﻿ Sample Problems > Usage > Misc > polar_coordinates

# polar_coordinates

Navigation:  Sample Problems > Usage > Misc >

# polar_coordinates   {  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