Mathematics 447(647)
Analytic Methods of Applied Mathematics
Syllabus
Spring 2006
Instructor: Prof. Xuefeng Wang
Office: Gibson Hall 305
Office Hour: MWF 11:00-11:50 or
by appointment
Phone: 862-3451
email: xdw@math.tulane.edu
- Course Description
This course is an introduction to the theory and techniques of partial
differential equations(PDEs), which are widely used to model natural
phenomena. It is a necessity for analytic approaches to problems
in Physics, Chemistry, Biology, Engineering, Economics and
Finance (PDE is a must-take for students in Mathematical Finance
in universities such as NYU, Columbia and
Chicago). The prerequisites are Math 221 (Calculus III) and Math 224 (
the
ability to solve linear and separable ordinary differential equations
is
sufficient for the present purpose). We shall cover
- derivations of some basic partial differential equations
arising in physics, chemistry, biology and other areas. Examples of
such are the transport equation, wave equation (modeling the vibration
of strings, sound and electromagnetic fields), diffusion/heat equation
(distribution of diffusive entities such as heat, particles, even
biological species), Laplace's equation and Poisson's equation
(electrostatics, steady fluid flow, etc.);
- basic analytic methods for solving these equations ("good-ole
fashioned formulas"); a systematic treament of Fourier series;
Sturm-Liouville theory and eigen-expansions.
- qualitative analysis and physical interpretations of
mathematical results: existence, uniqueness, long time
behavior/stability solutions,
etc, by using explicit solution formulas, maximum principles, energy
methods.
- visualization of mathematical results/principles via PDE
softwares/websites.
Textbook
Partial Differential Equations, by Walter Strauss
Course Grade
The semester letter grade will be given based on your performance
in one in-class-test (25%), homework(25%), a group project (15%) and
the final exam (35%).
Test, Project and Final Exam
There will be a one-hour long mid-term exam. There
will be generally no make-up tests. However, in some extreme cases, if
you are forced to miss the test, you must notify me within three (3)
days of the date of the test, preferably before the
test is given. For the Final Exam, you must notify me before or on the
date
of the exam. An excuse from your doctor or other appropriate
authorities
must be presented at that time. The project should be done in groups,
each having no more than 5 people. The final exam will be
comprehensive.
Homework
Homework will be assigned and collected weekly. You are welcome
to discuss homework problems with the instructor.
Matlab PDE toolbox
The MathWorks -
Partial Differential Equation Toolbox
PDE Toolbox
PDE graphics on Internet
Java applet simulating the basic equations
Wave equations in 1-D
Joel
Feldman's Wave Equation
Joel
Feldman's
Telegraph Equation
Wave equations in 2-D
Mode n contributes (1/ n)^2 to the solution.
Mode n contributes 1/ n to the solution.
Odd mode n contributes 1/ n to the solution
with an alternating sign.
Approximation to the drums being "plucked" at their centers.
Heat/Diffusion equations
Joel
Feldman's
Heat Equation
Fourier Series
Joel
Feldman's Fourier Series