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Kevin Weed

A solution to the two dimensional diffusion equation PDE can be written in terms of error functions. For the case heat flow on an infinite plane that is initially at temperature zero, except for the unit square which is initially at temperature 100, the solution is

      y = ymin
      do 1 n = 0,nmax
      x = xmin
      do 2 m = 0,mmax
      u(m,n) = 25.*(erf((x+1)/sqrt(t))-erf((x-1)/sqrt(t)))*
     1             (erf((y+1)/sqrt(t))-erf((y-1)/sqrt(t)))
      x = x + dx 
2     continue
      y = y + dy
1     continue

View a sequence of images of the heat distribution on the square -5 < x,y < 5 at times of t = 0.01, to t = 2.01, or download the FORTRAN source file.