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Random Walk Solutions to the Dirichlet Problem for the Laplace Equation

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boundary values
top
25
bottom
50
left
75
right
100
random walk settings
seed
0
scale
1
0.75
0.5
0.25
walk
10
endpoint counts
top
bottom
left
right
12
2
2
9
values at locator
predicted
actual
difference
58.
61.1
3.1
The Dirichlet problem seeks to find the solution to a partial differential equation inside a domain
D
, with prescribed values on the boundary of
D
. In 1944, Kakutani showed that the Dirichlet problem for the Laplace equation
f
xx
+
f
yy
=0
can be solved using random walks as follows. Given a point
P
in the interior of
D
, generate random walks that start at
P
and end when they reach the boundary of
D
. Then compute the average of the values of the given function at these boundary points. This average value is approximately equal to the value of the solution to the Dirichlet problem at the point
P
.
This Demonstration illustrates this method in the case when the domain
D
is a square and the function takes on prescribed values on each of the four sides. Choose boundary values and use the locator to select a point
P
inside the square. The Demonstration then generates 20 random walks starting at
P
and ending at the boundary of the square and computes the average value at the endpoints of these random walks. This is shown, along with the exact value of the solution at the point
P
.
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