Syllabus: M374M

MATHEMATICAL MODELING IN SCIENCE AND ENGINEERING

Text: J. David Logan, Applied Mathematics, 4th Edition

Responsible party: Oscar Gonzalez November 2015

Description of the Course: This course is for students interested in mathematical modeling and analysis.  The goals are to develop tools for stuying differential equation models that arise in applications, and to illustrate how the derivation and analysis of models can be used to gain insight and make predictions about physical systems.  Emphasis should be placed on examples and case studies, and a broad range of applications from the engineering and physical sciences should be considered.

Prerequisites: A grade of at least C- in differential equations (M427K or M427J) and in linear algebra (M341 or M340L); basic programming skills; and familiarity with the software package Matlab.

Topics Covered

The following outline is a
list of relevant concepts for each core topic.
Instructors should carefully choose and balance
the concepts depending on the case studies they
have in mind.  Note that a well-designed case
study will likely occupy 1-2 class days, and can
be continued in an associated homework assignment.
The number of class days listed below is for a
standard MWF schedule.  A typical semester has
43 MWF days, and the schedule below contains
material for 41 days, allowing time for 2
midterm exams. The suggested text provides some coverage of all
the core topics; instructors may find it necessary to
employ supplementary material to increase the depth of
coverage in areas of interest, and to support their
case studies. 

1) Dimensional analysis and scaling (6 days)
-Fundamental physical dimensions, units
-Dimensional vs dimensionless quantities
-Unit-free equations and their properties
-Buckingham Pi Theorem
-Characteristic scales for a function
-Transforming equations to dimensionless form
-Scaling to expose dominant/small effects

2) Dynamical systems in one dimension (4 days)
-Properties of solutions
-Phase line diagrams
-Equilibrium solutions
-Stability of equilibria
-Classification via linearization
-Classification via Lyapunov functions
-Bifurcation of equilibria
-Basic types of bifurcations, hysteresis

3) Dynamical systems in two dimensions (9 days)
-Properties of solutions
-Phase plane diagrams, nullclines, direction fields
-Equilibrium solutions, stability
-Stability in linear systems, eigenvalues
-Phase diagrams for linear systems
-Stability in nonlinear systems, linearization thm
-Stability in nonlinear systems, Lyapunov thm
-Bifurcations in linear and nonlinear systems
-Closed orbits and limit cycles, Hopf bifurcation
-Poincare-Bendixson thm

4) Regular perturbation methods (6 days)
-Perturbed equations, regular vs singular
-Characteristics of regular problems
-Approximation via asymptotic series
-Regular method for algebraic equations
-Typical error bounds
-Regular method for differential equations
-Typical error bounds, issue of uniformity
-Poincare-Lindstedt method for oscillatory problems

5) Singular perturbation methods (5 days)
-Characteristics of singular algebraic problems
-Rescaling method for algebraic problems
-Characteristics of singular differential problems
-Boundary layers, failure of regular method
-Two-scale method for problems with boundary layers
-Idea of inner, outer and matched expansions
-WKB method for oscillatory, exponential problems

6) Calculus of variations (11 days)
-Function spaces and functionals
-Absolute extrema of a functional
-Local extrema of a functional, issue of norms
-Concept of admissible variations
-Necessary conditions for local extrema
-Fundamental lemma, Euler-Lagrange equations
-Fixed-endpoint and free-endpoint problems
-Multiple-function and higher-order problems
-Isoperimetric constraints, multiplier rule
-Convexity and sufficient conditions