12 04/10 F

1. Write the complete (Sturm-Liouville) eigenvalue problem for the eigenfunctions of 7.27. Find the eigenfunctions of that problem. Make very sure you do not miss one. Write a symbolic expression for the eigenfunctions in terms of an index, and identify all the values that that index takes.

2. Continuing the previous homework, write $f=x$ and $g=1$ in terms of the eigenfunctions you found for the case $\ell=1$. Be very careful with one particular eigenfunction. Note that sometimes you need to write a term in a sum or sequence out separately from the others.

3. Continuing the previous homework, substitute $u(x,t)=\sum_nu_n(t)X_n(x)$ into the PDE to convert it into an ordinary differential for each separate coefficient $u_n(t)$. Solve the ODE. Be very careful with one particular case. By writing the initial conditions in terms of the eigenfunctions, identify the integration constants. Write out a complete summary of the solution. Make sure to identify the values of your numbering index in each expression.

4. (6 pts) Reconsider the separation of variables solution you derived. Using some programming language, evaluate the found solution at 101 equally spaced $x$-values from 0 to $\ell$ at times 0, 0.25, 0.5, and 1.25. Take $\ell=1$ and $a=1$. Include at least 50 nonzero terms in the summations. Plot the solution at these four times. Compare with the D'Alembert solution of the previous homework, which must be the same. (Check your D'Alembert solution first against the posted solution). Show also what happens if you only include 10 terms in the summations.

   To help you get started, a Matlab program that plots the solution to problem 7.28 is provided as an example. You need both p7_28.m and p7_28u.m. This program is valid for the PDE and BC solved in class, with the additional data
   

   $$a={\textstyle\frac{1}{2}},\quad \ell={\textstyle\frac{1}{2}... ...n = \frac{1}{(2n-1)^2}, \qquad g(x) = 0 \Rightarrow g_n = 0.$$
   
   
   These may of course not apply for your problem.

   To run the program, enter matlab and type in p7_28. If you do not have matlab, a free replacement is octave. Or you can use some other programming and plotting facilities.

   Include your code.