University Mathematics Handbook (2015)
XIV. Partial Differential Equations (PDE)
Chapter 4. Second-Order Linear PDE in Two Variables
4.1 Basic Concepts
a. The equation
(*)
when 
are given functions and 
is the unknown function, is the second-order PDE.
b. For every two functions 
and 
from class C2, and for every two constants 
there holds
c. A linear combination of n functions 
for which exist 
, holds
d. The Superposition Principle:
1. If 
are 
solutions of the equation 
, then every linear combination of 
is also a solution of that equation.
2. Let 
be an infinite sequence of solutions of . If the series 
:
I. Uniformly converges to function 
.
II. Twice differentiable and the derivative of the sum is equal to the sum of derivatives, (see VI.4.4), then also is a solution of and
4.2 Classification
a. The expression 
is called the principal part of (*)
b. The equations
(**)
are characteristic equations, the solutions of which, 
and 
are characteristic curves.
c. If 
on 
, then (*) is said to be a hyperbolic equation on .
d. If 
on , then (*) is said to be a parabolic equation on .
e. If 
on , then (*) is said to be an elliptic equation on .
4.3 Reducing PDE into a Canonical Form
a. Change of Variables
Let 
be continuous functions, with continuous first-order partial derivatives and the Jacobian 
in domain .
Changing variables, we denote: 
Using the Chain rule, we get
b. If , the characteristic equations (**) have two sets of characteristic curves 
, 
.
By change of variables 
, 
, and substitution in (*), we get 
, a canonical form of a hyperbolic equation.
c. If in , then equation (*) has one characteristic curve, 
. We choose new variables and 
such that the Jacobian will be nonzero in every point of . Substituting in (*), we get 
, a canonical form of a parabolic equation.
d. If in , then the characteristic equations have two complex solutions. If 
is a solution of (**), we choose new variables , . Substituting in (*), we get 
, a canonical form of an elliptic equation.