Asynchronous Studies in Undergraduate Partial Differential Equations

These lecture notes were constructed in preparation for the presentation of a course in which the students will progress in their studies asynchronously. Many of these students will be scientists and engineers and will have responsibilities which come with their work. Beginning the studies that go with these notes is an added requirement and may have a lower priority for them than ongoing responsibilities. Thus, it is necessary that the students participate at a pace which is suitable for their schedules. The instructor and the administration of Continuing Education at Georgia Tech will try to assist the student's management of schedules.

There are four parts of the course:

  1. These notes form the outline for a series of lectures.
  2. The lectures are accessed thorough the Office of Continuing Education at Georgia Tech.
  3. Each lecture has an accompanying Maple worksheet which parallels the lecture notes.
  4. Questions can be presented to the instructor via e-mail.

The notes will be visible during the presentation of lectures to form a skeleton on which to hang ideas and to give direction. If the students makes a paper copy of these lecture notes, then the notes can be annotated as the presentation is made.

The format for delivering the lectures will be determined by the technology available to the student and to Georgia Tech. It was requested of the instructor that the lectures average about twenty minutes. The students should spend more time with the Maple Worksheets to amplify the lecture.

The Maple Worksheets should be viewed as active documents. The student can modify examples, animate graphs, and create new problems. The Worksheets will form the basis for conversations with the instructor.

The instructor will check e-mail with regularity. Public answers will be displayed to public questions. New, as well as revised Maple worksheets will be created in response to student questions.

The ideas of this course are 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.

What follows is a presentation of the lecture titles. Lectures notes are pdf files; Worksheets are Maple files.

 

Welcoming Lecture, with accompanying worksheet  .

Part I. Preliminaries: Spaces, Geometry, and Ordinary Differential Equations.

Lecture 1: Linear Spaces , with accompanying worksheet .

Lecture 2: Geometry, with accompanying worksheet .

Lecture 3: Orthogonal Families , with accompanying worksheet .

Lecture 4: The Gramm-Schmidt Process, with accompanying worksheet .

Lecture 5: Projections , with accompanying worksheet .

 Lecture 6: Examples: Computations and Graphs, with accompanying worksheet .

 Lecture 7: Periodic Extensions , with accompanying worksheet .

 Lecture 8: General Convergence , with accompanying worksheet .

 Lecture 9: Fourier Series Convergence , with accompanying worksheet .

Lecture 9 Addendum : Examples: Convergence of Series, with accompanying worksheet

 Lecture 10: Calculus on Fourier Series , with accompanying worksheet .

 Lecture 11: Generation of Orthogonal Functions , with accompanying worksheet .

 Lecture 12: Elementary Differential Equations I , with accompanying worksheet .

 Lecture 13: Elementary Differential Equations II , with accompanying worksheet .

 Lecture 14: Non-homogeneous Initial Value Problems, with accompanying worksheet .

 Lecture 15: Complete Orthonormal Families , with accompanying worksheet .

 

Part II. The One Dimensional Diffusion Equation.

Lecture 16: The Simple Heat Equation , with accompanying worksheet .

Lecture 17: Radiation Cooling , with accompanying worksheet .

Lecture 17 Addendum : Steady State Solutions for Diffusion Equations, with accompanying worksheet .

Lecture 18: Insulated Boundary Conditions , with accompanying worksheet .

Lecture 19: Convection Boundary Conditions , with accompanying worksheet .

Lecture 20: Structure of Solutions , with accompanying worksheet .

Lecture 20 Addendum : Newtonís Point-Mass Cooling, with accompanying worksheet .

Material Review : Pulling it together, with accompanying worksheet .

 

Part III. The One Dimensional Wave Equation.

Lecture 21: The Wave Equation in One Dimension , with accompanying worksheet .

Lecture 22: d'Alembert's Solution on the real line, with accompanying worksheet .

Lecture 23: d'Alembert's Solution on intervals, with accompanying worksheet .

Lecture 24: The Structure of Solutions, with accompanying worksheet .

Lecture 25: A Viscous Medium, with accompanying worksheet .

Lecture 25b: You will like this solution for the problem of Lecture 25 better.

Lecture 26: Different Boundary Condtions, with accompanying worksheet .

 

Part IV. Laplace's Equation.

Lecture 27: Laplace's Equation on a Rectangle, with accompanying worksheet .

Lecture 28: Insulated Boundaries, with accompanying worksheet .

Lecture 29: The Structure of Solutions, with accompanying worksheet .

Lecture 30: The Laplacian in Polar Coordinates, with accompanying worksheet .

Lecture 31: On a Ring and On a Half-Disk, with accompanying worksheet .

 

Part V. Time and Space

Lecture 32: The Heat Equation on a Rectangle, with accompanying worksheet .

Lecture 32b: Two Dimensional Diffusion, Neumann Boundary Conditions: An Example

Lecture 33: The Heat Equation on a Disk , with accompanying worksheet .

Lecture 34: Recipe for a Cheese Cake , with accompanying worksheet .

Lecture 35: Warm Spheres, with accompanying worksheet .

Lecture 36: Vibrations of a Drum, with accompanying worksheet .

 

Part VI. Laplace Transforms.

Lecture 37: Laplace Transforms and ODE's, with accompanying worksheet .

Lecture 38: The Heaviside and Dirac Functions, with accompanying worksheet .

Lecture 39: Periodic Forcing Functions, with accompanying worksheet .

Lecture 40: First Order PDE's, with accompanying worksheet .

Lecture 41: The Diffusion Equation, with accompanying worksheet .

Lecture 42: The Wave Equation, with accompanying worksheet .

 

Part VII. Miscellaneous Problems.

Lecture 43: Vibrations of a Rigid Beam, with accompanying worksheet .

Lecture 44: Forcing Functions in the Diffusion Equation, with accompanying worksheet .

Lecture 45: Periodic Forcing Functions, with accompanying worksheet .

Lecture 46: Time Dependent Boundary Conditions, with accompanying worksheet .