With these values of we see that there are solutions for the equation that are given by

Thus there are solutions to the heat equation of the form

.

We can instruct Maple to verify this as follows:

> u[k] := (x,t) -> c[k]*exp(-((2*k-1)/2)^2*t)*cos((2*k-1)*x/2);

> D[2](u[k])(x,t) - D[1,1](u[k])(x,t);

We have one remaining condition to satisfy, namely .

Since we have an infinite set of solutions, each with an unspecified coefficient, , can we find a linear

combination, which yields the solution? That is, can we write ?

The answer is yes , provided we can find coefficients , such that

.

But this is just the answer to Problem 1 - or is it?

Unfortunately, the answer to Problem 1 does NOT give us the answer to this problem

because the frequency term in Problem 1 is simply while the frequency term here is .

The preceding work constitutes the answer I was looking for on the test. Curiosity demands that we

say something about the coefficients of the series that we just derived. As the book mentions in

Section 16.1 on page 791, the Sturm-Liouville theorem (Section 6.1) asserts that this will work and

that the coefficients can be calculated from

Note that the term in the denominator is exactly what we need if we want to reproduce a single term of

the series.

One immediate observation is that the Sine series converges much more quickly than the Cosine series.

The maximum error is almost 100 times smaller with only half as many terms. This is because the function and

the first derivative are continuous under the odd extension.

> d[k] := int((cos((2*k-1)*x/2))^2,x=0..Pi);

> c[k] := (1/d[k])*int(f(x)*cos((2*k-1)*x/2),x=0..Pi);

> ccoef := unapply(c[k],k);

> c[1] := ccoef(1);c[2] := ccoef(2);c[3] := ccoef(3);c[4] := ccoef(4);

> HF := (x,N) -> sum(ccoef(k)*cos((2*k-1)*x/2),k=1..N);

> plot([f(x),HF(x,1),HF(x,2),HF(x,3),HF(x,4)],x=0..Pi);

Notice that each of the partial sums has a zero derivative at , which is one of the requirements of the solution.

However, since has a derivative of at , the series converges slowly at . On the other hand it

definitely seems to be converging. This is further supported by the error plot for 25 terms.

> plot(f(x)-HF(x,25),x=0..Pi);

An approximation to the solultion using 25 terms is thus

> u[25] := (x,t) -> sum(ccoef(k)*exp(-((2*k-1)/2)^2*t)*cos((2*k-1)*x/2),k=1..25);

We can get a plot of this approximate solution. It shows the temperature rising at the insulated end with a zero derivative.

It also shows that the total heat in the rod is decreasing as the temperature in the rod decays to zero, the temperature at

the far end.

> plot3d(u[25],0..Pi,0..1,axes=BOXED,labels=['x','t','u']);

We can also use Maple's plot package to show this solution in a movie.

> with(plots): animate(u[25](x,t), x=0..Pi, t=0..4);