14 HW 14

1. (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.

2. Refer to problem 7.19. Find a function $u_0(x,t)$ that satisfies the inhomogeneous boundary conditions. Define $v=u-u_0$. Find the PDE, BC and IC satisfied by $v$.

3. Find suitable eigenfunctions in terms of which $v$ may be written, and that satisfy the homogeneous boundary conditions. Write the relevant known functions in terms of these eigenfunctions and give the expressions for their Fourier coefficients. Work out these integrals for arbitrary $\ell$, $f$, $g_0$, and $g_1$ as far as possible.

4. Continuing the previous question, solve for $v$ using separation of variables in terms of integrals of the known functions $f(x)$, $g_0(t)$, and $g_1(t)$. Write the solution for $u$ completely, worked out for arbitrary $\ell$, $f$, $g_0$, and $g_1$ as far as possible.

5. Assume that $f=0$, $k=\ell=1$, and that $u_x=t$ at both $x=0$ and $x=\ell$. Work out the solution for that case completely. Do not use the above values for earlier homeworks.

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

7. Consider a simple problem of unsteady, axisymmetric, heat conduction in a ring (or unsteady axial flow between concentric pipes) of radii $a$ and $b$:
   
   $$u_t = \kappa \left( u_{rr} + \frac{1}{r} u_r\right) \qquad u(r,0) =f(r) \quad u(a,t)=0 \quad u(b,t)= 0$$
   
   
   Find the eigenvalue problem for the eigenfunctions $R(r)$. Do not try to solve it (or look under Bessell functions in our math handbook). Given an arbitrary function $g(r)$, figure out how to obtain the coefficients $g_n$ in
   
   $$g(r) = \sum_{{\rm all }n} g_n R_n(r)$$