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DirichletCircle.mw

Plate 1: Dirichlet Problem for a Circle

Maplesoft, a division of Waterloo Maple Inc., 2004

> restart;

> with(plots):

Warning, the name changecoords has been redefined

> setoptions3d(scaling=constrained, projection=0.5, style=patchnogrid);

> f1:=(x, y)->0.5*sin(10*x*y):

> f2:=t->f1(cos(t), sin(t)):

> a0:=evalf(Int(f2(t), t=-Pi..Pi)/Pi):

> a:=seq(evalf(Int(f2(t)*cos(n*t), t=-Pi..Pi)/Pi), n=1..50):

> b:=seq(evalf(Int(f2(t)*sin(n*t), t=-Pi..Pi)/Pi), n=1..50):

> L:=(r, s)->a0/2+sum('r^n*(a[n]*cos(n*s)+b[n]*sin(n*s))', 'n'=1..50):

> q:=plot3d([r*cos(s), r*sin(s), L(r, s)], r=0..1, s=0..2*Pi, color=[L(r, s), -L(r, s), 0.2], grid=[29, 100], numpoints=5000):

> p:=tubeplot([cos(t), sin(t), f2(t), t=-Pi..Pi, radius=0.015], tubepoints=70, numpoints=750):

> display3d({q, p}, orientation=[3, 89], lightmodel=light3);

[Plot]

To create a smoother result, try the following commands.  This may take some time to produce a result

> q:=plot3d([r*cos(s), r*sin(s), L(r, s)], r=0..1, s=0..2*Pi, color=[L(r, s), -L(r, s), 0.2], grid=[29, 100], numpoints=10000):

> p:=tubeplot([cos(t), sin(t), f2(t), t=-Pi..Pi, radius=0.015], tubepoints=70, numpoints=1500):

> display3d({q, p}, orientation=[3, 89], lightmodel=light3);

[Plot]

>