Classification of Partial Differential Equations and Canonical Forms

[Pages:38]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

2u A x2

+

2u B x y

+

2u C y2

+

D

u x

+

E

u y

+

Fu

=

G

(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

A(x,

y)

2u x2

+

B(x,

y)

2u x y

+

C(x,

y)

2u y2

=

x,

y,

u,

u x

,

u y

(2a)

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 ? 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

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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

(x0, y0)

B 2A 2C B

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

(3)

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 x2

+

x

2u y2

=

0

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

=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 xx + w x2 + w xx + w xx

(6)

uyy = w y2 + 2w yy + w y2 + w yy + w yy

uxy = w xy + w (xy + yx) + w xy + w xy + w xy

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

aw + bw + cw = , , 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 + Bxy + Cy2

b = 2Axx + B(xy + yx) + 2Cyy

(8)

c = Ax2 + Bxy + 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 T x y

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

B/2 C

J2

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 = J2(B2 - 4AC) = = J2

(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 = 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.

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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( , )w = , , 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:

w = , , w, w , w

(11)

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 + w = , , 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

w = , , w, w , w

(11)

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

w + w = , , w, w , w

(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 + Bxy + Cy2 = 0 c = Ax2 + Bxy + Cy2 = 0

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

(13a) (13b)

A

x y

2

+B

x y

A

x y

2

+B

x y

+C = 0 +C = 0

(14a) (14b)

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

?1(x, y)

=

-B - B2 - 4AC 2A

?2(x, y) = -B +

B2 - 4AC 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,

?1(x, y)

=

x y

=

-B - B2 - 4AC 2A

?2(x, y)

=

x y

= -B +

B2 - 4AC 2A

(15a) (15b)

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The above equations lead to the following two first-order differential equations

x - ?1(x, y)y = 0 x - ?2(x, y)y = 0

(16a) (16b)

These are the equations that define the new coordinate variables and that are necessary to make a = c = 0 in (7).

As the total derivative of along the coordinate line (x, y) = constant, d = 0. It follows that

d = xdx + ydy = 0

and hence, the slope of such curves is given by

dy dx

=

-

x y

We also have a similar result along coordinate line (x, y) = constant, i.e.,

dy dx

=

-

x y

Using these results, equation (14) can be written as

A

dy

2

-B

dy

+C = 0

(17)

dx

dx

This is called the characteristic polynomial of the PDE (2) and its roots are given by

dy dx

=

B + B2 - 4AC 2A

=

1(x, y)

dy = B- dx

B2 - 4AC 2A

=

2(x, y)

(18a) (18b)

The required variables and are determined by the respective solutions of the two ordinary differential equations (18a) and (18b), known as the characteristic equations of the PDE (2). They are ordinary differential equations for families of curves in the xy-plane along which = constant and = constant. Clearly, these families of curves depend on the coefficients A, B, and C in the original PDE (2).

Integration of equation (18a) leads to the family of curvilinear coordinates (x, y) = c1 while the integration of (18b) gives another family of curvilinear coordinates (x, y) = c2, where c1 and c2 are arbitrary constants of integration. These two families of curvilinear coordinates (x, y) = c1 and (x, y) = c2 are called characteristic curves of the hyperbolic equation (2) or, more simply, the characteristics of the equation. Hence, second-order hyperbolic equations have two families of characteristic curves. The fact that > 0 means that the characteristic are real curves in xy-plane.

6

If the coefficients A, B, and C are constants, it is easy to integrate equations (18a) and

(18b) to obtain the expressions for change of variables formulas for reducing a hyperbolic PDE

to the canonical form. Thus, integration of (18) produces

y = B+

B2 - 2A

4AC

x

+

c1

and

y = B-

B2 - 2A

4AC

x

+

c2

(19a)

or

y - B+

B2 - 4AC 2A

x

=

c1

and

y - B-

B2 - 4AC 2A

x

=

c2

(19b)

Thus, when the coefficients A, B, and C are constants, the two families of characteristic curves

associated with PDE reduces to two distinct families of parallel straight lines. Since the families

of curves = constant and = constant are the characteristic curves, the change of variables

are given by the following equations:

=

y-

B + B2 - 4AC x

2A

=

y - 1x

(20)

= y - B-

B2 - 4AC 2A

x

=

y

-

2x

(21)

The first canonical form of the hyperbolic is:

w = , , w, w , w

(22)

where = /b and b is calculated from (8)

b = 2Axx + B(xy + yx) + 2Cyy

= 2A

B2 - (B2 - 4AC) 4A2

+B - B - B 2A 2A

+ 2C

= 4C - B2 = -

(23)

A

A

Each of the families (x, y) = constant and (x, y) = constant forms an envelop of the domain of the xy-plane in which the PDE is hyperbolic.

The transformation = (x, y) and = (x, y) can be regarded as a mapping from the xy-plane to the -plane, and the curves along which and are constant in the xy-plane become coordinates lines in the -plane. Since these are precisely the characteristic curves, we conclude that when a hyperbolic PDE is in canonical form, coordinate lines are characteristic curves for the PDE. In other words, characteristic curves of a hyperbolic PDE are those curves to which the PDE must be referred as coordinate curves in order that it take on canonical form.

We now determine the Jacobian of transformation defined by (20) and (21). We have

J=

-1 1 -2 1

= 2 - 1

We know that 1 = 2 only if B2 - 4AC = 0. However, for an hyperbolic PDE, B2 - 4AC = 0. Hence Jacobian is nonsingular for the given transformation. A consequence of 1 = 2 is that at no point can the particular curves from each family share a common tangent line.

7

It is easy to show that the hyperbolic PDE has a second canonical form. The following

linear change of variables

= + ,

= -

converts (22) into

w - w = , , w, w , w

(24)

which is the second canonical form of the hyperbolic equations.

Example 1

Show that the one-dimensional wave equation

2u t2

-

c2

2u x2

=

0

is hyperbolic, find an equivalent canonical form, and then obtain the general solution.

Solution To interpret the results for (2) that involve the independent variables x and y in terms of the wave equation utt - c2uxx = 0, where the independent variables are t and x, it will be necessary to replace x and y in (2) and (6) by t and x. It follows that the wave equation is

a constant coefficient equation with

A = 1, B = 0, C = -c2

We calculate the discriminant, = 4c2 > 0, and therefore the PDE is hyperbolic. The roots of

the characteristic polynomial are given by

1

=

B+ 2A

=

c

and

2

=

B- 2A

=

-c

Therefore, from the characteristic equations (18a) and (18b), we have

dx = c, dt

dx = -c dt

Integrating the above two ODEs to obtain the characteristics of the wave equation

x = ct + k1,

x = -ct + k2

where k1 and k2 are the constants of integration. We see that the two families of characteristics

for the wave equation are given by x - ct = constant and x + ct = constant. It follows, then,

that the transformation

= x - ct,

= x + ct

reduces the wave equation to canonical form. We have,

a = 0,

c = 0,

b = - = -4c2 A

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