Computer Algebra Recipes

5.2. DIFFUSION AND LAPLACE'S EQUATION MODELS 239
For an initial neutron density satisfying the boundary conditions, Hugo takes
f to be of the following simple structure:
> f:=(1-r^2/a^2)*sin(Pi*z/h);
f :=
μ
1 ¡ r2
a2
³
¼z
´
sin
h
The coe±cients Cm;n are calculated, and then the Fourier term Nm;n.
> C[m,n]:=int(int(r*f*g,z=0..h),r=0..a)
/int(int(r*g^2,z=0..h),r=0..a):
> N[m,n]:=C[m,n]*N2;
For integer n>1, Nm;n should be equal to zero, since only an n =1term
occurs in f. Hugo checks that this is the case.
> simplify(N[m,n]) assuming n::integer,n>1;
0
To determine Nm;1 greater care must be exercised, and the limit taken as n ! 1.
The result then is simpli¯ed with respect to the exponentials.
> N[m,1]:=simplify(limit(N[m,n],n=1),exp);
Nm; 1 := 8 sin
³
¼z
´ μ
BesselJ 0;
h
BesselJZeros(0; m) r
a
e (¡ (¡¯ a2 h 2 +d BesselJZeros(0;m) 2 h 2 +d¼ 2 a 2 ) t
a2 h2 )
Á
BesselJ(1; BesselJZeros(0; m)) BesselJZeros(0; m) 3
The total neutron density at time t then is P 1
m=1 Nm;1. To plot the density,
Hugo takes the nominal values a =1,h =1,andd = 1. In the time-dependent
part of the density, the coe±cient of t is ¯ ¡ d (¸ 2 m =a2 + ¼ 2 =h 2 ). The largest
possible positive value of this coe±cient occurs when m = 1, corresponding to
the ¯rst zero of J0. Inthiscase¸1 ¼ 2:405. If
¯