Computer Algebra Recipes

226 CHAPTER 5. LINEAR PDE MODELS. PART 1
The complete series solution is T = P1 n=1 Tn, withtheformofAnto be determined
from the initial condition. At t = 0, one must have
1X
T (x; 0) = An sin(n¼x)=100x (1 ¡ x):
n=1
If T (x; 0) is multiplied by sin(m¼x) and integrated from x =0tox = L =1,
the lhs will integrate to zero for every term in the series except for the term
corresponding to m = n. This is a result of the independence, or orthogonality,
of the sine terms in the series. The resulting equation can then be solved for
An. Russell will now apply this approach. He evaluates Tn at t =0,andenters
the initial temperature pro¯le, T0 .
> X:=eval(T[n],t=0); T0:=100*x*(1-x);
X := An sin(n¼x) T0 := 100 x (1 ¡ x)
Since only the nth term in the series survives, Russell forms T0 ¡ X, multiplies
this by sin(n¼x), and integrates from x =0to1ineq.Theneq is set equal to
zero and solved for An.
> eq:=int((T0-X)*sin(n*Pi*x),x=0..1);
> A[n]:=solve(eq=0,A[n]);
200 (¡2+n¼sin(n¼)+2cos(n¼))
An := ¡
n2 ¼2 (n¼¡ sin(n¼)cos(n¼))
Assuming that n is an integer, the nth Fourier term Tn is simpli¯ed, An having
been automatically substituted. The double colon in assuming is a \type
match" command.
> T[n]:=simplify(T[n]) assuming n::integer;
Tn := ¡ 400 (¡1+(¡1)n ) e (¡n2 ¼ 2 t) sin(n¼x)
n3 ¼3 Thus, the temperature distribution in the rod satisfying the boundary and
initial conditions is given by the following in¯nite Fourier series:
> Temp:=Sum(T[n],n=1..infinity);
1X
Ã
Temp := ¡ 400 (¡1+(¡1)n ) e (¡n2 ¼ 2 t) sin(n¼x)
n3 ¼3 !
n=1
The sum of the ¯rst ¯ve terms gives a good approximation to T (x; t), because
the contribution of higher-order terms drops very rapidly with increasing time.
> TT:=sum(T[n],n=1..5);
TT := 800 e(¡¼2 t) sin(¼x)
¼3 + 800 e
27
(¡9 ¼2 t) sin(3 ¼x)
¼3 + 32 e
5
(¡25 ¼2 t) sin(5 ¼x)
¼3 Then TT is animated so its spatial and temporal evolution can be clearly seen.
> animate(plot,[TT,x=0..1],t=0..1,frames=50);
Russell is feeling sleepy, so is going back to bed. If you wish to see how the
initial parabolic temperature pro¯le decays to zero everywhere inside the rod,
execute the work sheet, click on the plot, and then on the play arrow.