Introduction - Heat Conduction: Mathematical Models and Analytical Solutions - page 6

The matrix has two characteristic roots which can be determined by its characteristic
equation

=

a 11 − λ a 12

.

0

a 21 a 22 − λ

The two characteristic roots have the same sign when Δ < 0, and the opposite sign

when 0, there is a vanished characteristic root. This can be easily

verified. Therefore, the equation (1.2) is elliptic, parabolic, or hyperbolic accord-
ingly as two characteristic roots have the same sign, there is a vanished characteristic
root, or two characteristic roots have opposite signs.
An analogous classification can be made in the case of second-order linear equa-
tions with three independent variables for the three characteristic roots of the matrix
of the quadratic form defined by

Δ > 0. When Δ =

3
∑
i , j =

A ( λ )= a ij λ i λ j .

1

+ + =

For instance, the three-dimensional Laplace equation u xx u yy u zz 0 is elliptic

( λ = λ = λ = )

because the three characteristic roots 1 are with the same sign.

1 2 3

The two-dimensional heat-conduction equation

a 2 a 2

positiveconstant
is parabolic because there is a vanished characteristic root

u t = ( u xx + u yy ) , =

a 2 .

λ = 0 , λ = λ =

1 2 3

The three- dimensional nonhomogeneous wave equation

a 2 a 2

u tt = Δ u + f ( x , y , z , t ) , = positive constant

contains four independent variables and is of hyperbolic type because there are both

negative and positive characteristic roots λ 1 = − a 2 and three 1 , λ 2 = λ 3 = λ 4 =

of them are with the same sign.
In general, the second-order linear partial differential equation (1.1) in n inde-
pendent variables is elliptic, parabolic, or hyperbolic at a point P 0 accordingly as
all n characteristic roots are with the same sign, there is a vanished characteristic
root, or the n characteristic roots have different signs but n 1 of them share the

same sign. Here the characteristic roots are those of the matrix of the quadratic form
defined by

−

n
∑
i , j = 1

A ( λ )= a ij ( p 0 ) λ i λ j .

If this is true at all points in a domain D , the equation is of elliptic, parabolic, or
hyperbolic type in D .
It should be remarked here that this classification is only for the second-order
linear equations. Whose types are fixed only by the coefficients of second-order
partial derivatives.