Class Notes for Math 4348

Winter Quarter, 1998
Instructor: Professor Jim Herod
email: herod@math.gatech.edu
Phone: (404) 894 9240 Fax: (404) 894 4409

Catalog Description: MATH 4348.

Solutions of boundary value problems for partial differential equations by Green's functions. Representative solutions for potential and diffusion equations. Applications.

Electronic Text

The text which follows is provided as an Adobe Acrobat file.

Each section is followed by a Maple construction to illustrate graphs and computations.

Preface

Introduction

Chapter 1: Integral Equations

Section 1 Geometry and a Linear Function

   Maple construction for Section 1

Section 2 The Fredholm Alternative Theorems

   Maple construction for Section 2

Section 3 Separable Kernels

   Maple construction for Section 3

Section 4 The Kernel is Small

   Maple construction for Section 4

Section 5 Alternate "K is Small"

   Maple construction for Section 5

Section 6 Neither Small nor Separable

   Maple construction for Section 6

Problems A Compendium of Problems

Chapter 2: Ordinary Differential Equations

Section 1 More About the Space

   Maple construction for Section 1

Section 2 Differential Operators and Their Adjoints

   Maple construction for Section 2

Section 3 The Fredholm Alternative Theorems

   Maple construction for Section 3

Section 4 G(x,t) in the First Alternative

   Maple construction 1, construction 2, and construction 3 for Section 4

Section 5 G(x,t) and the Dirac Delta Distribution

   Maple construction for Section 5

Section 6 G(x,t) in the Second Alternative

   Maple construction for Section 6

Problems A Compendium of Problems

Chapter 3: Partial Differential Equations

Section 1 Classification for Second Order Equations

   Maple construction for Section 1

Section 2 A Standard Form for Second Order Equations

   Maple construction for Section 2

Section 3 A Calculus Review

Section 4 Green's Identities

Section 5 Adjoints of Differential Operators

Section 6 Green's Functions for Dirichlet Problems in the Plane

Section 7 A Maximum Principle

Section 8 An Example: The Shape of a Drum

Appendix

   Maple construction for the Appendix

Class Project