MATH3078/3978 - Week 1 Lecture 2
Classification of 2nd-Order Linear PDEs
Compiled by F. Viera from the given references
© Draft date July 31, 2016
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Contents
1 Classification of 2nd-Order Linear PDEs
1
1.1
The Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Coordinate Transformations . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.3
Canonical Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.4
The Hyperbolic Case – Characteristics . . . . . . . . . . . . . . . . . . .
5
1.5
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 1
Classification of 2nd-Order Linear
PDEs
There are two major steps in the process of classification of 2nd-order linear PDEs in
two or more independent variables:
(1) The classification process itself, based on the sign of the expression known as the
discriminant ∆ of the PDE, which is a function of the coefficients of the principal
part of the PDE, and
(2) The coordinate transformations that leads to the simpler forms of the PDE
known as the canonical or standard forms and the justification of the role of ∆ in
the classification.
1.1
The Classification
Definitions
The general form of a 2nd order linear PDE in two independent variables (x, y) may be
written in the form
A uxx + B uxy + C uyy + D ux + E uy + F u = G,
(1.1)
where all the coefficients and the function G are allowed to be functions of (x, y).
It is a fact of the theory of linear PDEs that the properties of their solutions depend only
on the form of the highest-order terms appearing in the equation and not on the lowerorder derivatives. It is therefore customary to move the higher-order terms together to
one side of the equation and all the lower-order derivatives to the other side and write
(1.1) in the form
A(x, y) uxx + B(x, y) uxy + C(x, y) uyy = Φ(x, y, u, ux , uy ).
The expression on the left side of (1.2),
A(x, y) uxx + B(x, y) uxy + C(x, y) uyy ,
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(1.2)
is called the principal part of the PDE and Φ = G − D ux − E uy − F u. The calculations
below will show that the expression
∆(x, y) = B 2 − 4 A C,
(1.3)
called the discriminant of the PDE (1.2) appears naturally when applying a coordinate
transformation to new independent variables. We introduce the following classes:
Definition – At a given point (x, y) the 2nd-order linear PDE (1.2) is said to be:
-
Hyperbolic
Parabolic
Elliptic
Singular
if
if
if
if
∆(x, y) > 0. It usually describes wave propagation processes.
∆(x, y) = 0, but A2 + B 2 + C 2 6= 0. Governs diffusion processes
∆(x, y) < 0. Describes equilibrium phenomena.
A = B = C = 0.
Examples
- The wave equation utt − uxx = 0 has ∆ = 4 > 0 and is therefore hyperbolic.
- The heat equation ut − uxx = 0 has ∆ = 0 and hence is parabolic.
- Laplace’s equation uxx + uyy = 0 has ∆ = −4 < 0 and is therefore elliptic.
Auxiliary conditions
It is a fact that the solutions of PDEs belonging to the same class have many qualitative
properties in common, including the type of auxiliary conditions they have to satisfy for
the problem to be well-posed.
Parabolic and hyperbolic PDEs are typically evolution equations that represent dynamical processes, and so one of the independent variables is identified with time. They are
associated with initial and initial-boundary problems.
Elliptic PDEs usually model equilibrium processes, therefore they contain only space
variables and are associated with boundary value problems.
Equation of a Conic
The terms hyperbolic, parabolic and elliptic are just convenient names that come from
a formal comparison between the coefficients A, B and C of the PDE (1.2) and the
corresponding coefficients of the quadratic terms of the general 2nd-degree algebraic
equation
Ax2 + Bxy + Cy 2 = F (x, y).
(1.4)
Recall that the type of plane geometrical curve represented by (1.4) is determined by its
discriminant ∆ = B 2 − 4AC. The conic is a hyperbola if ∆ > 0, a parabola if ∆ = 0,
and an ellipse if ∆ < 0. It is because of this similarity that the same names are used to
classify the PDE and not because of any geometrical implications.
1.2
Coordinate Transformations
The classification described above is based on the possibility of reducing (1.2) to a simpler
form, called a canonical or standard form, by means of a coordinate transformation of
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the form
ξ = ξ(x, y),
η = η(x, y),
(1.5)
where ξ and η are both continuous and twice differentiable functions with respect to x
and y, and such that their Jacobian J does not vanish in the domain of definition of the
given PDE, that is,
∂(ξ, η)
= ξx ηy − ξy ηx 6= 0.
(1.6)
J=
∂(x, y)
The nonvanishing of the Jacobian of the transformation ensures that the transformation
(x, y) → (ξ, η) is one-to-one and therefore (ξ, η) can be used as new independent variables
without ambiguity. Straightforward application of the chain rule gives:
ux = uξ ξx + uη ηx , uy = uξ ξy + uη ηy
uxx = uξξ ξx2 + 2uξη ξx ηx + uηη ηx2 + uξ ξxx + uη ηxx
uyy = uξξ ξy2 + 2uξη ξy ηy + uηη ηy2 + uξ ξyy + uη ηyy
(1.7)
uxy = uξξ ξx ξy + uξη (ξx ηy + ηx ξy ) + uηη ηx ηy + uξ ξxy + uη ηxy .
Substituting these results into (1.2) gives the transformed PDE in the new coordinates,
a(ξ, η)uξξ + b(ξ, η)uξη + c(ξ, η)uηη = Φ̂(ξ, η, u, uξ , uη ),
(1.8)
where Φ becomes Φ̂ after the transformation, and
a(ξ, η) = Aξx2 + Bξx ξy + Cξy2
b(ξ, η) = Aξx ηx + B(ξx ηy + ξy ηx ) + Cξy ηy
c(ξ, η) = Aηx2 + Bηx ηy + Cηy2 .
(1.9)
(1.10)
(1.11)
One of the aims of these calculations is to allow us to prove the following theorem that
guarantees that the type of a PDE does not change if we make a change of variables,that
is, the classification is robust.
Theorem
The change of variables (1.5) with Jacobian J = ξx ηy − ξy ηx 6= 0 does not change the
type of the original PDE.
The proof is straightforward and involves substituting the expressions for the new coefficients in (1.9), (1.10) and (1.11), into the new discriminant ∆new = b2 − 4ac of the
transformed equation (1.8) and show that its sign is the same as that of ∆old = B 2 −4AC.
Indeed, a simple algebraic calculation shows that
∆new = b2 − 4ac = (B 2 − 4AC) J 2 = ∆old J 2 ,
(1.12)
where J is the Jacobian introduced in equation (1.6). Since J 6= 0 by assumption, it
means that the sign of the discriminant is unchanged by the transformation and therefore
the class of the PDE is also unchanged.
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1.3
Canonical Forms
We have seen how to apply an arbitrary transformation
ξ = ξ(x, y),
η = η(x, y),
(1.13)
to the linear PDE
A(x, y) uxx + B(x, y) uxy + C(x, y) uyy = Φ(x, y, u, ux , uy ),
(1.14)
to obtain a new PDE of the same class in the form
a(ξ, η)uξξ + b(ξ, η)uξη + c(ξ, η)uηη = Φ̂(ξ, η, u, uξ , uη ),
(1.15)
The next step is to show that for each of the classes defined earlier, namely, hyperbolic,
parabolic and elliptic, we can actually find a transformation (1.13) that reduces the PDE
in (1.15), to one of the following canonical forms,
uξη = Φ̂(ξ, η, u, uξ , uη ) first canonical form for hyperbolic PDEs,
uξξ − uηη = Φ̂(ξ, η, u, uξ , uη ) second canonical form for hyperbolic PDEs,
uξ − uηη = Φ̂(ξ, η, u, uξ , uη ) canonical form for parabolic PDEs,
uξξ + uηη = Φ̂(ξ, η, u, uξ , uη ) canonical form for elliptic PDEs.
As before, the function Φ̂(ξ, η, u, uξ , uη ) involves only the lower-order tems and is unimportant since it does not contribute to the classification process.
We can see that (1.15) can be reduced to one of the canonical forms if the transformation
(1.13) can be chosen in such a way that:
(i) a = c = 0, leading to the first hyperbolic canonical form;
(ii) The second canonical form for hyperbolic PDEs can easily be obtained from the
first one using the simple linear change of variables
X = x + y,
Y = x − y;
(iii) a = b = 0, giving the parabolic canonical form;
(iv) a = c, b = 0, corresponding to the elliptic canonical form.
Up to this point the material in these notes is common to both 3078/3978
The rest is for math3978 (advanced).
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1.4
The Hyperbolic Case – Characteristics
Since the majority of wave propagation problems we encounter are described by hyperbolic equations we now show how the PDE
A(x, y) uxx + B(x, y) uxy + C(x, y) uyy = Φ(x, y, u, ux , uy ),
(1.16)
given by (1.2) or (1.14) and written here again for convenience, can be transformed to
the first hyperbolic canonical form if ∆ = B 2 − 4AC > 0.
Case A = C = 0
If A = C = 0 already, then ∆ = B 2 6= 0 and so we may divide both sides of (1.16) by B
to obtain
uxy = Ψ(x, y, u, ux , uy ),
(1.17)
where Ψ = Φ/b. Since this is already in the standard (canonical) form of a hyperbolic
PDE, there if no need to perform a change of coordinate to simplify its structure in this
special case. Applying the linear change of variables
Y = x − y,
X = x + y,
(1.18)
to (1.17), leads to the second hyperbolic standard form
uXX − uY Y = Ψ̂(X, Y, u, uX , uY ),
(1.19)
where Ψ̂ is the transformed function Ψ after the linear change of variables (1.18).
Case A 6= 0
If A 6= 0 (a similar argument applies if C 6= 0), we must use the transformed equation
(1.8) or (1.15), namely,
a(ξ, η)uξξ + b(ξ, η)uξη + c(ξ, η)uηη = Φ̂(ξ, η, u, uξ , uη ),
(1.20)
and find the specific transformation that makes a(ξ, η) = c(ξ, η) ≡ 0. Using expressions
(1.9) and (1.11), it follows that ξ and η must be solutions of the PDEs
a(ξ, η) ≡ Aξx2 + Bξx ξy + Cξy2 = 0
c(ξ, η) ≡ Aηx2 + Bηx ηy + Cηy2 = 0.
and
(1.21)
The first equation may be factored in the form
(ξx + λ1 ξy )(ξx + λ2 ξy ) = 0,
√ √ λ1 (x, y) = B + ∆ /2A, λ2 (x, y) = B − ∆ /2A,
∆ = B 2 − 4 A C.
Note that since the discriminant ∆ > 0, the two roots λ1 and λ2 are real and distinct.
The second equation in (1.21) can be factored in exactly the same way, so between the
two equations, there are only two possible distinct factorizations, which lead to the two
1st-order PDEs
ξx + λ1 (x, y) ξy = 0 ηx + λ2 (x, y) ηy = 0.
(1.22)
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These are the equations defining the new coordinate system
ξ = ξ(x, y),
η = η(x, y),
that will make a = c = 0 in (1.20) and transform it into the standard canonical form
(1.17), namely
uxy = Ψ(x, y, u, ux , uy ).
The 1st-order PDEs (1.22) can be solved using the method of characteristics by transforming the two PDEs into the two ODEs
dy
= λ2 (x, y),
dx
dy
= λ1 (x, y),
dx
(1.23)
called the characteristic equations of the original PDE in (1.16). Integration of these
two ODEs leads to the family of curvilinear coordinates
ξ(x, y) = c1 ,
η(x, y) = c2 ,
respectively, where c1 and c2 are arbitrary constants that can be regarded as parameters.
These families of curves are called the characteristic curves of the hyperbolic PDE (1.16)
or simply, its characteristics.
Notes
(1) Because the discriminant ∆ > 0, the two families of characteristic curves are real
curves in the (x, y) plane.
(2) If the PDE (1.16) is a constant coefficient equation, its characteristics reduce to two
different families of straight lines.
Example
Show that the homogeneous wave equation
utt − c2 uxx = 0
(1.24)
is unconditionally hyperbolic. Reduce it to the first standard canonical form and hence
find its general solution.
Solution: To interpret the results above that involve x and y in terms of the wave
equation (1.24) that involves t and x, it is necessary to replace x, y in (1.7) and (1.16)
by t and x.
The wave equation is a constant coefficient equation with A = 1, B = 0 and C = −c2 ,
therefore ∆ = c2 > 0, showing that it is unconditionally hyperbolic, in other words, its
classification is independent of x and t.
The characteristic equations (1.23) become
dx
= −c.
dt
dx
= c,
dt
Integrating these two ODEs gives the characteristic curves of the wave equation,
ξ = x − c t and η = x + c t,
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where ξ and η are constants of integration. Substituting these expressions for ξ and η
into (1.7) gives,
utt = c2 uξξ − 2c2 uξη + c2 uηη ,
uxx = uξξ + 2uξη + uηη ,
so in terms of the characteristic coordinates ξ and η the wave equation (1.24) reduces to
uξη = 0.
The general solution of this simple PDE is
u = f (ξ) + g(η),
where f and g are arbitrary twice differentiable functions of their arguments. In terms
of the original variables x, t we obtain
u(x, t) = f (x − ct) + g(x + ct).
This is one of the few cases where a general solution of an important PDE can be
obtained in closed form.
1.5
References
1. R. Haberman, Elementary Applied Partial Differential Equations Prentice-Hall,
NJ, 1998.
2. A. Jeffrey, Applied Partial Differential Equations: An Introduction, Academic
Press, 2003.
3. W.E. Boyce and R.C. DiPrima, Elementary Differential Equations and Boundary Value Problems, 10th Edition, John Wiley & Sons, New York, 2012.
4. J. Kevorkian, Partial Differential Equations: Analytical Solution Techniques,
2nd edition, Springer, 2000.
5. P.J. Olver, Introduction to Partial Differential Equations, Springer, 2014.
6. Y. Pinchover and J. Rubinstein, An Introduction to Partial Differential Equations, Cambridge University Press, 2005.
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