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{  POLAR_COORDINATES.PDE  

 

  This example demonstrates the use of functional parameter definitions

   to pose equations in polar-coordinate form.  The function definitions

   expand polar derivatives in cartesian (XY) geometry.

}

title 'Polar Coordinates'

Variables

   u

definitions

   k = 1

   u0 = 1-r^2

   s = 4

   dr(f) = (x/r)*dx(f) + (y/r)*dy(f) { functional definition of polar derivatives... }

   dphi(f) = (-y)*dx(f) + x*dy(f)     {... in cartesian coordinates }

equations { equation expressed in polar coordinates

              (Multiplied by r^2 to clear the r=0 singularity) }

   U: r*dr(r*dr(u)) + dphi(dphi(u)) + r*r*s = 0    

boundaries

  Region 1

      start(0,0)

      natural(u) = 0 line to (1,0)

      value(u)=u0   arc(center=0,0) angle=90

      natural(u)=0   line to close

monitors

  grid(x,y) as "Computation Mesh"

  contour(u) as "Solution"

  contour(u-u0) as "Error (u-u0)"

 

plots

  grid(x,y) as "Computation Mesh"

  contour(u) as "Solution"

  contour(u-u0) as "Error (u-u0)"

end