Myint Tyn U., Debnath L. Linear Partial Differential Equations for Scientists and Engineers
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4
Classification of Second-Order Linear
Equations
“When we have a good und erstandin g of the problem, we are able to clear
it of all auxiliary notions and to reduce it to simplest element.”
Ren´e Descartes
“The first process ... in the effectual study of sciences must be one of sim-
plification and reduction of the results of previous investigations to a form
in which the mind can grasp them.”
James Clerk Maxwell
4.1 Second-Order Equations in Two Independent
Variables
The general linear second-order partial differential equation in one depen-
dent variable u may be written as
n
i,j=1
A
ij
u
x
i
x
j
+
n
i=1
B
i
u
x
i
+ Fu = G, (4.1.1)
in which we assume A
ij
= A
ji
and A
ij
, B
i
, F ,andG are real-valued
functions defined in some region of the space (x
1
,x
2
,...,x
n
).
Here we shall be concerned with second-order equations in the depen-
dent variable u and the independent variables x, y. Hence equation (4.1.1)
can be put in the form
Au
xx
+ Bu
xy
+ Cu
yy
+ Du
x
+ Eu
y
+ Fu = G, (4.1.2)
where the coefficients are functions of x and y and do not vanish simulta-
neously. We shall assume that the function u and the coefficients are twice
continuously differentiable in some domain in R
2
.
92 4 Classification of Second-Order Linear Equations
The classification of partial differential equations is suggested by the
classification of the quadratic equation of conic sections in analytic geom-
etry. The equation
Ax
2
+ Bxy + Cy
2
+ Dx + Ey + F =0,
represents hyperbola, parabola, or ellipse accordingly as B
2
− 4AC is pos-
itive, zero, or negative.
The classification of second-order equations is based upon the possibility
of reducing equation (4.1.2) by coordinate transformation to canonical or
standard form at a point. An equation is said to b e hyperbolic, parabolic,
or elliptic at a point (x
0
,y
0
) accordingly as
B
2
(x
0
,y
0
) − 4A (x
0
,y
0
) C (x
0
,y
0
) (4.1.3)
is positive, zero, or negative. If this is true at all points, then the equation
is said to be hyperbolic, parab oli c, or elliptic in a domain. In the case of
two independent variables, a transformation can always be found to reduce
the given equation to canonical form in a given domain. However, in the
case of several independent variables, it is not, in general, possible to find
such a transformation.
To transform equation (4.1.2) to a canonical form we make a ch ange of
independent variables. Let the new variables be
ξ = ξ (x, y) ,η= η (x, y) . (4.1.4)
Assuming that ξ and η are twice continuously differentiable and that the
Jacobian
J =
ξ
x
ξ
y
η
x
η
y
, (4.1.5)
is nonzero in the region under consideration, then x and y can be deter-
mined uniquely from th e system (4.1.4). Let x and y be twice continuou sly
differentiable functions of ξ and η.Thenwehave
u
x
= u
ξ
ξ
x
+ u
η
η
x
,u
y
= u
ξ
ξ
y
+ u
η
η
y
,
u
xx
= u
ξξ
ξ
2
x
+2u
ξη
ξ
x
η
x
+ u
ηη
η
2
x
+ u
ξ
ξ
xx
+ u
η
η
xx
, (4.1.6)
u
xy
= u
ξξ
ξ
x
ξ
y
+ u
ξη
(ξ
x
η
y
+ ξ
y
η
x
)+u
ηη
η
x
η
y
+ u
ξ
ξ
xy
+ u
η
η
xy
,
u
yy
= u
ξξ
ξ
2
y
+2u
ξη
ξ
y
η
y
+ u
ηη
η
2
y
+ u
ξ
ξ
yy
+ u
η
η
yy
.
Substituting these values in equation (4.1.2) we obtain
A
∗
u
ξξ
+ B
∗
u
ξη
+ C
∗
u
ηη
+ D
∗
u
ξ
+ E
∗
u
η
+ F
∗
u = G
∗
, (4.1.7)
where
4.2 Canonical Forms 93
A
∗
= Aξ
2
x
+ Bξ
x
ξ
y
+ Cξ
2
y
,
B
∗
=2Aξ
x
η
x
+ B (ξ
x
η
y
+ ξ
y
η
x
)+2Cξ
y
η
y
,
C
∗
= Aη
2
x
+ Bη
x
η
y
+ Cη
2
y
,
D
∗
= Aξ
xx
+ Bξ
xy
+ Cξ
yy
+ Dξ
x
+ Eξ
y
, (4.1.8)
E
∗
= Aη
xx
+ Bη
xy
+ Cη
yy
+ Dη
x
+ Eη
y
,
F
∗
= F, G
∗
= G.
The resulting equation (4.1.7) is in the same form as the original equation
(4.1.2) under the general transformation (4.1.4). The nature of the equation
remains invariant under such a transformation if the Jacobian does not
vanish. This can be seen from the fact that the sign of the discriminant
does not alter under the transformation, that is,
B
∗2
− 4A
∗
C
∗
= J
2
B
2
− 4AC
, (4.1.9)
which can be easily verified. It should be noted here that the equation can
be of a different type at different points of the domain, but for our purpose
we shall assume that the equation under consideration is of the single type
in a given domain.
The classification of equation (4.1.2) depends on the co efficients A (x, y),
B (x, y), and C (x, y) at a given point (x, y). We shall, therefore, rewrite
equation (4.1.2) as
Au
xx
+ Bu
xy
+ Cu
yy
= H (x, y, u, u
x
,u
y
) , (4.1.10)
and equation (4.1.7) as
A
∗
u
ξξ
+ B
∗
u
ξη
+ C
∗
u
ηη
= H
∗
(ξ,η,u,u
ξ
,u
η
) . (4.1.11)
4.2 Canonical Forms
In this section we shall consider the problem of reducing equation (4.1.10)
to canonical form.
We suppose first that none of A, B, C, is zero. Let ξ and η be new
variables such that the coefficients A
∗
and C
∗
in equation (4.1.11) vanish.
Thus, from (4.1.8), we have
A
∗
= Aξ
2
x
+ Bξ
x
ξ
y
+ Cξ
2
y
=0,
C
∗
= Aη
2
x
+ Bη
x
η
y
+ Cη
2
y
=0.
Thesetwoequationsareofthesametypeandhencewemaywritethemin
the form
Aζ
2
x
+ Bζ
x
ζ
y
+ Cζ
2
y
=0, (4.2.1)
94 4 Classification of Second-Order Linear Equations
in which ζ stand for either of th e functions ξ or η. Dividing through by ζ
2
y
,
equation (4.2.1) becomes
A
ζ
x
ζ
y
2
+ B
ζ
x
ζ
y
+ C =0. (4.2.2)
Along the curve ζ = constant, we have
dζ = ζ
x
dx + ζ
y
dy =0.
Thus,
dy
dx
= −
ζ
x
ζ
y
, (4.2.3)
and therefore, equation (4.2.2) may be written in the form
A
dy
dx
2
− B
dy
dx
+ C =0, (4.2.4)
the roots of which are
dy
dx
=
B +
B
2
− 4AC
/2A, (4.2.5)
dy
dx
=
B −
B
2
− 4AC
/2A. (4.2.6)
These equations, which are known as the characteristic equations, are or-
dinary differential equations for families of curves in the xy-plane along
which ξ = constant and η = constant. Th e integrals of equations (4.2.5)
and (4.2.6) are called the characteristic curves. Since the equations are
first-order ordinary differential equations, the solutions may be written as
φ
1
(x, y)=c
1
,c
1
= constant,
φ
2
(x, y)=c
2
,c
2
= constant.
Hence the tr ansformation s
ξ = φ
1
(x, y) ,η= φ
2
(x, y) ,
will transform equation (4.1.10) to a canonical form.
(A) Hyperbolic Type
If B
2
−4AC > 0, then integration of equations (4.2.5) and (4.2.6) yield
two real and distinct families of characteristics. Equation (4.1.11) reduces
to
u
ξη
= H
1
, (4.2.7)
4.2 Canonical Forms 95
where H
1
= H
∗
/B
∗
. It can be easily shown that B
∗
=0.Thisformiscalled
the first canonical f orm of the hyperbolic equation.
Now if new in dependent variables
α = ξ + η, β = ξ − η, (4.2.8)
are introduced, then equation (4.2.7) is transformed into
u
αα
− u
ββ
= H
2
(α, β, u, u
α
,u
β
) . (4.2.9)
This form is called the second canonical form of the hyperbolic equation.
(B) Parabolic Type
In this case, we have B
2
− 4AC = 0, and equations (4.2.5) and (4.2.6)
coincide. Thus, there exists one real family of characteristics, and we obtain
only a single integral ξ = constant (or η = constant).
Since B
2
=4AC an d A
∗
= 0, we find that
A
∗
= Aξ
2
x
+ Bξ
x
ξ
y
+ Cξ
2
y
=
√
Aξ
x
+
√
Cξ
y
2
=0.
From this it follows that
A
∗
=2Aξ
x
η
x
+ B (ξ
x
η
y
+ ξ
y
η
x
)+2Cξ
y
η
y
=2
√
Aξ
x
+
√
Cξ
y
√
Aη
x
+
√
Cη
y
=0,
for arbitrary values of η (x, y) which is functionally independent of ξ (x, y);
for instance, if η = y, the Jacobian does not vanish in the domain of parabol-
icity.
Division of equation (4.1.11) by C
∗
yields
u
ηη
= H
3
(ξ,η,u,u
ξ
,u
η
) ,C
∗
=0. (4.2.10)
This is called the canonical form of the parabolic equation.
Equation (4.1.11) may also assume the form
u
ξξ
= H
∗
3
(ξ,η,u,u
ξ
,u
η
) , (4.2.11)
if we choose η = constant as the integral of equation (4.2.5).
(C) Elliptic Type
For an equation of elliptic type, we have B
2
− 4AC < 0. Consequently,
the quadratic equation (4.2.4) has no real solutions, but it has two complex
conjugate solutions which are continuous complex-valued functions of the
real variables x and y. Thus, in this case, there are no real characteristic
curves. However, if the coefficients A, B,andC are analytic functions of
x and y, then one can consider equation (4.2.4) for complex x and y.A
function of two real variables x and y is said to be analytic in a certain
domain if in some neighborhood of every point (x
0
,y
0
) of this domain, the
96 4 Classification of Second-Order Linear Equations
function can be represented as a Taylor series in the variables (x − x
0
)and
(y − y
0
).
Since ξ and η are complex, we intro du ce new real variables
α =
1
2
(ξ + η) ,β=
1
2i
(ξ − η) , (4.2. 12)
so that
ξ = α + iβ, η = α − iβ. (4.2.13)
First, we transform equation s (4.1.10). We then have
A
∗∗
(α, β) u
αα
+ B
∗∗
(α, β) u
αβ
+ C
∗∗
(α, β) u
ββ
= H
4
(α, β, u, u
α
,u
β
) ,
(4.2.14)
in which the coefficients assume th e same form as the coefficients in equation
(4.1.11). With the use of (4.2.13), the equati ons A
∗
= C
∗
= 0 become
Aα
2
x
+ Bα
x
α
y
+ Cα
2
y
−
Aβ
2
x
+ Bβ
x
β
y
+ Cβ
2
y
+i [2Aα
x
β
x
+ B (α
x
β
y
+ α
y
β
x
)+2Cα
y
β
y
]=0,
Aα
2
x
+ Bα
x
α
y
+ Cα
2
y
−
Aβ
2
x
+ Bβ
x
β
y
+ Cβ
2
y
−i [2Aα
x
β
x
+ B (α
x
β
y
+ α
y
β
x
)+2Cα
y
β
y
]=0,
or,
(A
∗∗
− C
∗∗
)+iB
∗∗
=0, (A
∗∗
− C
∗∗
) − iB
∗∗
=0.
These equations are satisfied if and only i f
A
∗∗
= C
∗∗
and B
∗∗
=0.
Hence, equation (4.2.14) transforms into the form
A
∗∗
u
αα
+ A
∗∗
u
ββ
= H
4
(α, β, u, u
α
,u
β
) .
Dividing through by A
∗∗
, we obtain
u
αα
+ u
ββ
= H
5
(α, β, u, u
α
,u
β
) , (4.2.15)
where H
5
=(H
4
/A
∗∗
). This is called the canonical form of the elliptic
equation.
We close this discussion of canonical forms by adding an important
comment. From mathematical and physical points of view, characteristics
or characteristic coordinates play a very important physical role in hyper-
bolic equations. However, t hey do n ot play a particularly physical role in
parabolic and elliptic equations, but their role is somewhat mathematical
4.2 Canonical Forms 97
in solving th ese equations. In general, first-order partial differential equa-
tions such as advection-reaction equations are regarded as hyperbolic be-
cause they describe pr opagation of waves like the wave equation. On the
other hand, second-order linear partial differential equations with constant
coefficients are sometimes classified by the associated dispersion relation
ω = ω (κ
κ
κ) as defined in Section 13.3. In one-dimensional case, ω = ω (k).
If ω (k)isrealandω
′′
(k) = 0, the equation is called dispersive.Theword
dispersive simply means that the phase velocity c
p
=(ω/k) of a p lane wave
solution, u (x, t)=A exp [i (kx − ωt)] depends on the wavenumber k.This
means that waves of different wavelength propagate with different phase ve-
locities and h ence, disperse in the medium. If ω = ω (k)=σ (k)+iµ (k)is
complex, the associated partial differential equation is called diffusive.From
a physical point of view, such a classification of equations is p articularly
useful. Both dispersive and diffusive equations are physically important,
and such equations will be discussed in Chapter 13.
Example 4.2.1. Consider the equation
y
2
u
xx
− x
2
u
yy
=0.
Here
A = y
2
,B=0,C= −x
2
.
Thus,
B
2
− 4AC =4x
2
y
2
> 0.
The equation is hyperbolic everywhere except on the coor din ate axes x =0
and y = 0. From the characteristic equations (4.2.5) and (4.2.6), we have
dy
dx
=
x
y
,
dy
dx
= −
x
y
.
After integration of these equations, we obtain
1
2
y
2
−
1
2
x
2
= c
1
,
1
2
y
2
+
1
2
x
2
= c
2
.
The first of these curves is a family of hyperbolas
1
2
y
2
−
1
2
x
2
= c
1
,
and the second is a family of circles
1
2
y
2
+
1
2
x
2
= c
2
.
To transform the given equation to canonical form, we consider
98 4 Classification of Second-Order Linear Equations
ξ =
1
2
y
2
−
1
2
x
2
,η=
1
2
y
2
+
1
2
x
2
.
From the relations (4.1.6), we have
u
x
= u
ξ
ξ
x
+ u
η
η
x
= −xu
ξ
+ xu
η
,
u
y
= u
ξ
ξ
y
+ u
η
η
y
= yu
ξ
+ yu
η
,
u
xx
= u
ξξ
ξ
2
x
+2u
ξη
ξ
x
η
x
+ u
ηη
η
2
x
+ u
ξ
ξ
xx
+ u
η
η
xx
= x
2
u
ξξ
− 2x
2
u
ξη
+ x
2
u
ηη
− u
ξ
+ u
η
.
u
yy
= u
ξξ
ξ
2
y
+2u
ξη
ξ
y
η
y
+ u
ηη
η
2
y
+ u
ξ
ξ
yy
+ u
η
η
yy
= y
2
u
ξξ
+2y
2
u
ξη
+ y
2
u
ηη
+ u
ξ
+ u
η
.
Thus, the given equation assumes the canonical form
u
ξη
=
η
2(ξ
2
− η
2
)
u
ξ
−
ξ
2(ξ
2
− η
2
)
u
η
.
Example 4.2.2. Consider the partial differential equation
x
2
u
xx
+2xy u
xy
+ y
2
u
yy
=0.
In this case, the discriminant is
B
2
− 4AC =4x
2
y
2
− 4x
2
y
2
=0.
The equation is therefore parabolic everywhere. The characteristic equation
is
dy
dx
=
y
x
,
and hence, the characteristics are
y
x
= c,
which is the equation of a family of straight lines.
Consider the transformation
ξ =
y
x
,η= y,
where η is chosen arbitrarily. The given equation is then reduced to the
canonical form
y
2
u
ηη
=0.
Thus,
u
ηη
=0 for y =0.
4.3 Equations with Constant Coefficients 99
Example 4.2.3. The equation
u
xx
+ x
2
u
yy
=0,
is elliptic everywhere except on the coordinate axis x = 0 because
B
2
− 4AC = −4x
2
< 0,x=0.
The characteristic equations are
dy
dx
= ix,
dy
dx
= −ix.
Integration yields
2y − ix
2
= c
1
, 2y + ix
2
= c
2
.
Thus, if we write
ξ =2y − ix
2
,η=2y + ix
2
,
and hence,
α =
1
2
(ξ + η)=2y, β =
1
2i
(ξ − η)=−x
2
,
we obtain the canonical form
u
αα
+ u
ββ
= −
1
2β
u
β
.
It should be remarked here that a given partial differential equation may
be of a different type in a different domain. Thus, for example, Tricomi’s
equation
u
xx
+ xu
yy
=0, (4.2.16)
is elliptic for x>0 and hyperbolic for x<0, since B
2
− 4AC = −4x.For
a detailed treatment, see Hellwig (1964).
4.3 Equations with Constant Coefficients
In this case of an equation with real constant coefficients, the equation is of
a single type at all points in the domain. This is because the discriminant
B
2
− 4AC is a constant.
From the characteristic equations
dy
dx
=
B +
B
2
− 4AC
/2A, (4.3.1)
100 4 Classification of Second-Order Linear Equations
we can see that the characteristics
y =
B +
√
B
2
− 4AC
2A
x + c
1
,y=
B −
√
B
2
− 4AC
2A
x + c
2
, (4.3.2)
are two families of straight lines. Consequently, the characteristic coordi-
nates take the form
ξ = y − λ
1
x, η = y − λ
2
x, (4.3.3)
where
λ
1,2
=
B+
√
B
2
− 4AC
2A
. (4.3.4)
The linear second-order partial differential equation with constant coeffi-
cients may be written in the general form as
Au
xx
+ Bu
xy
+ Cu
yy
+ Du
x
+ Eu
y
+ Fu = G (x, y) . (4.3.5)
In particular, the equation
Au
xx
+ Bu
yy
+ Cu
yy
=0, (4.3.6)
is called the Euler equation.
(A) Hyperbolic Type
If B
2
− 4AC > 0, the equation is of hyperbolic type, in which case the
characteristics form two distinct families.
Using (4.3.3), equation (4.3.5) becomes
u
ξη
= D
1
u
ξ
+ E
1
u
η
+ F
1
u + G
1
(ξ,η) , (4.3.7)
where D
1
, E
1
,andF
1
are constants. Here, since the coefficients are con-
stants, the lower order terms are expressed explicitly.
When A = 0, equation (4.3.1) does not hold. In this case, the charac-
teristic equation may be put in the form
−B (dx/dy)+C (dx/dy)
2
=0,
which may again be rewritten as
dx/dy =0, and − B + C (dx/dy)=0.
Integration gives
x = c
1
,x=(B/C ) y + c
2
,
where c
1
and c
2
are integration constants. Thus, the characteristic coordi-
nates are