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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 2

2 2

+

2

+

2

+

+

+

=

(1)

where the coefficients , , and are functions of and and do not vanish simultaneously, because in that case the second-order PDE degenerates to one of first order. Further, the coefficients , , and are also assumed to be functions of and . We shall assume that the function ( , ) 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   2 2 2 (2a) , , , , ( , ) + ( , ) + ( , ) 2 = 2

or using the short-hand notations for partial derivatives, ( , )

+ ( , )

+ ( , )

=

( , , ,

,

)

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 1

(2b)

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 ( 0 , 0 ) depends on the sign of the discriminant defined as     2 2   (3) ( 0, 0) ≡  = ( 0 , 0 ) − 4 ( 0 , 0 ) ( 0 , 0 ) 2

The classification of second-order linear PDEs is given by the following: If ( 0 , 0 ) > 0, the equation is hyperbolic, ( 0 , 0 ) = 0 the equation is parabolic, and ( 0 , 0 ) < 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 2

2 2

+

2

=0

is hyperbolic in the left half-plane < 0, parabolic for = 0, and elliptic in the right half-plane > 0, since = −4 . 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, 2 −4 , for PDEs and the form of the discriminant, 2 −4

, which classifies conic sections given by 2

+

+

2

+

+

+

=0

The type of the curve represented by the above conic section depends on the sign of the discriminant, ≡ 2 − 4 . 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 and to the new independent variables and through the change of variables = ( , ), = ( , ) (4) where both

and

are twice continuously differentiable and that the Jacobian     ( , )   = ( , ) =  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 ( , ) = ( ( , ), ( , )). Then ( , ) = ( ( , ), ( , )) and,

2

apply the chain rule to compute the terms of the equation (2) in terms of

and

as follows:

+

=

+

=

2

=

2

=

+2 +2

2

+

+

+

2

+

+ +

+

(

+

=

+

)+

(6)

+

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



=

+

, , ,



,

(7)

where becomes and the new coefficients of the higher order terms , , and via the original coefficients and the change of variables formulas as follows: 2

=

+

2

=

2

+

+

+ (

=2 +

are expressed

)+2

(8)

2

+

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 

/2



/2 =





/2 /2





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      /2  2 /2  =     /2   /2 where is the Jacobian of the change of variables given by (5). Expanding the determinant and multiplying by the factor, −4, to obtain 2

−4

=

2

(

2

−4

)

=⇒

=

2

(9)

where = 2 − 4 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 ( 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 ( , )

+ ( , )

+ ( , )

=

( , , ,

,

)

(2)

and the corresponding transformed PDE ( , )

+ ( , )

+ ( , )



=

, , ,

,



(7)

We again mention here that for the PDE (2) (or (7)) to remain a second-order PDE, the coefficients , , and (or , , and ) do not vanish simultaneously. By definition, a PDE is hyperbolic if the discriminant = 2 − 4 > 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 2 − 4 > 0. The simplest case of satisfying this condition is = = 0. So, if we try to chose the new variables and such that the coefficients and vanish, we get the following canonical form of hyperbolic equation:   = , , , , (10a)

where = / . This form is called the first canonical form of the hyperbolic equation. We also have another simple case for which 2 − 4 > 0 condition is satisfied. This is the case when

= 0 and

= − . In this case (9) reduces to  = , , , −

,



(10b)

which is the second canonical form of the hyperbolic equation. By definition, a PDE is parabolic if the discriminant = 2 − 4 = 0. It follows that for 2 a parabolic PDE, we should have − 4 = 0. The simplest case of satisfying this condition is

(or ) = 0. In this case another necessary requirement = 0 will follow automatically (since such that the coefficients = 0). So, if we try to chose the new variables and and vanish, we get the following canonical form of parabolic equation:   = , , , , (11) 2 −4

where = / . By definition, a PDE is elliptic if the discriminant = 2 − 4 < 0. It follows that for 2 a elliptic PDE, we should have − 4 < 0. The simplest case of satisfying this condition is such that vanishes and = 0 and = . So, if we try to chose the new variables and = , we get the following canonical form of elliptic equation:   = , , , , +

(12)

where = / . In summary, equation (7) can be reduced to a canonical form if the coordinate transformation = ( , ) and = ( , ) can be selected such that: 4

•

= = 0 corresponds to the first canonical form of hyperbolic PDE given by   = , , , ,

•

= 0,

•

= = 0 corresponds to the canonical form of parabolic PDE given by   = , , , ,

•

1.2

= 0,

(10a)

= − corresponds to the second canonical form of hyperbolic PDE given by   = , (10b) , , , −

=

corresponds to the canonical form of elliptic PDE given by   = , , , , +

(11)

(12)

Hyperbolic equations

For a hyperbolic PDE the discriminant (= 2 − 4 ) > 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 and vanish in (7). Thus, from (8), we have = =

2

+

2

+

2

+

2

+

=0

(13a)

=0

(13b)

Dividing equation (13a) and (13b) throughout by 2 and 2 respectively to obtain  2   + =0 + 

2

+





+

=0

(14a) (14b)

Equation (14a) is a quadratic equation for ( / ) whose roots are given by √ 2 −4 − − 1( , ) = √2 2 −4 − + 2( , ) = 2 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, √ 2 −4 − − = (15a) ( , ) = 1 2 √ 2 −4 − + = (15b) ( , ) = 2 2 5

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

−

1(

, )

=0

(16a)

2(

, )

=0

(16b)

These are the equations that define the new coordinate variables and that are necessary to make = = 0 in (7). As the total derivative of along the coordinate line ( , ) = constant, = 0. It follows that = + =0 and hence, the slope of such curves is given by =− We also have a similar result along coordinate line

( , ) = constant, i.e.,

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

2

−





+

=0

This is called the characteristic polynomial of the PDE (2) and its roots are given by √ 2 −4 + = 1( , ) = √2 2 −4 − = 2( , ) = 2

(17)

(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 -plane along which = constant and = constant. Clearly, these families of curves depend on the coefficients , , and in the original PDE (2). Integration of equation (18a) leads to the family of curvilinear coordinates ( , ) = 1 while the integration of (18b) gives another family of curvilinear coordinates ( , ) = 2 , where 1 and 2 are arbitrary constants of integration. These two families of curvilinear coordinates ( , ) = 1 and ( , ) = 2 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

-plane.

6

If the coefficients , , and 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 √ √ 2 −4 2 −4 − + + 1 and = = + 2 (19a) 2 2 or √ √ 2 −4 2 −4 + − − = 1 and − = 2 (19b) 2 2 Thus, when the coefficients , , and 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: √ 2 −4 + = − = − 1 (20) √2 2 −4 − = − (21) = − 2 2 The first canonical form of the hyperbolic is:   = , , , , (22) where

= / and

is calculated from (8)

=2

+

+ (

=2 

=4 −

2 −( 2 −4

4

2

2

)



)+2   +2 + − − 2 2 (23)

=−

Each of the families ( , ) = constant and ( , ) = constant forms an envelop of the domain of the -plane in which the PDE is hyperbolic. The transformation = ( , ) and = ( , ) can be regarded as a mapping from the -plane, and the curves along which and are constant in the -plane -plane to the 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    − 1 1    = = 2− 1 − 2 1 

We know that 1 = 2 only if 2 − 4 = 0. However, for an hyperbolic PDE, 2 − 4 6= 0. Hence Jacobian is nonsingular for the given transformation. A consequence of 1 6= 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 

=

, , ,



,

(24)

− which is the second canonical form of the hyperbolic equations. Example 1 Show that the one-dimensional wave equation 2

2

2

2

−

2

=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 and in terms of the wave equation − 2 = 0, where the independent variables are and , it will be necessary to replace and in (2) and (6) by and . It follows that the wave equation is a constant coefficient equation with = 1,

= 0,

=−

2

We calculate the discriminant, = 4 2 > 0, and therefore the PDE is hyperbolic. The roots of the characteristic polynomial are given by √ √ + − = and 1 = 2 = =− 2 2 Therefore, from the characteristic equations (18a) and (18b), we have =−

= ,

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

+

1,

=−

+

2

where 1 and 2 are the constants of integration. We see that the two families of characteristics for the wave equation are given by − = constant and + = constant. It follows, then, that the transformation = − , = + reduces the wave equation to canonical form. We have, = 0,

= 0,

=− 8

= −4

2

So in terms of characteristic variables, the wave equation reduces to the following canonical form =0 For the wave equation the characteristics are found to be straight lines with negative and positive slopes as shown in Fig. 1. The characteristics form a natural set of coordinates for the hyperbolic equation.

+

=

−

=

Figure 1: The pair of characteristic curves for wave equation.

The canonical forms are simple because they can be solved directly by integrating twice. For example, integrating with respect to gives = where the ‘constant of integration’ respect to to obtain ( , )=

Z

= ( )

0

is an arbitrary function of

Z

. Next, integrating with

+ ( ) = ( )+ ( )

( )

where and are arbitrary twice differentiable functions and is just the integral of the arbitrary function . The form of the general solutions of the wave equation in terms of its original variable and are then given by = ( − )+ ( + ) Note that

is constant on “wavefronts” = + that travel towards right, whereas is constant on wavefronts = − + that travel towards left. Thus, any general solution can be expressed as the sum of two waves, one travelling to the right with constant velocity and the other travelling to the left with the same velocity . This is one of the few cases where the general solution of a PDE can be found. We also note that hyperbolic PDE has an alternate canonical form with the following linear change of variables = + and = − , given by =0

− 9

Example 2 In steady or unsteady transonic flow around wings and airfoils with thickness to chord ratios of a few percent, we can generally consider that the flow is predominantly directed along the chordwise direction, taken as the -direction. In this case, the velocities in the transverse direction can be neglected and the potential equation reduces to the so-called small disturbance potential equation: 2  2  =0 + 1− 2 2 2

Historically, this was the form of equation used by Murman and Cole (1961) to obtain the first numerical solution for a transonic flow around an airfoil with shocks. Show that, depending on the Mach number, the small disturbance potential equation is elliptic, parabolic, or hyperbolic. Find the characteristic variables for the hyperbolic case and hence write the equation in canonical form. Solution

The given equation is of the form (2) where 2,

= 1− The discriminant,

=

2 −4

=...