for the planar heat equation is given by the linear superposition formula
u
(
t, x, y
) =
1
4
πγ t
integraldisplayintegraldisplay
f
(
ξ, η
)
e
−
[(
x
−
ξ
)
2
+(
y
−
η
)
2
]
/
(4
γ t
)
dξ dη.
(12
.
125)
We can interpret the solution formula (12.125) as a two-dimensional
convolution
u
(
t, x, y
) =
F
(
t, x, y
)
∗
f
(
x, y
)
(12
.
126)
of the initial data with a one-parameter family of progressively wider and shorter Gaussian
filters
F
(
t, x, y
) =
F
(
t, x, y
; 0
,
0) =
1
4
πγ t
e
−
[
x
2
+
y
2
]
/
(4
γ t
)
.
(12
.
127)
As in (8.51), such a convolution can be interpreted as a weighted averaging of the function,
and has the effect of smoothing out the initial signal
f
(
x, y
).
Example 12.14.
If our initial temperature distribution is constant on a circular
region, say
u
(0
, x, y
) =
braceleftbigg
1
x
2
+
y
2
<
1
,
0
,
otherwise
,
then the solution can be evaluated using (12.125), as follows:
u
(
t, x, y
) =
1
4
π t
integraldisplayintegraldisplay
D
e
−
[(
x
−
ξ
)
2
+(
y
−
η
)
2
]
/
(4
t
)
dξ dη,
where the integral is over the unit disk
D
=
{
ξ
2
+
η
2
≤
1
}
. Unfortunately, the integral
cannot be expressed in terms of elementary functions.
On the other hand, numerical
evaluation of the integral is straightforward.
A plot of the resulting radially symmetric
solution appears in Figure 12.8. One could also interpret this solution as the diffusion of an
animal population in a uniform, isotropic environment, or bacteria in a similarly uniform
large petri dish, that is initially confined to a circular region.
12.6.
The Planar Wave Equation.
Let us next consider the two-dimensional wave equation
∂
2
u
∂t
2
=
c
2
Δ
u
=
c
2
parenleftbigg
∂
2
u
∂x
2
+
∂
2
u
∂y
2
parenrightbigg
,
(12
.
128)
which models the unforced vibrations of a homogeneous two-dimensional membrane, e.g., a