8 03/07 F

1. 2.22b,g. Draw the characteristics very neatly in the $xy$-plane,

2. 2.28d. (20 pt) First find a particular solution. Next convert the remaining homogeneous problem to characteristic coordinates. Show that the homogeneous solution satisfies
   
   $$2 u_{h,\xi\eta} = u_{h,\eta}$$
   
   
   Put, say, $v=u_{h,\eta}$. Solve this ODE to find $v=u_{h,\eta}$, then integrate $u_{h,\eta}$ with respect to $\eta$ to find $u_h$. Finally find the complete $u$, in terms of $x$ and $y$. Watch any integration constants; they might not be constants.

3. 2.28f. (20 pt) In this case, leave the inhomogeneous term in there, don't try to find a particular solution for the original PDE. Transform the full problem to characteristic coordinates. Show that the solution satisfies
   
   $$4 u_{\xi\eta} - 2 u_{\xi} \pm e^\eta = 0$$
   
   
   where $\pm$ indicates the sign of $xy$, or
   
   $$4 \xi\eta u_{\xi\eta} - 2\xi u_{\xi} + \eta = 0$$
   
   
   or
   
   $$4 \xi\eta u_{\xi\eta} + 2\xi u_{\xi} - \frac{1}{\eta} = 0$$
   
   
   or equivalent, depending on exactly how you define the characteristic coordinates. Solve this ODE for $v=u_{\xi}$, then integrate with respect to $\xi$ to find $u$. Write the solution in terms of $x$ and $y$. Watch any integration constants; they might not be constants.

4. 2.28c. (20 pt) 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 $y$. Watch any integration constants; they might not be constants.

5. Notes 1.5.3.1

6. 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?