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Applied Fourier Series and Boundary Value Problems

Alfred Clark Jr.
Organization: University of Rochester
Department: Departments of Mechanical Engineering and Mathematics
URL: http://www.me.rochester.edu/~clark
Education level


In general terms the objective of the course is to teach engineers and scientists how to use this kind of mathematics to learn more about the world. In more specific terms, the course has the following five objectives:
  • to provide students with a basic knowledge of Fourier series and separation of variables
  • to show students how to formulate physical problems in terms of partial differential equations
  • to show students how to find insight into the physical behavior of systems from the mathematical solutions
  • to provide practice for students in using Mathematica to calculate and visualize solutions of partial differential equations
  • to motivate students to learn mathematics by presenting a number of interesting case studies.

All of the materials for this course except the class lecture notes are available on the course web site. This includes the Mathematica tutorial, the homework, the homework solutions, review problems for the exams, the exams, the exam solutions, the Star Trek computer project, and about 16 examples in the form of Mathematica notebooks.

This is a first course in Fourier series and boundary value problems, taken scientists and engineers. Some students are sophomores and some are graduate students, but most are juniors. A typical enrollment is about 50. The prerequisites are calculus, ordinary differential equations, and vector calculus. The first homework is a review assignment on these subjects. Mathematica is used throughout the course. The initial assignment is to work through parts of a Mathematica tutorial that is available on the course web site. Students are then asked to carry out some Mathematica work on every assignment, beginning with simple graphs and ending the semester with a difficult and lengthy computer project. Many sample notebooks are handed out in class, shown in class with a computer projector, and put on the web site in downloadable form. Student assignments include 10 homeworks, two 75-minute exams, a computer project for which they have 10 days, and a three-hour final exam.
  • examples of partial differential equations describing physical systems
  • Fourier series
  • basic separation of variables
  • Sturm Liouville Theory
  • separation of variables with more complicated boundary conditions or sources
  • Fourier transform
  • similarity methods
  • power series solutions of ordinary differential equations
  • separation of variables in spherical coordinates
  • Legendre polynomials and Fourier-Legendre expansions
  • regular singular points and the method of Frobenius
  • separation of variables in cylindrical coordinates
  • Bessel functions and Fourier-Bessel expansions.

Some of the case studies used in the course to illustrate this material are centered on the following questions:
  • Why do you stir your coffee?
  • Why are muscle cells the size they are?
  • What does it mean to call diffusion an irreversible process?
  • Why does a guitar string sound tinny when plucked near the bridge?
  • How do we determine the critical size of a nuclear reactor?
  • How many resonant acoustic modes are in the audible range in a typical classroom, and what does this have to do with the ultraviolet catastrophe of classical physics?
  • How did Lord Kelvin estimate the age of the earth?
  • Why does a cup of coffee spill over if you walk too fast with it?

*Mathematics > Calculus and Analysis > Differential Equations
*Mathematics > Calculus and Analysis > Harmonic Analysis