Computer Algebra Recipes

232 CHAPTER 5. LINEAR PDE MODELS. PART 1
Pi
2/3*Pi 1/3*Pi
4/3*Pi
0
1/4
1/2
5/3*Pi
3/4
Figure 5.7: Constant r and μ lines in polar coordinates.
The resulting picture is reproduced in Figure 5.7, circles being produced for
r =0; 1 1 3 ; ; ; 1 and polar lines for μ =0;¼=3; 2 ¼=3; ¼;::::
4 2 4
Benny now enters Laplace's equation (LE ) in polar coordinates.
> LE:=expand(Laplacian(phi(r,theta),'polar'[r,theta]))=0;
LE :=
@
Á(r; μ)
@r
r
μ 2 @
+
Á(r; μ)
@r2
+
1
@2 Á(r; μ)
@μ2 The pdsolve commandwiththeHINT=f(r)*g(theta), INTEGRATE and build
options is used to ¯nd the general product solution of Laplace's equation.
> sol:=pdsolve(LE,HINT=f(r)*g(theta),INTEGRATE,build);
r 2
=0
sol := Á(r; μ) = C3 sin( p c1 μ) C1 r (p c1) + C3 sin( p c1 μ) C2
r (p c1)
+ C4 cos( p c1 μ) C1 r (p c1) + C4 cos( p c1 μ) C2
r (p c1)
Benny notes that the solution must reduce to a cos μ form on one boundary
and a sin μ form on the other, so he accordingly sets the separation constant
p c1 = 1 on the rhs of sol.
> phi:=subs(sqrt(_c[1])=1,rhs(sol));
C3 sin(μ) C2
C4 cos(μ) C2
Á := C3 sin(μ) C1 r + + C4 cos(μ) C1 r +
r
r
The terms are then grouped by successively collecting cos μ, sinμ, andrterms.