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a) parabolic, if ∆(x , y ) = 0, e.g., heat flow and diffusion-type
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0
problems;
b) hyperbolic, if ∆(x , y ) > 0, e.g., vibrating systems, wave motion;
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0
c) elliptic, if ∆(x , y ) < 0, e.g., steady-state, potential-type prob-
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0
lems.
Notice that in general a second order equation may be of one type at a specific point,
and of another type at some other point. In order to illustrate the significance of the
2
discriminant ∆ = B − AC, we next describe a method of reducing the equation (3.1)
to a canonical form. Similar to the second order algebraic equations, we use change of
coordinates to reduce the equation to a simpler form. Define the new variables as

ξ = ξ(x, y) ξ x ξ y
with J = 6= 0. (3.3)
ψ = ψ(x, y) ψ x ψ y
We then use the chain rule to compute the terms of the equation (3.1) in these new
variables:
u x = u ξ ξ x + u ψ ψ x ,
u y = u ξ ξ y + u ψ ψ y .
To express the second order derivatives in terms of the (ξ, ψ) variables, differentiate the
above expressions for the first derivatives using the chain rule again:

2
2
u xx = u ξξ ξ + 2u ξψ ξ x ψ x + u ψψ ψ + l.o.t ,
x x
u xy = u ξξ ξ x ξ y + u ξψ (ξ x ψ y + ξ y ψ x ) + u ψψ ψ x ψ y + l.o.t ,
2
2
u yy = u ξξ ξ + 2u ξψ ξ y ψ y + u ψψ ψ + l.o.t .
y
y
Here l.o.t. stands for the low order terms, which contain only one derivative of the
unknown u. Using these expressions for the second order derivatives of u, we can rewrite
equation (3.1) in these variables as
∗
∗
∗
∗
A u ξξ + 2B u ξψ + C u ψψ = F (ξ, ψ, u, u ξ , u ψ ), (3.4)
∗
∗
∗
where the new coefficients of the higher order terms A , B and C are expressed via the
original coefficients and the change of variables formulas as follows:
2
2
∗
A = Aξ + 2Bξ x ξ y + Cξ , (3.5)
x
y
∗
B = Aξ x ψ x + B(ξ x ψ y + ξ y ψ x ) + Cξ y ψ y , (3.6)
2
2
∗
C = Aψ + 2Bψ x ψ y + Cψ . (3.7)
x
y
It is obvious that the discriminant for the equation in the new variables via the new
coefficients is equal:
∗ 2
∗
∗
∗
∆ = (B ) − A C .
We need to guarantee that the reduced equation will have the same type as the original
one. Otherwise, the classification given by the definition is meaningless, since in that case
the same physical event will be described by equations of different types, depending on
the particular coordinate system. The following statement provides such a guarantee.
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