Notes for Studies in Classical Linear Partial Differential Equations

In the Winter of 2003, I was commissioned to write a set of notes for studies in classical linear partial differential equations and to publish these notes on the web site for Maple Waterloo Software, Inc. I had taught a course in this subject many times before retiring from Georgia Tech. Even after retiring, I continued to teach such a course in the summers. The 4000 level summer course was for working engineers who wanted to get a graduate degree in engineering while continuing to work. The course fulfilled one of their math requirements for this degree. Naturally, I had constructed notes during these years of experience and relied on these to help in satisfying the commission from MWS, Inc.

These notes are now available three ways. They are available from the MWS site for persons not having the software package Maple on their computer and as a Maple worksheet for persons having Maple. In this latter form, the examples can be modified to create alternate examples. The notes are also available below with minor modifications from what appears at the Maple site. The modifications occur as small changes are made to the text or to the models, and as additions are created to amplify these ideas.

Having the notes accessible to me from my web page keeps them alive. I can easily make changes. The student is encouraged to use them freely. Any faculty is welcomed to reference these for their student’s use.

These notes are both more and less than what is offered in MATH 4581 at Georgia Tech.

The ideas of these notes are mostly linear ideas even though we live in a nonlinear world. Recall that the calculus begins one dimensional and progresses to multidimensional. The multidimensional calculus stands on a firm foundation laid down in the simpler one dimensional situation. So, too, the nonlinear models for physical phenomena will step off from this introduction to the structure of linear boundary value problems in partial differential equations.

The history of Science, Engineering, and Mathematics swirl around the ideas invoked here.

Linear Partial Differential Equations and Boundary Value Problems

Chapter 1: A Geometry for Linear Spaces

Section 1.1: Linear Spaces.

Section 1.2: Geometry in Linear Spaces ,

Section 1.3: Orthogonal Families .

Section 1.4: The Gramm-Schmidt Process.

Section 1.5: Projections . 

Section 1.6:: A Maximal Orthonormal Family .

Chapter 2: Linear Spaces of Functions

 Section 2.1: Convergence .

 Section 2.2: Extensions .

 Section 2.3: Fourier Series Convergence .

Section 2.4: Calculus on Fourier Series .

Chapter 3: A Review of Ordinary Differential Equations

Section 3.1: Ordinary Differential Equations Review .

Section 3.2: Homogeneous and Nonhomogeneous Differential Equations .

Section 3.3: Eigenvalues and Eigenfunctions .

Chapter 4: The Heat Equation

Section 4.1: The Simple Heat Equation .

Section 4.2: Diffusion with Radiation Cooling .

Section 4.3: Insulated Boundary Conditions .

Section 4.4: Convection Across Boundaries .

Section 4.5: A Structure for Solutions of the Diffusion Equation .

Section 4.6: Internal Heating .

Section 4.7: Periodic Forcing Functions .

Section 4.8: Time Dependent Boundary Conditions .

Chapter 5: The Wave Equation

Section 5.1: The One Dimensional Wave Equation .

Section 5.2: The Solution of d’Alembert .

Section 5.3: The Solution of d’Alembert on Intervals .

Section 5.4: A String in a Viscous Medium .

Section 5.5: Different Boundary Conditions .

Chapter 6: The Equation of Laplace

Section 6.1: Laplace’s Equation on a Rectangle .

Section 6.2: Laplace’s Equation with Neumann Boundary Conditions .

Section 6.3: The Structure of Solutions for Laplace’s Equation .

Section 6.4: Laplace’s Equation on a Disk .

Section 6.5: Laplace’s Equation on a Ring or Half Disk .

Chapter 7: Time and Space

Section 7.1: The Heat Equation on a Rectangle .

Section 7.2: Two Dimensional Diffusion with Neumann Boundary Conditions .

Section 7.3: The Heat Equation on a Disk .

Section 7.4: Partial Differential Equations in the Recipe for a Cheese Cake .

Section 7.5: Warm Spheres .

Section 7.6: Vibrations of a Circular Drum .

Chapter 8: Numerical Methods

Section 8.1: Difference Methods for Ordinary Differential Equations .

Section 8.2: Numerical Methods for the One Dimensional Heat Equation .

Section 8.3: Numerical Methods for the One Dimensional Wave Equation .

Chapter 9: A Brief Look at the Methods of Characteristics

Section 9.1: An Introduction to First Order Partial Differential Equations .

Section 9.2: Characteristics for First Order Partial Differential Equations .

Conclusion: It Stops Here

Epilog: An Overview for the Method of Separation of Variables Using Computer Technology