7 HW 7

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. Then find a particular solution. 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 $u_{h,\eta}=v$. Solve the ODE to find $v=u_{h,\eta}$, noting that the integration constant is not really a constant. Next restore $v=u_{h,\eta}$ and integrate with respect to $\eta$ to find $u_h$, watching again the integration constant. Finally write down the complete $u$, including particular solution, in terms of $x$ and $y$.

4. 2.28f expanded. Solve this much like the previous problem, but in this case, leave the inhomogeneous term in there, don't try to find a particular solution for the original PDE. So transform the full problem to characteristic coordinates. (I think it is easiest to leave the logarithms in the coordinates you get, 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 $y$.

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 really 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, assume that the “spatial coordinate” $\eta$ is restricted to some finite interval $\xi_1\le\xi\le\xi_2$, (like heat conduction in a finite bar). Assume that you have been given some suitable Dirichlet or Neumann boundary conditions at the ends $\xi_1$ and $\xi_2$. Finally assume that you have been given an initial condition at some value $\eta_0$ of the “time coordinate $\eta$. Under these conditions, determine whether the PDE be numerically solved to find $u$ at large positive $\eta$. Can it be if $\eta_0$ is positive? If $\eta_0$ is a small negative number? If $\eta_0$ is a large negative number?

7. 2.28b. After transforming, describe a typical properly posed problem for the original equation. What sort of conditions could be prescribed on what sort of curves in the $\xi,\eta$-plane?