8 10/23 F

1. 2.24 continued. Take the result of the previous homework question, as posted, and convert the principal part to the Laplacian. Then get rid of the first order derivatives to get a Helmholtz equation (with an imaginary $k$).

2. 2.22b,g. Draw the characteristics very neatly in the $xy$-plane,

3. 2.28d expanded. First find a particular solution. Try a quadratic
   
   $$u_p = A x^2 + B xy +Cy^2 + Dx + Ey +F$$
   
   
   But obviously you do not need $F$, nor do you need B to create the $x$ and $y$ terms, so take these zero. Next convert the remaining homogeneous problem for $u_h$ 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.

4. 2.28f expanded. 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. (I think it is easiest to leave the logarithms in the coordinates, but you can try it either way.) 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.

5. 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 $y$. Watch any integration constants; they might not be constants.

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?

7. 2.28b. Describe a typical properly posed problem for the original equation.