14 04/24 F

1. Solve 7.26, by Laplace transforming the problem as given in time. This is a good way to practice back transform methods. Note that one factor in $\widehat u$ is a simpler function at a shifted value of coordinate $s$.

2. Solve 7.35 by Laplace transform in time. Clean up completely; only the given function may be in your answer, no Heaviside functions or other weird stuff. There is a minor error in the book’s answer.

3. Consider a simple problem of unsteady, axisymmetric, heat conduction in a ring (or unsteady unidirectional 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
   
   $$g(r) = \sum_{{\rm all\ }n} g_n R_n(r)$$