8 03/04 F

1. 2.28c. Use the 2D procedure. Show that the equation may be simplified to
   
   $$u_{\xi\xi} = 0$$
   
   
   Solve this equation and write the solution in terms of $x$ and $y$.

2. Notes 1.5.3.1

3. 2.28k. Reduce the PDE to the form
   
   $$u_\eta = \left(e^{-\xi} + \frac1{\eta}\right) u_{\xi\xi}$$
   
   
   Now discuss the properly posedness for the initial value problem, recalling from the class notes that the backward heat equation is not properly posed. In particular, given an interval $\xi_1\le\xi\le\xi_2$, with an initial condition at some value of $\eta_0$ and boundary conditions at $\xi_1$ and $\xi_2$, can the PDE be numerically solved to find $u$ at large $\eta$? If $\eta_0$ is positive? If $\eta_0$ is a small negative number? If $\eta_0$ is a large negative number?

4. 3.44. This is mostly the uniqueness proof given in class, which can also be found in the notes and more generally in solved problems 3.14-3.16. However, here you will want to write out the two parts of the surface integral separately since the boundary conditions are a mixture of the two cases 3.14 and 3.15 (with $c=0$).

5. Notes 1.7.1.1

6. Notes 1.7.1.2