Computer Algebra Recipes

248 CHAPTER 6. LINEAR PDE MODELS. PART 2
Professor Simon Legree, 1 who is teaching the course, has a reputation for
assigning large numbers of often di±cult problems, so Vectoria decides once
again to let Maple help her in deriving the solutions. Although she could do
the problems by hand, it seems smarter in the long run to develop a computer
algebra approach to lighten the workload and to avoid mathematical mistakes.
On talking to the professor, she ¯nds out that, surprisingly, Legree not only
agrees but suggests that it might be wise to start with a relatively simple
problem, before tackling the more di±cult ones. Amazed that Legree, despite
his hard-nosed reputation, has been so helpful, Vectoria decides to follow his
advice and selects a problem involving the transverse motion of a string that
has been plucked and released from rest.
Since she doesn't yet know how to include sti®ness in the string, she opts
to make use of the linear wave equation that neglects sti®ness. The horizontal
string is ¯xed at its endpoints, x =0andx = L>0, and is given an initial
transverse displacement Ã(x; t =0)=2hx=L for 0 · x · L=2 andÃ(x; 0) =
2 h (L ¡ x)=L for L=2 · x · L. Vectoria recognizes that this is a triangular
pro¯le with a maximum displacement h at the center (x = L=2) of the string.
Professor Legree has also stressed that his marking assistant has been instructed
not to give full credit for a problem solution unless it is accompanied by some
sort of meaningful plot as well as some pertinent discussion.
So Vectoria, anticipating that she will animate the vibrational motion of the
string, begins her recipe with a call to the plots package, and enters the wave
equation WE for the tranverse displacement Ã(x; t) ofthestringattimet.
> restart: with(plots):
> WE:=diff(psi(x,t),x,x)=(1/c^2)*diff(psi(x,t),t,t);
WE := @2
@
Ã(x; t) =
@x2 2
Ã(x; t)
@t2 c2 On applying the pdsolve command to WE without any options,
> pdsolve(WE);
Ã(x; t) = F1 (ct+ x)+ F2 (ct¡ x)
Vectoria ¯nds that the solution is given by Ã(x; t) = F 1(ct+ x)+ F 2(ct¡ x),
where F 1and F 2 are arbitrary functions. This is the well-known general
solution of the wave equation, which is not too useful for solving the present
problem with speci¯ed boundary and initial conditions. Clearly, a hint should
be provided. She could be quite general and enter HINT=f(x)*g(t), butshe
realizes that sine and cosine functions clearly satisfy the wave equation. Since
sin(kx), where k is an undetermined constant, is equal to zero at x =0andthus
satis¯es the boundary condition there, she provides the HINT=sin(k*x)*g(t)
and integrates and builds up the solution.
1 Unfortunately, Mrs. Legree, when naming her newborn son, chose Simon as a ¯rst name.
She was unaware that Simon Legree was the cruel plantation owner in Harriet Beecher Stowe's
novel, Uncle Tom's Cabin.