10
I. Introduction
When parabolic partial differential equations describe time-dependent physical systems, such as the diffusion of heat along a rod, the second-order partial
derivative is usually computed with respect to a spatial coordinate. Letting x represent the spatial coordinate and y the time coordinate, the one-dimensional heat
equation becomes
a
2
1/t _ a1/l _ 0
ax 2
ay - ,
whieh is in the general form (1.11) with b = c = O. According to (1.19), the
characteristie curves for the heat equation have slope dy ldx = 0, i.e., they are
lines parallel to the spatial coordinate (which in contrast to the hyperbolic example
is now x).
If the partial differential equation is elliptie, then b
2 - ac < 0, and there are no
real -valued functions that satisfy (1.17) and (1.18). Provided that a, b, and c are
analytic' a transformation can always be found that zeros Band sets A = C = I,
thereby obtaining the canonieal form (1.14) . (See Carrier and Pearson 1988 or
Kevorkian 1990 for further details.) If a, b, and c are constant, the transformation
to canonieal form may be accomplished by choosing
_
bx -ay
- (ac -
and dividing the resulting equation by a.
Tl = x,
Since elliptic partial differential equations do not have real-valued characteristies, their solutions do not generally include wave-like perturbations that
propagate through the domain at well-defined velocities. Nevertheless, elliptic
equations describing the spatial distribution of a physical parameter such as pressure can be coupled with other time-dependent equations to yield a problem
with wave-like solutions. As noted by Whitham (1974), linearized surface gravity waves in a flat-bottomed basin of infinite horizontal extent and depth H are
govemed by the elliptie partial differential equation
a 2p
a 2p
a 2p
ax2 + a y2 + az2 = 0,
subject to the upper and lower boundary conditions
(1.20)
a 2p
ap
at2 + g az = 0 at z = 0,
3Let Z = x + iy be a complex variable in which x and y are real-valued. The function f(z) is
analytic if its derivative
df = !im fez + 6z) - fez)
dz
6z
exists and is uniquely defined as 6z goes to zero along any arbitrary path in the complex plane. If
f = u + iv where u and v are real-valued, a necessary condition for f to be analytic is that u and v
satisfy the Cauchy-Riernann conditions
Ux = vy.
u
I. Introduction
When parabolic partial differential equations describe time-dependent physical systems, such as the diffusion of heat along a rod, the second-order partial
derivative is usually computed with respect to a spatial coordinate. Letting x represent the spatial coordinate and y the time coordinate, the one-dimensional heat
equation becomes
a
2
1/t _ a1/l _ 0
ax 2
ay - ,
whieh is in the general form (1.11) with b = c = O. According to (1.19), the
characteristie curves for the heat equation have slope dy ldx = 0, i.e., they are
lines parallel to the spatial coordinate (which in contrast to the hyperbolic example
is now x).
If the partial differential equation is elliptie, then b
2 - ac < 0, and there are no
real -valued functions that satisfy (1.17) and (1.18). Provided that a, b, and c are
analytic' a transformation can always be found that zeros Band sets A = C = I,
thereby obtaining the canonieal form (1.14) . (See Carrier and Pearson 1988 or
Kevorkian 1990 for further details.) If a, b, and c are constant, the transformation
to canonieal form may be accomplished by choosing
_
bx -ay
- (ac -
and dividing the resulting equation by a.
Tl = x,
Since elliptic partial differential equations do not have real-valued characteristies, their solutions do not generally include wave-like perturbations that
propagate through the domain at well-defined velocities. Nevertheless, elliptic
equations describing the spatial distribution of a physical parameter such as pressure can be coupled with other time-dependent equations to yield a problem
with wave-like solutions. As noted by Whitham (1974), linearized surface gravity waves in a flat-bottomed basin of infinite horizontal extent and depth H are
govemed by the elliptie partial differential equation
a 2p
a 2p
a 2p
ax2 + a y2 + az2 = 0,
subject to the upper and lower boundary conditions
(1.20)
a 2p
ap
at2 + g az = 0 at z = 0,
3Let Z = x + iy be a complex variable in which x and y are real-valued. The function f(z) is
analytic if its derivative
df = !im fez + 6z) - fez)
dz
6z
exists and is uniquely defined as 6z goes to zero along any arbitrary path in the complex plane. If
f = u + iv where u and v are real-valued, a necessary condition for f to be analytic is that u and v
satisfy the Cauchy-Riernann conditions
Ux = vy.
u