Department of Mathematics

Pomona College

 

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