CATALOG DESCRIPTION:

1701.386 Introduction to Partial Differential Equations 3 s.h.
(Prerequisite: 1701.231 Ordinary Differential Equations)

This course is a study of partial differential equations and their applications. Topics include a derivation of the wave equation, Laplace’s equation, heat equation, Fourier series and integrals, boundary value problems, Bessel functions and Legendre polynomials.
 

OBJECTIVES:

Students in this course will become familiar with the three main types of partial differential equations (PDEs) and how they arise from physical problems. The important technique of separation of variables will be used to reduce the PDE to a system of ODEs (ordinary differential equations). The use of Fourier series and integrals will be explained. Solutions in other orthogonal functions will be examined. The use of a high-level mathematics programming language (such as Mathematica) to simplify the analytical computations will be encouraged.
 

CONTENT:

1. Partial Differential Equations of Physics

Linear Boundary Value Problems

The Vibrating String

Other examples of Wave Equations

Conduction of Heat

Laplace's Equation

Cylindrical and Spherical Coordinates

Types of Equations and Conditions
 

2. Superposition of Solutions

Linear Combinations

Series Solutions

Separation of Variables

A Plucked String
 

3. Fourier Series

The Basic Series

Examples

Fourier Sine and Cosine Series
 

4. Orthogonal Sets of Functions

Functions as Vectors

Inner Products and Orthonormal Sets

Generalized Fourier Series

Sturm-Liouville Problems
 

5. Fourier Integrals

The Fourier Integral Formula

Sine and Cosine Forms

Exponential Form
 

6. Boundary Value Problems

Formal and Rigorous Solutions

The Vibrating String, Initially Displaced

Nonhomogeneous Differential Equations

Elastic Bar

Temperatures in a Bar

A Dirichlet Problem
 

7. Bessel Functions and Applications

Bessel's Equation

Bessel Functions

Differentiation and Recurrence Formulae

Zeroes of the Bessel Functions

Temperatures in a Long Cylinder

Vibration of a Circular Membrane
 

8. Legendre Polynomials and Applications

Derivation of Legendre Polynomials

Legendre's Series

Temperatures in a Hemisphere
 

TEXTS: The following might be possible texts for this course:

1. Strauss, Walter A., Partial Differential Equations: An Introduction, John Wiley & Sons, 1992.

2. Haberman, Richard, Elementary Applied Partial Differential Equations, 3rd ed., Prentice-Hall, 1998.