Examples.Probability
Class DirichletProblem

java.lang.Object
  extended byExamples.Probability.DirichletProblem

public class DirichletProblem
extends java.lang.Object

We solve the Dirichlet problem on the unit disk with boundary function * h(x,y)=x*y (which is harmonic on the entire plane). Thus the solution * is given by the same formula on the interior of the disc.

* *

Fixing the point x=(1/4,1/4) we compute the solution f(x) as * f(x)=E(h(B_tau)), where B is a two dimensional Brownian motion * started at the point x and tau the first time B hits the boundary * of the disc.

* *

As a proxy for the point at which the Brownian motion hits the boundary * we take the first point a past the boundary projected radially back * on the circle, ie.: a/||a||.

* * @author Michael J. Meyer


Constructor Summary
DirichletProblem()
           
 
Method Summary
static double[] boundaryProjection(double[] x)
          projects the point (u,v) radially on the unit circle
static void main(java.lang.String[] args)
           
 
Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
 

Constructor Detail

DirichletProblem

public DirichletProblem()
Method Detail

boundaryProjection

public static double[] boundaryProjection(double[] x)
projects the point (u,v) radially on the unit circle


main

public static void main(java.lang.String[] args)