Computer Algebra Recipes

242 CHAPTER 5. LINEAR PDE MODELS. PART 1
To solve Laplace's equation, Hugo loads the VectorCalculus package,
> restart: with(plots): with(VectorCalculus):
and inputs Laplace's equation in spherical coordinates. He notes that the problem
has rotational symmetry about the z-axis, the direction of °uid °ow, so
that there should be no Á dependence in the ¯nal solution. Accordingly, he
takes U = U(r; μ).
LE:=
> LE:=expand(Laplacian(U(r,theta),'spherical'[r,theta,phi]))=0;
μ
@
2 U (r; μ)
@r
r
μ 2 @
+
U (r; μ)
@r2
+
μ
@
cos(μ) U (r; μ)
@μ
r 2 sin(μ)
+
@2 U (r; μ)
@μ2 With no HINT provided, a general product solution is built with the pdsolve
command, the answer involving one separation constant, c1, and four arbitrary
constants. The result is then expanded.
> sol:=expand(pdsolve(LE,INTEGRATE,build));
sol := U (r; μ) =
+
+
+
C1 r (1=2 p μ
1+4 c1) 1
C3 LegendreP
C1 r (1=2 p 1+4 c1) C4 LegendreQ
p r
μ 1
2
C2 r (¡1=2 p μ
1+4 c1) 1
C3 LegendreP
p r
C2 r (¡1=2 p μ
1+4 c1) 1
C4 LegendreQ
p r
2
p r
2
r 2
=0
p
1+4 c1 ¡ 1
; cos(μ)
2
p
1+4 c1 ¡ 1
; cos(μ)
2
2
p
1+4 c1 ¡ 1
; cos(μ)
2
p
1+4 c1 ¡ 1
; cos(μ)
2
The solution is expressed in terms of Legendre functions of the ¯rst kind
(LegendreP) and of the second kind (LegendreQ). The former are well-behaved
polynomial functions of cos μ, but the latter diverge at the ends of the angular
range and therefore must be removed on physical grounds.
> U:=remove(has,rhs(sol),LegendreQ);
U :=
C1 r (1=2 p μ
1+4 c1) 1
C3 LegendreP
p r
C2 r (¡1=2 p μ
1+4 c1) 1
C3 LegendreP
2
p
1+4 c1 ¡ 1
; cos(μ)
2
p
1+4 c1 ¡ 1
; cos(μ)
2
+
p
r
2
In standard math notation, the Legendre polynomials of the ¯rst kind would