11 04/04 F

1. Solve the ODE of the previous question. Apply the initial conditions to find the integration constants. Write the complete solution to the PDE, BC, and IC.

2. Plot the solution using a computer and compare with the D'Alembert solution you got earlier. You should get the same results.

   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.

3. Refer to problem 7.19. Find a function $u_0(x,t)$ that satisfies the inhomogeneous boundary conditions.

4. Continuing the previous problem, define $v=u-u_0$. Find the PDE, BC and IC satisfied by $v$.

5. Find eigenfunctions in terms of which $v$ may be written, and that satisfy the homogeneous boundary conditions. Work out the Fourier coefficients of the relevant functions in the problem for $v$ as far as possible.

6. Solve for $v$ using separation of variables in terms of expressions in terms of the known functions $f(x)$, $g_0(t)$, and $g_1(t)$. Write the solution for $u$ completely.

7. Assume that $f=0$, $k=\ell=1$, and that $u_x=t$ at both $x=0$ and $x=\ell$. Work out the solution completely.

8. Plot the solution numerically at some relevant times. I suspect for large times the solution is given by
   
   $$u = (x-{\textstyle\frac{1}{2}})t + {\textstyle\frac{1}{6}}(... ...{1}{2}})^3-{\textstyle\frac{1}{8}}(x-{\textstyle\frac{1}{2}})$$
   
   
   Do your results agree?