Computer Algebra Recipes

5.2. DIFFUSION AND LAPLACE'S EQUATION MODELS 225
> pdsolve(heateq,HINT=S(x)*F(t));
·½
d
(T (x; t) =S (x) F (t)) &where
dt F (t) = c1 F (t); d2
dx2 S(x) = c1
¾¸
S(x)
The heat equation has been separated into two ODEs, c1 being the separation
constant. If the INTEGRATE option is also included, the general solution of each
ODE will be generated as Russell now illustrates.
> pdsolve(heateq,HINT=S(x)*F(t),INTEGRATE);
(T (x; t) =S (x) F (t)) &where
[ffF (t) = C3 e ( c1 t) g; fS (x) = C1 e ( p c1 x) + C2 e (¡ p c1 x) gg]
There are three unknown constants in the output, namely C1 , C2 ,and C3 .
Finally, including the build option produces the general product solution, sol.
> sol:=pdsolve(heateq,HINT=S(x)*F(t),INTEGRATE,build);
sol := T (x; t) = C3 e ( c1 t) C1 e (p c1 x) + C3 e ( c1 t) C2
e (p c1 x)
Russell then substitutes c1 = ¡k 2 on the rhs of sol and applies the simplify
command with the symbolic option, which is equivalent to assuming that k>0.
> temp:=simplify(subs(_c[1]=-k^2,rhs(sol)),symbolic);
temp := C3 ( C1 e (k (¡tk+xI)) + C2 e (¡k (tk+xI)) )
The resulting form is complex in appearance, so the complex evaluation command
is applied to separate temp into real and imaginary terms.
> temp:=evalc(temp);
temp := C3 ( C1 e (¡k2 t) (¡k cos(kx)+ C2 e 2 t) cos(kx))
+ C3 ( C1 e (¡k2 t) (¡k sin(kx) ¡ C2 e 2 t) sin(kx)) I
The temperature has been expressed in terms of cosine, sine, and exponential
terms, which are now successively collected.
> temp:=collect(%,[cos,sin,exp]);
temp := C3 ( C1 + C2) e (¡k2 t) cos(kx)+ C3 ( C1 ¡ C2 ) e (¡k 2 t) sin(kx) I
Since T (x =0;t) = 0 for all times and cos(kx) does not vanish at x =0,Russell
removes it from temp. This is the ¯rst boundary condition (bc1).
> temp:=remove(has,temp,cos); #bc1
temp := C3 ( C1 ¡ C2 ) e (¡k2 t) sin(kx) I
Similarly, the temperature must also vanish at x = L =1forallt (the second
boundary condition). Therefore sin(kL)=0,sok = n¼=Lwith n =1; 2; 3;:::
The general solution must involve a linear combination of terms involving all
possible n values. With the coe±cient labeled An, thenth term in the Fourier
series representation of the temperature will be of the following form.
> T[n]:=A[n]*subs(k=n*Pi/L,exp(-k^2*t)*sin(k*x)); #bc2
Tn := An e (¡n2 ¼ 2 t) sin(n¼x)