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PCMI Undergraduate Faculty Program:

Harmonic Analysis and Partial Differential Equations:

 An Introduction

Facilitated by
Andrew J. Bernoff, Harvey Mudd College

Course Description

Harmonic analysis and partial differential equations arise naturally in the application of mathematics to physical problems such as the oscillation of a drumhead, the conduction of heat in a metal bar, or the shape of a soap film.  In this course, we will derive some basic PDEs (Heat equation, Wave equation, Laplace's equation), and  discuss methods of solution, concentrating on separation of variables, which leads naturally to the topics of Fourier series,  eigenfunction expansions, Sturm-Liouville problems and special functions (in particular Bessel functions).This course will use MAPLE as an algebraic and graphical companion -- the hope is that students will come away from the course comfortable with their ability to use MAPLE to validate and visualize solutions.

Lecture Topics

Week 1: Meet the Partial Differential Equations

Lecture 1: What is a PDE?

Lecture 2: Cooling of a Hot Bar: The Diffusion Equation

Lecture 3: Laplace's Equation and Harmonic Functions

MAPLE Unplugged 

This was a MAPLE orientation for the UFP Faculty.
Here is the MAPLE worksheet unplugged.mws

Lecture 4: MAPLE and PDE Lab

Week 2: Separation of Variables and Orthogonal Functions

Lecture 5: The Diffusion Equation and Fourier Series

Lecture 6: Convergence of Fourier Series

Lecture 7: The Wave Equation and Separation of Variables

Lecture 8: Sturm-Liouville Problems I

Lecture 9: Sturm-Liouville Problems II

Week 3: Higher Dimensions and Special Functions

Lecture 10:  An Introduction to The Fourier Transform

Lecture 11: Laplace's Equation in a Circle

Lecture 12: Heat Transfer in the Ball

Lecture 13: Sturm-Liouville Problems -- Numerics

Lecture 14: Playing the Timpani: Vibrations of a Circular Membrane 

Additional Resources:

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