Chapter 1
INTRODUCTION
1.1 Partial differential equations
1.2 Examples
1.3 Strategies for studying PDE
1.4 Overview
1.5 Problems
1.6 References
This chapter surveys the principal theoretical issues concerning the solv-
ing of partial differential equations.
To follow the subsequent discussion, the reader should first of all turn
to Appendix A and look over the notation presented there, particularly the
multiindex notation for partial derivatives.
1.1. PARTIAL DIFFERENTIAL EQUATIONS
A partial differential equation (PDE) is an equation involving an unknown
function of two or more variables and certain of its partial derivatives.
Using the notation explained in Appendix A, we can write out symbol-
ically a typical PDE, as follows. Fix an integer k ≥ 1 and let U denote an
open subset of
Rn.
DEFINITION. An expression of the form
(1) F
(Dku(x), Dk−1u(x),
. . . , Du(x), u(x), x) = 0 (x ∈ U)
is called a kth-order partial differential equation, where
F :
Rnk
×
Rnk−1
× · · · ×
Rn
× R × U → R
1
