**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