Computer Algebra Recipes

6.2. SEMI-INFINITE AND INFINITE DOMAINS 265
Then, the ordinary di®erential equation FCT is analytically solved for F (t),
subject to the initial condition F (0) = F0 ,
> eq:=dsolve(fFCT,F(0)=F0g,F(t));
eq := F(t) = A p 2cos(sa) e (¡ds2 t)
p
¼
and the inverse transform performed on the rhs of eq.
> T:=fouriercos(rhs(eq),s,x);
r
¼
A
td
T :=
e(¡ a2 +x 2
4 td ) ³
ax
´
cosh
2 td
¼
Looking at the analytic result for T , Vectoria is pleased that she has spent time
learning how to use the Maple computer algebra system. Once she has set up
a template for a certain type of problem, then any other problem of that type
is generally trivial to tackle. In the real world, certain integrals and other steps
may not be carried out analytically, but traditionally in the world of academia,
problems are assigned by instructors for which analytic answers exist.
To ¯nish o® the problem and complete the assignment, Vectoria evaluates
T with the given parameter values (d =1,a =5,A =5),
> T:=eval(T,fd=1,a=5,A=5g);
r μ
¼ 25+x2
5 e(¡
) 5 x
4 t cosh
t 2 t
T :=
¼
and animates the solution over the suggested time interval t = 1 to 200. Clearly,
the solution cannot be plotted at t = 0, because the initial pro¯le has been
assumed to be a Dirac delta function. The number of points is controlled to
product a smooth curve.
> animate(plot,[T,x=0..20],t=1..200,numpoints=200,
frames=50,thickness=2);
As she watches the animation on the computer screen, Mike arrives to whisk
her o® on their date.
PROBLEMS:
Problem 6-11: Fourier cosine transform
Calculate the Fourier cosine transform of each f(x) belowandplottheanswers
where possible:
(a) f(x) =e ¡3 jxj ; (b) f(x) =cos(2x); (c) f(x) =sin(x) 2 ;
(d) f(x) =x=(x2 +1).
Problem 6-12: Di®erent initial condition
Modify the text recipe to ¯nd the temperature distribution inside a semi-in¯nite
rod (0 · x ·1) that is insulated at x = 0 and has the initial temperature
distribution T (x >0; 0) = 25 x2 =(x2 +25)andd =1. AnimateT (x; t).