Math 01.386 - Introduction to Partial Differential Equations
Math 01.386 Introduction to Partial Differential Equations 3 s.h.
(Prerequisite: Math 01.231 Ordinary Differential Equations with a grade of C- or better)
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.
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.
1. Partial Differential Equations of Physics
Linear Boundary Value Problems
The Vibrating String
Other examples of Wave Equations
Conduction of Heat
Cylindrical and Spherical Coordinates
Types of Equations and Conditions
2. Superposition of Solutions
Separation of Variables
A Plucked String
3. Fourier Series
The Basic Series
Fourier Sine and Cosine Series
4. Orthogonal Sets of Functions
Functions as Vectors
Inner Products and Orthonormal Sets
Generalized Fourier Series
5. Fourier Integrals
The Fourier Integral Formula
Sine and Cosine Forms
6. Boundary Value Problems
Formal and Rigorous Solutions
The Vibrating String, Initially Displaced
Nonhomogeneous Differential Equations
Temperatures in a Bar
A Dirichlet Problem
7. Bessel Functions and Applications
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
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.