Department of Mathematics
Course Outline for Mathematics 180
Introduction to Partial Differential
Equations
Spring 2018
Time MWF 10:00 AM - 10:50 AM
Place: Millikan Room 2131
Instructor:
Dr. Adolfo J. Rumbos
Office: Andrew 2287.
Phone/e-mail: ext. 18713 /
Courses
Website: http://pages.pomona.edu/~ajr04747/
Office
Hours: TR 9:00 am - 9:50 am, or by appointment
Text: Introduction to Partial Differential Equations and Hilbert
Space Methods, by Karl E. Gustafson.
Published by Dover.
Prerequisites: Ordinary Differential Equations and
some Real 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: separation of variables, Fourier series
expansions, Hilbert space methods (orthogonal functions expansions), Green's
function methods, transform methods (e.g., Fourier transform) 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, the maximum principle and energy methods. We will also delve into the theory of Sturm-Liouville boundary value problems and eigenvalue problems.
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 an 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. 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 180 Spring
2018
Tentative Schedule of Lectures and
Examinations
Date Topic
W Jan.
17 Introduction to PDEs: The
vibrating string equation
F Jan.
19 Solving the vibrating
string equation
M Jan. 22 Separation
of variables
W Jan.
24 Fourier series
expansion
F Jan.
26 Convergence of Fourier
series
M Jan. 29 Existence
and uniqueness for the one-dimensional wave equation
W Jan.
31 The diffusion equation
F Feb.
2 Solving
the one-dimensional heat equation in a bounded interval
M Feb. 5 Existence and uniqueness for the heat
equation
W Feb.
7 Laplace’s
equation and the Dirichlet problem
F Feb.
9 Solving the Dirichlet problem in
a square
M Feb. 12 The
Dirichlet problem in a disc
W Feb.
14 The Poisson kernel
F Feb.
16 Fundamental solution
of Laplace’s equation
M Feb. 19 Review
W Feb.
21 Exam 1
F Feb.
23 Green’s function
M Feb. 26 The
maximum principle
W Feb.
28 Existence and
uniqueness for the Dirichlet problem
F Mar.
2 Sturm-Liouville eigenvalue problems
M Mar. 5 Eigenvalue
problems
W Mar.
7 Eigenvalues of the Laplacian
F Mar.
9 Eigenvalues
of the Laplacian (continued)
M Mar. 12 Spring Recess!
W Mar.
14 Spring Recess!
F Mar.
16 Spring Recess!
Date Topic
M Mar. 19 Equations
in infinite domains
W Mar.
21 The heat kernel
F Mar.
23 Transform methods
M Mar. 26 The
Fourier Transform
W Mar.
28 Problems
F Mar.
30 Cesar Chavez Day
M Apr. 2 Conservation
equations
W Apr.
4 Method of
characteristic curves (continued)
F Apr.
6 Method
of characteristic curves (continued)
M Apr. 9 Review
W Apr.
11 Exam 2
F Apr.
13 Special Topic
M Apr. 16 Special
Topic
W Apr.
18 Special Topic
F Apr.
20 Special Topic
M Apr. 23 Special
Topic
W Apr.
25 Special Topic
F Apr, 27
Special Topic
M Apr. 30 Special
Topic
W May
2 Special Topic