Computer Algebra Recipes

238 CHAPTER 5. LINEAR PDE MODELS. PART 1
N := C5 C1 BesselJ(0; ¹r) e (¡td¹2 ¡tdº 2 +t¯) ( C3 cos(ºz)+ C4 cos(ºz))
+ C5 C1 BesselJ(0; ¹r) e (¡td¹2 ¡tdº 2 +t¯) ( C3 sin(ºz) ¡ C4 sin(ºz)) I
Hugo assumes that the neutron density at the surface of the cylindrical mass is
zero. Thus, since it doesn't go to zero at z =0,thecos(ºz) term is removed
(a second boundary condition) from N and the result factored.
> N:=factor(remove(has,N,cos)); #bc2
N := C5 C1 BesselJ(0; ¹r) e (¡t (¡¯+d¹2 +dº 2 )) sin(ºz)(¡ C4 + C3) I
In the next command line, Hugo removes the \ugly" Maple coe±cient combination
from N by forming the product of the 4th, 5th, and 6th operands of N,
the other operands corresponding to the various constants.
> N2:=op(4,N)*op(5,N)*op(6,N);
N2 := BesselJ(0; ¹r) e (¡t (¡¯+d¹2 +dº 2 )) sin(ºz)
Since the neutron density is zero on the cylindrical surface r = a and at the end
z = h, two more boundary conditions must be imposed. At r = a, J0(¹a)=0,
so that ¹ = ¸m=a, where ¸m is the mth zero of J0. These zeros may be
found with the command BesselJZeros(0,m), wherem =1; 2;:::: At z = h,
sin(ºh)=0,soº = n¼=h,withn a positive integer. This pair of boundary
condition relations is substituted into N2.
> N2:=subs(fmu=BesselJZeros(0,m)/a,nu=n*Pi/hg,N2); #bcs
μ
N2 := BesselJ 0;
BesselJZeros(0; m) r
a
d BesselJZeros(0;m)2
e
(¡t (¡¯ + a2 + dn2 ¼ 2
h2 ³
)) n¼z
´
sin
h
By the principle of linear superposition, the complete solution will be
1X 1X
1X 1X
N(r; z; t) = Cm;n N2 ´ Nm;n;
m=1 n=1
m=1 n=1
with the Cm;n determined by the initial neutron density N(r; z; 0) = f. Evaluating
N2 at t = 0, and labeling the result g,
> g:=eval(N2,t=0);
μ
BesselJZeros(0; m) r
³
n¼z
´
g := BesselJ 0; sin
a
h
one has
1X 1X
N(r; z; 0) = Cm;n g = f:
m=1 n=1
If we multiply both sides of N(r; z; 0) by rJ0(¸m0 r=a)sin(n0¼z=h)andinte grate r from 0 to a and z from 0 to h, the lhs will yield zero unless m = m 0
and n = n 0 . Solving for Cm 0 ;n 0, and dropping the primes, the coe±cients can
be calculated from
R a R h
0 0
Cm;n =
rfgdrdz
R a R h
0 0 rg2 dr dz :