuronal systems results from the
cooperative activity of the participating electric currents, both intrinsic and
synaptic. An important task of the modeling effort is to reproduce experimental
results as a mean to understanding the link between the dynamic information
contained in experimental data and the underlying biophysics. Mathematical models
play a key role in this process. This goal of this project is to learn and use the
necessary tools to address these issues.
Theoretical component:
(1) Biophysical (conductance-based) models of neurons and neuronal networks:
Hodgkin-Huxley formalism for single neurons and synaptic connections
(2) Dynamical systems tools for the understanding of the mechanisms underlying
spike generation
(3) Data analysis tools for the understanding of experimental data
(4) Firing rate models
Computational component:
(1) Development of numerical algorithms to simulate biophysical (conductancebased) models of Hodgkin-Huxley type for single neurons and networks of
interconnected neurons
(2) Development of numerical algorithms to simulate firing rate type models
(3) Development of numerical algorithms for the analysis of data including spike
trains and firing rates, spike train statistics, spectral analysis, smoothing, spiketriggered average.
(4) Development of numerical algorithms to fit model parameters to data.
Experimental component: Carrying out electrophysiological experiments involving
spiking and bursting neurons and small neuronal networks