14 12/04 F

1. 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 $v$.

2. 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.

3. 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.

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

5. 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?

6. 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
   
   $$g(r) = \sum_{{\rm all\ }n} g_n R_n(r)$$