pde - Introduction to Partial Dierential Equations John...

Introduction to Partial Differential Equations
John Douglas Moore
May 21, 2003

Preface
Partial differential equations are often used to construct models of the most
basic theories underlying physics and engineering. For example, the system of
partial differential equations known as Maxwell’s equations can be written on
the back of a post card, yet from these equations one can derive the entire theory
of electricity and magnetism, including light.
Our goal here is to develop the most basic ideas from the theory of partial dif-
ferential equations, and apply them to the simplest models arising from physics.
In particular, we will present some of the elegant mathematics that can be used
to describe the vibrating circular membrane. We will see that the frequencies
of a circular drum are essentially the eigenvalues from an eigenvector-eigenvalue
problem for Bessel’s equation, an ordinary differential equation which can be
solved quite nicely using the technique of power series expansions. Thus we
start our presentation with a review of power series, which the student should
have seen in a previous calculus course.
It is not easy to master the theory of partial differential equations. Unlike
the theory of ordinary differential equations, which relies on the “fundamental
existence and uniqueness theorem,” there is no single theorem which is central
to the subject. Instead, there are separate theories used for each of the major
types of partial differential equations that commonly arise.
However, there are several basic skills which are essential for studying all
types of partial differential equations. Before reading these notes, students
should understand how to solve the simplest ordinary differential equations,
such as the equation of exponential growth dy/dx = ky and the equation of
simple harmonic motion d
2
y/dx
2
+ ωy = 0, and how these equations arise in
modeling population growth and the motion of a weight attached to the ceiling
by means of a spring. It is remarkable how frequently these basic equations
arise in applications. Students should also understand how to solve first-order
linear systems of differential equations with constant coefficients in an arbitrary
number of unknowns using vectors and matrices with real or complex entries.
(This topic will be reviewed in the second chapter.) Familiarity is also needed
with the basics of vector calculus, including the gradient, divergence and curl,
and the integral theorems which relate them to each other. Finally, one needs
ability to carry out lengthy calculations with confidence. Needless to say, all
of these skills are necessary for a thorough understanding of the mathematical
i

language that is an essential foundation for the sciences and engineering.