Classification of Partial Differential Equations and ...

Classification of Partial Differential Equations and

Canonical Forms

A. Salih

Department of Aerospace Engineering

Indian Institute of Space Science and Technology, Thiruvananthapuram

22 December 2014

1 Second-Order Partial Differential Equations

The most general case of second-order linear partial differential equation (PDE) in two independent variables is given by

A

? 2u

? 2u

? 2u

?u

?u

+

B

+

C

+D

+E

+ Fu = G

2

2

?x

? x? y

?y

?x

?y

(1)

where the coefficients A, B, and C are functions of x and y and do not vanish simultaneously,

because in that case the second-order PDE degenerates to one of first order. Further, the

coefficients D, E, and F are also assumed to be functions of x and y. We shall assume that the

function u(x, y) and the coefficients are twice continuously differentiable in some domain ?.

The classification of second-order PDE depends on the form of the leading part of the

equation consisting of the second order terms. So, for simplicity of notation, we combine the

lower order terms and rewrite the above equation in the following form





? 2u

? 2u

? 2u

?u ?u

(2a)

+ C(x, y) 2 = x, y, u, ,

A(x, y) 2 + B(x, y)

?x

? x? y

?y

?x ?y

or using the short-hand notations for partial derivatives,

A(x, y)uxx + B(x, y)uxy + C(x, y)uyy = (x, y, u, ux , uy )

(2b)

As we shall see, there are fundamentally three types of PDEs C hyperbolic, parabolic, and

elliptic PDEs. From the physical point of view, these PDEs respectively represents the wave

propagation, the time-dependent diffusion processes, and the steady state or equilibrium processes. Thus, hyperbolic equations model the transport of some physical quantity, such as

fluids or waves. Parabolic problems describe evolutionary phenomena that lead to a steady

state described by an elliptic equation. And elliptic equations are associated to a special state

of a system, in principle corresponding to the minimum of the energy.

Mathematically, these classification of second-order PDEs is based upon the possibility of

reducing equation (2) by coordinate transformation to canonical or standard form at a point. It

may be noted that, for the purposes of classification, it is not necessary to restrict consideration

1

to linear equations. It is applicable to quasilinear second-order PDE as well. A quasilinear

second-order PDE is linear in the second derivatives only.

The type of second-order PDE (2) at a point (x0 , y0 ) depends on the sign of the discriminant

defined as

B 2A

= B(x0 , y0 )2 ? 4 A(x0 , y0 )C(x0 , y0 )

(3)

?(x0 , y0 )

2C B

The classification of second-order linear PDEs is given by the following: If ?(x0 , y0 ) > 0, the

equation is hyperbolic, ?(x0 , y0 ) = 0 the equation is parabolic, and ?(x0 , y0 ) < 0 the equation

is elliptic. It should be remarked here that a given PDE may be of one type at a specific point,

and of another type at some other point. For example, the Tricomi equation

? 2u

? 2u

+

x

=0

? x2

? y2

is hyperbolic in the left half-plane x < 0, parabolic for x = 0, and elliptic in the right half-plane

x > 0, since ? = ?4x. A PDE is hyperbolic (or parabolic or elliptic) in a region ? if the PDE

is hyperbolic (or parabolic or elliptic) at each point of ?.

The terminology hyperbolic, parabolic, and elliptic chosen to classify PDEs reflects the analogy between the form of the discriminant, B2 ?4AC, for PDEs and the form of the discriminant,

B2 ? 4AC, which classifies conic sections given by

Ax2 + Bxy + Cy2 + Dx + Ey + F = 0

The type of the curve represented by the above conic section depends on the sign of the

discriminant, ? B2 ? 4AC. If ? > 0, the curve is a hyperbola, ? = 0 the curve is an parabola,

and ? < 0 the equation is a ellipse. The analogy of the classification of PDEs is obvious. There

is no other significance to the terminology and thus the terms hyperbolic, parabolic, and elliptic

are simply three convenient names to classify PDEs.

In order to illustrate the significance of the discriminant ? and thus the classification of the

PDE (2), we try to reduce the given equation (2) to a canonical form. To do this, we transform

the independent variables x and y to the new independent variables and through the change

of variables

= (x, y),

= (x, y)

(4)

where both and are twice continuously differentiable and that the Jacobian

J=

? ( , )

=

? (x, y)

x y

x y

6= 0

(5)

in the region under consideration. The nonvanishing of the Jacobian of the transformation

ensure that a one-to-one transformation exists between the new and old variables. This simply

means that the new independent variables can serve as new coordinate variables without any

ambiguity. Now define w( , ) = u(x( , ), y( , )). Then u(x, y) = w( (x, y), (x, y)) and,

2

apply the chain rule to compute the terms of the equation (2) in terms of and as follows:

ux = w x + w x

uy = w y + w y

uxx = w x2 + 2w x x + wǦ x2 + w xx + w xx

(6)

uyy = w y2 + 2w y y + wǦ y2 + w yy + w yy

uxy = w x y + w (x y + y x ) + wǦ x y + w xy + w xy

Substituting these expressions into equation (2) we obtain the transformed PDE as



aw + bw + cwǦ = , , w, w , w

(7)

where becomes and 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:

a = Ax2 + Bx y + Cy2

b = 2Ax x + B(x y + y x ) + 2Cy y

(8)

c = Ax2 + Bx y + Cy2

At this stage the form of the PDE (7) is no simpler than that of the original PDE (2), but this

is to be expected because so far the choice of the new variable and has been arbitrary.

However, before showing how to choose the new coordinate variables, observe that equation

(8) can be written in matrix form as



a b/2

b/2 c



=



x y

x y



A B/2

B/2 C



x y

x y

T

Recalling that the determinant of the product of matrices is equal to the product of the determinants of matrices and that the determinant of a transpose of a matrix is equal to the

determinant of a matrix, we get

a b/2

b/2 c

=

A B/2 2

J

B/2 C

where J is the Jacobian of the change of variables given by (5). Expanding the determinant

and multiplying by the factor, ?4, to obtain

b2 ? 4ac = J 2 (B2 ? 4AC)

=?

= J 2?

(9)

where = b2 ? 4ac is the discriminant of the equation (7). This shows that the discriminant of

(2) has the same sign as the discriminant of the transformed equation (7) and therefore it is clear

that any real nonsingular (J 6= 0) transformation does not change the type of PDE. Note that

the discriminant involves only the coefficients of second-order derivatives of the corresponding

PDE.

3

1.1 Canonical forms

Let us now try to construct transformations, which will make one, or possibly two of the

coefficients of the leading second order terms of equation (7) vanish, thus reducing the equation

to a simpler form called canonical from. For convenience, we reproduce below the original PDE

A(x, y)uxx + B(x, y)uxy + C(x, y)uyy = (x, y, u, ux , uy )

(2)

and the corresponding transformed PDE

a( , )w + b( , )w + c( , )wǦ = , , w, w , w



(7)

We again mention here that for the PDE (2) (or (7)) to remain a second-order PDE, the

coefficients A, B, and C (or a, b, and c) do not vanish simultaneously.

By definition, a PDE is hyperbolic if the discriminant ? = B2 ? 4AC > 0. Since the sign of

discriminant is invariant under the change of coordinates (see equation (9)), it follows that for

a hyperbolic PDE, we should have b2 ? 4ac > 0. The simplest case of satisfying this condition

is a = c = 0. So, if we try to chose the new variables and such that the coefficients a and

c vanish, we get the following canonical form of hyperbolic equation:



w = , , w, w , w

(10a)

where = /b. This form is called the first canonical form of the hyperbolic equation. We

also have another simple case for which b2 ? 4ac > 0 condition is satisfied. This is the case

when b = 0 and c = ?a. In this case (9) reduces to



w ? w = , , w, w , w

(10b)

which is the second canonical form of the hyperbolic equation.

By definition, a PDE is parabolic if the discriminant ? = B2 ? 4AC = 0. It follows that for

a parabolic PDE, we should have b2 ? 4ac = 0. The simplest case of satisfying this condition is

a (or c) = 0. In this case another necessary requirement b = 0 will follow automatically (since

b2 ? 4ac = 0). So, if we try to chose the new variables and such that the coefficients a

and b vanish, we get the following canonical form of parabolic equation:



(11)

wǦ = , , w, w , w

where = /c.

By definition, a PDE is elliptic if the discriminant ? = B2 ? 4AC < 0. It follows that for

a elliptic PDE, we should have b2 ? 4ac < 0. The simplest case of satisfying this condition is

b = 0 and c = a. So, if we try to chose the new variables and such that b vanishes and

c = a, we get the following canonical form of elliptic equation:



w + wǦ = , , w, w , w

(12)

where = /a.

In summary, equation (7) can be reduced to a canonical form if the coordinate transformation

= (x, y) and = (x, y) can be selected such that:

4

? a = c = 0 corresponds to the first canonical form of hyperbolic PDE given by



w = , , w, w , w

(10a)

? b = 0, c = ?a corresponds to the second canonical form of hyperbolic PDE given by



w ? w = , , w, w , w

(10b)

? a = b = 0 corresponds to the canonical form of parabolic PDE given by



wǦ = , , w, w , w

? b = 0, c = a corresponds to the canonical form of elliptic PDE given by



w + wǦ = , , w, w , w

(11)

(12)

1.2 Hyperbolic equations

For a hyperbolic PDE the discriminant ?(= B2 ? 4AC) > 0. In this case, we have seen that, to

reduce this PDE to canonical form we need to choose the new variables and such that the

coefficients a and c vanish in (7). Thus, from (8), we have

a = Ax2 + Bx y + Cy2 = 0

(13a)

c = Ax2 + Bx y + Cy2 = 0

(13b)

Dividing equation (13a) and (13b) throughout by y2 and y2 respectively to obtain

 

 2

x

x

+B

+C = 0

A

y

y

 

 2

x

x

+B

+C = 0

A

y

y

(14a)

(14b)

Equation (14a) is a quadratic equation for (x /y ) whose roots are given by



?B ? B2 ? 4AC

?1 (x, y) =

2A

?B + B2 ? 4AC

?2 (x, y) =

2A

The roots of the equation (14b) can also be found in an identical manner, so as only two distinct

roots are possible between the two equations (14a) and (14b). Hence, we may consider ?1 as

the root of (14a) and ?2 as that of (14b). That is,



x

?B ? B2 ? 4AC

?1 (x, y) =

=

(15a)

y

2A



?B + B2 ? 4AC

x

?2 (x, y) =

=

(15b)

y

2A

5

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