13 04/18 F

1. A relatively basic problem for which the Sturm-Liouville theorem is helpful is for unsteady heat conduction in a bar, or acoustics in a pipe, or steady heat conduction in a plate, when there is a mixed boundary condition, like 7.21. Such problems will lead during separation of variables to a Sturm-Liouville problem maybe like
   
   $$X'' + \lambda X = 0 \qquad X(0) = 0 \qquad X(1)' + p X(1) = 0$$
   
   
   where $p>0$ is a positive constant, assuming the problem is stable. Show that there are no eigenfunctions for $\lambda$ negative nor for $\lambda$ zero. Also show that there are eigenfunctions $\sin\sqrt{\lambda}x$ for $\lambda$ positive, assuming that some nonlinear function of $\sqrt{\lambda}$ is zero.

2. Examine the zeros of the nonlinear function to establish the qualitative behavior of the eigenvalues. To examine the zeros of the function, you can plot it on a computer; there are also analytical ways to examine them, by formulating it into two functions that must be equal, then sketching both functions and examining their intersection points. Do the predictions of Sturm Liouville theory stand up, as found in chapter 6 or on the web page?

3. What is the orthogonality property? In particular, how can any arbitrary function defined from $x=0$ to $x=1$ be written in terms of the eigenfunctions? What is the expression for the Fourier coefficients? Would any of this have been self-evident without Sturm-Liouville theory? (These are sines, but the integration is over an irrational fraction of their period.)

4. Write out the first few terms of the eigenfunction expansion for $u$ for the unsteady heat conduction in a disk with insulated boundary explicitly (not using summation symbols). In particular, include $n=0$ and $1$, and the first two eigenvalues $\mu_{nm}>0$ for each $n$. Also include the constant term.

5. Identify the actual values of the four $\mu_{nm}$ using the tables in the Mathematical Handbook.

6. Consider the three modes $J_0(\sqrt{\mu_{01}}r)$, $J_1(\sqrt{\mu_{11}}r)\cos\theta$, and $J_1(\sqrt{\mu_{11}}r)\sin\theta$, where $\sqrt{\mu_{01}}$ and $\sqrt{\mu_{11}}$ are the first nonzero stationary points of $J_0$ respectivel $J_1$. In three separate circles, sketch the positions where each of these modes are zero.