Spring 2006

MATH 467.01 (3 credits), Tu_Th, 2:30PM-3:45PM, Osburn 119

**Prerequisites:**A grade of C- or better in MATH 365 (

*Ordinary Differential Equations*) is the prerequisite for this course.**Instructor:**Dr. Buchanan

Office: Chester 103, Phone: 872-3659, FAX: 871-2320

Office Hours: 10:00AM-10:50AM (MTuWThF), or by appointment

Email:`Robert.Buchanan@millersville.edu`

Course URL:`http://banach.millersville.edu/~bob/math467`**Textbook:***Applied Partial Differential Equations with Fourier Series and Boundary Value Problems*, 4th edition, Richard Haberman, Pearson Education, Inc., Upper Saddle River, NJ, 2004, ISBN: 0-13-065243-1**Objectives:**MATH 467 provides an introduction to partial differential equations and their applications. Upon completion of this course the student will:

- understand how partial differential equations arise in the mathematical description of heat flow and vibration,
- demonstrate the ability to solve initial boundary value problems,
- express and explain the physical interpretations of common forms of PDEs,
- understand issues related to existence and uniqueness of solutions,
- depict in series and graphical form the solutions to initial boundary value problems,
- appreciate the theory underlying the techniques of solution,
- be conversant with methods of applying partial differential equations in various applications.

**Course Contents:**- Topics covered in this course may include the following.
The material will be presented in a logical order, though not
necessarily in the order shown below.
Other topics will be added as time and interests allow.
- Introduction
*Extremely*brief review of topics from ordinary differential equations- Heat equation as model of heat conduction in a rod
- Separation of variables
- Fundamental solutions and superposition of solutions

- Fourier series
- Orthogonality and Euler-Fourier formulas
- Periodicity
- The Fourier Convergence Theorem
- Even and odd functions; sine and cosine series
- Extensions of functions to even and odd functions

- The Heat Equation
- Solution of initial/boundary value problems
- Homogeneous Dirichlet boundary conditions
- Nonhomogeneous boundary conditions and steady-state solutions
- Other boundary conditions
- A Maximum Principle and uniqueness of solution for the heat equation

- The Wave Equation
- Solution of initial/boundary value problems
- Characteristic coordinates and a general solution
- D'Alembert's solution of the initial value problem
- Energy integrals and uniqueness of solution for the wave equation

- Laplace's Equation
- Boundary value problems in rectangular coordinates
- Boundary value problems in polar coordinates
- Periodic boundary conditions

- Neumann problems and mixed boundary conditions
- Lack of uniqueness of solution
- Necessary conditions for the existence of a solution

- Uniqueness of solutions
- Mean Value Property
- Weak form of the Maximum Principle
- Uniqueness of solutions of the Dirichlet problem

- Sturm-Liouville Theory
- General two-point boundary value problem
- Eigenvalues and eigenfunctions
- Lagrange's identity and consequences
- Normalization of eigenfunctions and general eigenfunction expansions
- Nonhomogeneous boundary value problems

- Introduction
**Attendance:**Students are expected to attend all class meetings. Much of the material presented in class supplements the textbook, therefore it is very important for students to be in class every day. If you must be absent from class you are expected to complete class requirements (tests and/or homework assignments) prior to the absence. Students who miss a test should provide a valid excuse, otherwise you will not be allowed to make up the test. Tests should be made up within one week of their scheduled date. No final exam exemptions.

**Homework:**Students are expected to do their homework and participate in class. Homework problems will be collected frequently and graded. Complete solutions to the homework exercises will be posted on the web at the URL:

`http://banach.millersville.edu/~bob/math467/`. Late homework will not be accepted without a valid excuse. In no cases will late homework be accepted after an assignment has been graded and returned to the students. In addition to the written homework there will be a course project on a topic from PDEs. The project will have a written component (a short paper) which you will hand in to me. There will also be a brief (approximately 10 minutes) public presentation of your work to other interested students during Math Awareness Week (during April 2006). You will be graded on both your written work and public presentation.**Tests:**There will be two tests which are tentatively scheduled for

- Thursday, February 23, 2006
- Thursday, April 13, 2006

If you feel that an error was made in the grading of a test or homework assignment, you should explain the error on a separate sheet of paper and return both it and the test to me within three class periods after the test or homework is returned to you.

**Grades:**Course grade will be calculated as follows.

Tests 35% Homework 35% Project 10% Exam 20% I keep a record of students' test, homework, and exam scores. Students should also keep a record of graded assignments, tests, and other materials. The course letter grades will be calculated as follows. I will not ``curve'' course grades.

90-92 A 93-100 A 80-82 B 83-86 B 87-89 B 70-72 C 73-76 C 77-79 C 60-62 D 63-66 D 67-69 D 0-59 F **Course Repeat Policy**An undergraduate student may not take an undergraduate course of record more than three times. A course of record is defined as a course in which a student receives a grade of A, B, C, D, (including and ) F, U, Z or W. The academic department offering a course may drop a student from a course if the student attempts to take a course more than three times.

^{1}**Inclement Weather Policy:**If we should miss a class day due to a school closing because of weather, any activities planned for that missed day will take place the next time the class meets. For example, if a test is scheduled for a day that class is canceled on account of snow, the test will be given the next time the class meets.

**Final Word:**Mathematics is not a spectator sport. What you learn from this course and your final grade depend mainly on the amount of work you put forth. Daily contact with the material through homework assignments and review of notes taken during lectures is extremely important. Organizing and conducting regular study sessions with other students in this class will help you to understand the material better.