Computer Algebra Recipes

6.2. SEMI-INFINITE AND INFINITE DOMAINS 263
p
2 T0
eq := F (t) =
s p ¼ ¡ e(¡ds2 t) p 2 T0
s p ¼
The Fourier (inverse) sine transform of the rhs of eq with respect to s is taken,
> T:=fouriersin(rhs(eq),s,x);
μ
x
T := T0 ¡ T0 erf
2 p
dt
yielding the temperature T expressed in terms of the error function. Now T is
evaluated with the given parameter values (d=100, T0 =100) and simpli¯ed.
> T:=simplify(eval(T,fd=100,T0=100g));
μ
x
T := 100 ¡ 100 erf
20 p
t
Choosing the spatial range to be x = 0 to 400 cm, Vectoria animates T over
the time interval t = 0 to 150 seconds, 50 frames being taken.
> animate(plot,[T,x=0..400],t=0..150,frames=50,thickness=2);
On running the animation, Vectoria is pleased with the \sweetness" of the whole
computer algebra derivation.
Looking at her watch, she realizes that Mike will be picking her up at 5:00
p.m., so she had better ¯nish the last problem on Professor Legree's assignment
before he shows up. She will enjoy her dinner at Gira®es on the waterfront more
if she has ¯nished her work.
PROBLEMS:
Problem 6-9: Fourier sine transform
Calculate the Fourier sine transform of each f(x) belowandplottheanswers:
(a) f(x) =e ¡3 jxj ; (b) f(x) =cos(2x); (c) f(x) =sin(x) 2 ;
(d) f(x) =x=(x2 +1).
Problem 6-10: A di®erent T (x; 0)
Use the Fourier sine transform approach to solve the heat conduction problem
for d =10cm2 /s in a semi-in¯nite rod (0 · x ·1) that has the boundary
condition T (0;t) = 0 and initial interior temperature distribution T (x; 0) =
10 x=(x2 + 1). Animate the plot.
6.2.2 Assignment Complete!
It is not knowledge, but the act of learning, not possession, but the
act of getting there, which grants the greatest enjoyment.
Carl Friedrich Gauss, German mathematician (1777{1855)
The last problem on Vectoria's assignment is similar to the fourth one, but involves
a di®erent boundary condition and initial condition. It is now supposed
that the semi-in¯nite rod (0 · x ·1) has an initial temperature distribution