Catalog Description

Theory of partial differential equations and boundary value problems with applications to the physical sciences and engineering. Detailed analysis of the wave equation, the heat equation, and Laplace's equation.

Prerequisites

M 333 Differential Equations, or equivalent course elsewhere, or permission of the instructor. The rationale for the prerequisite is that ordinary differential equation techniques are used to study partial differential equations and students who are unacquainted with ordinary differential equations will have little chance of success in partial differential equations.

Jurisdiction

This course is not currently controlled by a departmental committee and individual instructors may choose different textbooks. Exams, homework, and grading system are left to the instructor.

Learning Objectives

As an applied mathematics course, the objectives of M 436 reflect three of the Department's teaching goals, that students be aware of nontrivial applications of mathematics, that students master suitable mathematical tools, and that students use mathematics as a language. M 436 does not stress the aesthetic side of mathematics or the idea of mathematics as the study of patterns.

Upon completion of this course, students should:

Be able to explain the meaning of a partial differential equation, both geometrically and analytically.
Be able to formulate a partial differential equation describing a situation in the sciences or engineering, given a clear statement of the scientific or engineering principles involved.
Be able to solve simple partial differential equations with a variety of elementary techniques.
Be able to apply numerical algorithms and the computer software implementing them to a partial differential equation for the approximation of its solution.

Assessment of Learning Objectives

Students will be assessed by ervaluating their ability to do problems based on the learning objectives. The problems will occur in several contexts:

Periodic problem sets for homework give students a chance to learn new concepts and give the instructor an assessment tool. Students will have the opportunity to discuss these problems during classtime both before and after the homework is due.
Problems given on an in-class exam will give the students the opportunity to demonstrate their understanding of the material while working reasonably simple and less time-consuming problems.
Problems given on a take-home portion of an exam will allow the students to demonstrate their understanding of the material while working more challenging problems as well as use technology that helps in solving these more challenging problems.
Writing assignments will give the students a chance to expand their knowledge of a topic that may not be seen in detail in the classroom and will also help them to become more competent at expressing information in a written fashion.

Topics and Approximate Timeline

The following table is based on a typical semester schedule-45 class meetings of 50 minutes each. The actual amount of time spent on each topic will vary slightly from semester to semester and instructor to instructor.


	Number of
Topic	Meetings
Introduction and General Modeling Considerations	2
Derivation of the heat equation	2
Solution of the heat equation by separation of variables	2
Fourier series	3
Different boundary conditions	3
Derivation of the wave equation	2
Solution of the wave equation by separation of variables	3
The method of characteristics	3
Derivation of Laplace's equation	2
Solution of Laplace's equation for various boundary conditions	4
Laplace transform methods	3
Introduction to numerical methods	8
Additional Topics in Modeling	6
Exams	2

Text

The most recently used text is Power? or Charley's? Other suitable texts include

Format, Student Activities, and Grades

Class meetings involve a combination of lecture, questions and discussion, and sometimes small group activity; the instructor chooses the appropriate mix. The computer algebra system, Maple, is used for laboratory activities and homework. Homework is an important part of the course; many exercises involve extensions of ideas in the text to new situations, rather than just routine applications. Some exams may be partially take-home. The instructor chooses the exact grading scheme, but a typical distribution might be:

Homework (including a writing assignment and project) 33 ¹/₃ %
2 Exams 33 ¹/₃%
Final Exam 33 ¹/₃%
Total 100%

Letter grades are based on a scale in which 90% of the total possible points guarantees an A, 80% a B, 70% a C, and 60% a D, with the instructor having the discretion to modify these cut-offs if warranted.

File translated from T_EX by T_TH, version 1.56.


Homework (including a writing assignment and project)	33 ¹/₃ %
2 Exams	33 ¹/₃%
Final Exam	33 ¹/₃%
Total	100%