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.

Week 1: Meet the Partial Differential Equations

Lecture 1: What is a PDE?

• What is a Partial Differential Equation?
• Linear vs. Nonlinear
• Definition of Linear, Homogeneous
• Superposition
• Transport Equation
• Lecture 1 Notes (.pdf),  LaTeX Source File for Lecture 1 Notes
• Lecture 1 Maple Script, MAPLE Script Handout (.pdf)
• Lecturer:  Prof. Andrew Bernoff

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

• An Introduction to Heat Flow
• Derivation of the Diffusion Equation
• Examples of Solution to the Diffusion Equation
• The Maximum Principle
• Energy Dissipation and Uniqueness
• Lecture 2 Notes (.pdf),  LaTeX Source File for Lecture 2 Notes
• Lecture 2 Maple Script, MAPLE Script Handout (.pdf)
• Lecturer:  Prof. Andrew Bernoff

Lecture 3: Laplace's Equation and Harmonic Functions

• Laplace's Equation and Harmonic Functions [e.g. Re {f(z)}, r^n cos(n \theta) ]
• The Mean Value Property
• Dirichlet's Principle
• Minimal Surfaces
• Lecture 3 Notes (.pdf),  Zipped LaTeX Source Files for Lecture 3 Notes
• Lecture 3 Maple Script, MAPLE Script Handout (.pdf)
• Lecturer:  Prof. Jon Jacobsen

MAPLE Unplugged 

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

Lecture 4: MAPLE and PDE Lab

• An Introductory Lab for MAPLE, oriented towards PDE
• Lab Notes (.pdf),  LaTeX Source Files for Lab Notes
• Maple Script for PDE Lab
• Facilitated by Prof. Andrew Bernoff
  

Week 2: Separation of Variables and Orthogonal Functions

Lecture 5: The Diffusion Equation and Fourier Series

• Separation of variables for  the Dirchlect problem
• The separation constant and corresponding solutions
• Incorporating the homogeneous boundary conditions
• Solving the general initial condition problem
• Lecture 5 Notes (.pdf),  LaTeX Source File for Lecture 5 Notes
• Lecture 5 Maple Script, MAPLE Script Handout (.pdf)
• Lecturer:  Prof. Janine Wittwer

Lecture 6: Convergence of Fourier Series

• Examples of Fourier Series
• Least square error, Bessel's  Inequality, Parseval's Theorem
• The Many Faces of Convergence
• Dirichlet kernel and Gibb's Phenomena
• Lecture 6 Notes (.pdf),  LaTeX Source File for Lecture 6 Notes
• Lecture 6 Maple Script, MAPLE Script Handout (.pdf)
• Lecturer:  Prof. Jorge Aarao

Lecture 7: The Wave Equation and Separation of Variables

• Separation of variables
• Applying initial conditions
• The plucked string revisited.
• Lecture 7 Notes (.pdf),  LaTeX Source File for Lecture 7 Notes
• Lecture 7 Maple Script, MAPLE Script Handout (.pdf)
• Lecturer:  Prof. Mihaela Vajiac

Lecture 8: Sturm-Liouville Problems I

• An example - a metal bar with a partially insulated boundary
• Orthogonality
• Long-term behavior
• Lecture 8 Notes (.pdf),  LaTeX Source File for Lecture 8 Notes
• Lecture 8 Maple Script, MAPLE Script Handout (.pdf)
• Lecturer:  Prof. Katherine Socha

Lecture 9: Sturm-Liouville Problems II

• The general theory: Sturm-Liouville form.
• Legendre Identity 
• Orthogonality and weight functions.
• Lecture 9 Notes (.pdf),  LaTeX Source File for Lecture 9 Notes
• Lecturer: Prof. Katherine Socha

Week 3: Higher Dimensions and Special Functions

Lecture 10:  An Introduction to The Fourier Transform

• Special Lecture by Prof. Gerald Folland (Univeristy of Washington) author of Fourier Analysis and Its Application and Introduction to Partial Differential Equations
  

Lecture 11: Laplace's Equation in a Circle

• The Laplacian in Cylindrical Coordinates
• Separation of Variables
• Poisson Kernel
• Lecture 11 Notes (.pdf),  Zipped LaTeX Source File for Lecture 11 Notes
• Lecture 11 Maple Script, MAPLE Script Handout (.pdf)
• Lecturer:  Prof. Juan Tolosa

Lecture 12: Heat Transfer in the Ball

• The diffusion equation in the sphere
  
• Solution by separation of variables
• Interchanging infinite sums and limits
  
• Lecturer:  Prof. John Polking
• Lecture 12 Notes (.pdf),  Zipped LaTeX Source File for Lecture 12 Notes
• Lecture 12 Maple Script, MAPLE Script Handout (.pdf)

Lecture 13: Sturm-Liouville Problems -- Numerics

• Revisiting an old problem
• Approximating the second derivative
• transforming to a matrix problem
• Automating the process
  
• Lecturer:  Prof. David Arnold
• Lecture 13 Notes (.pdf),  Zipped LaTeX Source File for Lecture 13 Notes

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

• Observations about playing the Timpani
  
• The wave equation in a disc
  
• Separation of Variables
  
• Axisymmetric vs. Non-axisymmetric
  
• Properties of Bessel functions 
• Lecturer:  Prof. Darryl Yong
• Lecture 14 Notes (.pdf),  Zipped LaTeX Source File for Lecture 14 Notes
• Some Vibrational Modes of the Timpani (.pdf)

• Style files for LaTeX Lecture Notes: pcms-l.cls,pcmslmod.tex