A∗ =Aξ2x+Bξxξy+Cξ2y,
B∗ = 2Aξxηx+B(ξxηy+ξyηx) + 2Cξyηy, C∗ =Aηx2+Bηxηy+Cηy2,
D∗ =Aξxx+Bξxy+Cξyy+Dξx+Eξy, (4.1.8) E∗ =Aηxx+Bηxy+Cηyy+Dηx+Eηy,
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∗=J2
B2−4AC , (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 coefficientsA(x, y), B(x, y), and C(x, y) at a given point (x, y). We shall, therefore, rewrite equation (4.1.2) as
Auxx+Buxy+Cuyy =H(x, y, u, ux, uy), (4.1.10) and equation (4.1.7) as
A∗uξξ+B∗uξη+C∗uηη=H∗(ξ, η, u, uξ, uη). (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 coefficientsA∗ and C∗ in equation (4.1.11) vanish.
Thus, from (4.1.8), we have
A∗=Aξx2+Bξxξy+Cξy2= 0, C∗=Aη2x+Bηxηy+Cη2y= 0.
These two equations are of the same type and hence we may write them in the form
Aζx2+Bζxζy+Cζy2= 0, (4.2.1)
in whichζ stand for either of the functionsξorη. Dividing through byζy2, equation (4.2.1) becomes
A ζx
ζy
2
+B ζx
ζy
+C= 0. (4.2.2)
Along the curveζ= constant, we have
dζ =ζxdx+ζydy= 0.
Thus,
dy dx =−ζx
ζy
, (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+
B2−4AC
/2A, (4.2.5)
dy dx =
B−
B2−4AC
/2A. (4.2.6)
These equations, which are known as thecharacteristic equations, are or- dinary differential equations for families of curves in the xy-plane along which ξ = constant and η = constant. The 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
φ1(x, y) =c1, c1= constant, φ2(x, y) =c2, c2= constant.
Hence the transformations
ξ=φ1(x, y), η=φ2(x, y), will transform equation (4.1.10) to a canonical form.
(A) Hyperbolic Type
IfB2−4AC >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
uξη=H1, (4.2.7)
4.2 Canonical Forms 95 whereH1=H∗/B∗. It can be easily shown thatB∗= 0. This form is called thefirst canonical form of the hyperbolic equation.
Now if new independent variables
α=ξ+η, β =ξ−η, (4.2.8)
are introduced, then equation (4.2.7) is transformed into
uαα−uββ =H2(α, β, u, uα, uβ). (4.2.9) This form is called thesecond canonical form of the hyperbolic equation.
(B) Parabolic Type
In this case, we have B2−4AC= 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 integralξ= constant (or η= constant).
SinceB2= 4AC andA∗= 0, we find that A∗ =Aξ2x+Bξxξy+Cξy2=√
A ξx+√ C ξy
2
= 0.
From this it follows that
A∗= 2Aξxηx+B(ξxηy+ξyηx) + 2Cξyηy
= 2√
A ξx+√ C ξy
√A ηx+√ C ηy
= 0,
for arbitrary values ofη(x, y) which is functionally independent ofξ(x, y);
for instance, ifη=y, the Jacobian does not vanish in the domain of parabol- icity.
Division of equation (4.1.11) byC∗ yields
uηη=H3(ξ, η, u, uξ, uη), C∗= 0. (4.2.10) This is called thecanonical form of the parabolic equation.
Equation (4.1.11) may also assume the form
uξξ =H3∗(ξ, η, u, uξ, uη), (4.2.11) if we chooseη= constant as the integral of equation (4.2.5).
(C) Elliptic Type
For an equation of elliptic type, we haveB2−4AC <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 xand y. Thus, in this case, there are no real characteristic curves. However, if the coefficients A, B, and C 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 (x0, y0) of this domain, the
function can be represented as a Taylor series in the variables (x−x0) and (y−y0).
Sinceξandη are complex, we introduce 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 equations (4.1.10). We then have
A∗∗(α, β)uαα+B∗∗(α, β)uαβ+C∗∗(α, β)uββ =H4(α, β, u, uα, uβ), (4.2.14) in which the coefficients assume the same form as the coefficients in equation (4.1.11). With the use of (4.2.13), the equationsA∗=C∗= 0 become
Aα2x+Bαxαy+Cα2y −
Aβx2+Bβxβy+Cβy2 +i[2Aαxβx+B(αxβy+αyβx) + 2Cαyβy] = 0, Aα2x+Bαxαy+Cα2y −
Aβx2+Bβxβy+Cβy2
−i[2Aαxβx+B(αxβy+αyβx) + 2Cαyβy] = 0, or,
(A∗∗−C∗∗) +iB∗∗= 0, (A∗∗−C∗∗)−iB∗∗= 0.
These equations are satisfied if and only if
A∗∗=C∗∗ and B∗∗= 0.
Hence, equation (4.2.14) transforms into the form A∗∗uαα+A∗∗uββ =H4(α, β, u, uα, uβ). Dividing through byA∗∗, we obtain
uαα+uββ =H5(α, β, u, uα, uβ), (4.2.15) where H5 = (H4/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, they do not play a particularly physical role in parabolic and elliptic equations, but their role is somewhat mathematical
4.2 Canonical Forms 97 in solving these equations. In general, first-order partial differential equa- tions such as advection-reaction equations are regarded as hyperbolic be- cause they describe propagation 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(κκκ) as defined in Section 13.3. In one-dimensional case, ω =ω(k).
Ifω(k) is real and ω′′(k)= 0, the equation is calleddispersive. The word dispersive simply means that the phase velocitycp= (ω/k) of a plane wave solution,u(x, t) =Aexp [i(kx−ωt)] depends on the wavenumberk. This means that waves of different wavelength propagate with different phase ve- locities and hence, disperse in the medium. Ifω=ω(k) =σ(k) +iµ(k) is complex, the associated partial differential equation is calleddiffusive. From a physical point of view, such a classification of equations is particularly 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
y2uxx−x2uyy = 0.
Here
A=y2, B= 0, C=−x2. Thus,
B2−4AC= 4x2y2>0.
The equation is hyperbolic everywhere except on the coordinate axesx= 0 andy= 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 2y2−1
2x2=c1, 1 2y2+1
2x2=c2. The first of these curves is a family of hyperbolas
1 2y2−1
2x2=c1, and the second is a family of circles
1 2y2+1
2x2=c2.
To transform the given equation to canonical form, we consider
ξ= 1 2y2−1
2x2, η= 1 2y2+1
2x2. From the relations (4.1.6), we have
ux=uξξx+uηηx=−xuξ+xuη, uy =uξξy+uηηy =yuξ+yuη,
uxx=uξξξx2+ 2uξηξxηx+uηηηx2+uξξxx+uηηxx
=x2uξξ−2x2uξη+x2uηη−uξ+uη. uyy =uξξξy2+ 2uξηξyηy+uηηη2y+uξξyy+uηηyy
=y2uξξ+ 2y2uξη+y2uηη+uξ+uη. Thus, the given equation assumes the canonical form
uξη = η
2 (ξ2−η2)uξ− ξ
2 (ξ2−η2)uη. Example 4.2.2.Consider the partial differential equation
x2uxx+ 2xy uxy+y2uyy = 0.
In this case, the discriminant is
B2−4AC= 4x2y2−4x2y2= 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
y2uηη = 0.
Thus,
uηη = 0 for y= 0.