
Chapter 1 4
Second-Order Linear PDEs:
Classification, Problems,
and Pa rticular Solutions
14.1 Classification of Second-Order Linear Partial
Differential Equations
14.1.1 Equations with Two Independent Variables
◮ Examples of equations encountered in applications.
Three basic types of partial differential equations are distinguished—parabolic, hyper-
bolic, and elliptic. The solutions of the equations pertaining to each of the types have
their own characteristic qualitative differences.
The simplest example of a parabolic equation is the heat equation
∂w
∂t
−
∂
2
w
∂x
2
= 0, (14.1.1.1)
where the variables t and x play the role of time and the spatial coordinate,