Department of Mathematics
Math 182. Partial
Differential Equations Spring 2014
Course Outline
Time and Place: MWF 10:00 am – 10:50 am Mudd Science Library 125
Instructor: Dr.
Adolfo J. Rumbos
Office:
Mudd Science Library 106
Phone/e-mail: ext. 18713 / arumbos@pomona.edu
Office Hours: MWF 11:05 am-11:55 am, TR 9:00 am – 10:00am,
or by
appointment
Text: Introduction to Partial Differential Equations and Hilbert
Space
Methods, by Karl E. Gustafson, Dover.
Course Website: http://pages.pomona.edu/~ajr04747/
Prerequisites: Ordinary
Differential Equations and some Analysis course
Course
Description
This
course is an introduction to the theory and applications of partial
differential equations (PDEs). PDEs are
expressions involving functions of several variables and its derivatives in
which we seek to find one of the functions, or a set of functions, subject to
some initial conditions (if time is involved as one of the variables) or
boundary conditions. They arise
naturally when modeling physical or biological systems in which assumptions of
continuity and differentiability are made about the quantities in
question. In this course we will discuss
several modeling situations that give rise to PDEs.
PDEs
are classified in various ways. PDEs
range from linear to nonlinear; single equations to systems; and from first
degree to higher degree. There is also a
further classification determined by the behavior of solutions of certain
classes of equations. Over the years researchers
have identified three major classes of PDEs: hyperbolic, elliptic and
parabolic. Archetypal instances of these
classes of PDEs are the classical equations of mathematical physics: the wave
equation, Laplace's or Poisson' equations, and the heat or diffusion equations,
respectively. In this course we will
provide examples of analysis for each of these types of equations.
In
problems involving PDEs we are mainly interested in the question of existence
of solutions. In a few cases, answering
these questions amounts to coming up with formulas for the solutions. In this course we will discuss a few
techniques for constructing solutions (e.g., separation of variables, series
expansions and Green's function methods) for the special case of linear
equations. In most cases, however,
explicit constructions of solutions are not possible. In these cases, the only
recourse we have is analytical proofs of existence, or nonexistence, and
qualitative analysis to deduce properties of solutions. We will discuss a few general approaches for
the analysis of PDE problems, including the method of characteristics for first
order PDEs and variational methods for a large class
boundary value problems for second order PDEs..
Course
Structure and Expectations
The structured of the coursed is centered on lectures and readings
on the topics listed in the attached tentative schedule of lecture and
examinations, homework assignments, two examinations and a term project.
Readings and problem sets will be assigned at every lecture and
collected on al alternate basis.
Students are strongly encouraged to work on every assigned problem. Late homework assignments will not be
graded.
The term project will consist of a paper and presentation on a topic not covered in the lectures. Ideas for topics in the term project may be
found in the text for the courses; possible topics may range from applications
of the theory and techniques learned in class to problems in various fields in
science to advanced analysis techniques that are not covered in the course. The term paper will be due on Wednesday, May 7, 2014.
Presentations will take place in the last three weeks of the semester
Grading
Policy
Grades
will be based on the homework, two examinations and a term project involving an
advanced topic in the analysis of PDE problems.
The overall score will be computed as follows:
homework 20%
Examinations 50%
term
project 30%
Math 182. Partial Differential
Equations Spring
2014
Tentative Schedule of Lectures and
Examinations
Date Topic
W Jan
22 Introduction: Where do PDEs arise?
F Jan
24 Conservation
principles
M Jan 27 The equations of fluid mechanics
W Jan 29 Euler’s
equations
F Jan 31 Diffusion
equation
M Feb 3 An application to traffic flow modeling
W Feb 5 Method of characteristics
F Feb
7 Method of
characteristics (continued)
M Feb 10 Types
of PDEs
W Feb
12 Classification of
second order PDEs
F Feb
14 Problems
M Feb 17 Solving the diffusion equation
W Feb 19 Existence
F Feb 21 The
heat kernel
M Feb 24 The principle of superposition
W Feb 26 Solutions
via Fourier series
F Feb 28
Solutions via Fourier
series (continued)
M Mar 3 Solutions
via Fourier transform
W Mar
5 The
eigenvalue problem for the Laplacian
F Mar
7 Solutions via eigenfunction expansion
M Mar 10 Probelms
W Mar 12 Review
F Mar 14 Exam
1
M Mar 17 Spring
Recess
W Mar 19 Spring Recess
F Mar 21 Spring Recess
M Mar 24 Application: vibrations of a strings
and membranes
W Mar 26 vibrations membranes (continued)
F Mar 28 César Chávez Day
Date Topic
M Mar 31 Elliptic boundary value problems
W Apr
2 The Green’s function
F Apr
4 Existence and
properties of solutions
M Apr 7 Variational problems
W Apr
9 Variational
problems (continued)
F Apr
11 Hilbert space methods
M Apr 14 Hilbert
space methods (continued)
W Apr
16 Review
F Apr
18 Exam 2
M Apr 21 Presentations
W Apr
23 Presentations
F Apr
25 Presentations
M Apr 28 Presentations
W Apr
30 Presentations
F May
2 Presentations
M May 5 Presentations
W May
7 Presentations and term
paper due