7 02/26 F

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

2. 2.28d. First find a particular solution. Next convert the remaining homogeneous problem to characteristic coordinates. Show that the homogeneous solution satisfies
   
   $$2 u_{h,\xi\eta} = u_{h,\eta}$$
   
   
   Solve this ODE to find $u_{h,\eta}$, then integrate $u_{h,\eta}$ with respect to $\eta$ to find $u_h$ and $u$. Write the total solution in terms of $x$ and $y$.

3. 2.28f. 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 $u_{\xi}$, then integrate with respect to $\xi$ to find $u$. Write the solution in terms of $x$ and $y$.

4. 2.28c. Show that the equation may be simplified to
   
   $$u_{\xi\xi} = 0$$
   
   
   Solve this equation and write the solution in terms of $x$ and $y$.

5. 2.28b. Reduce to canonical form by solving the characteristic equation. In 2.23 you diagonalized essentially the same equation by rotating the coordinate system; and you could then have stretched the coordinates to reduce it to the Laplace equation. Are the coordinates that you find now equivalent to those? In particular, are the lines of constant $\xi$ and $\eta$ orthogonal like in 2.23? If not, how come that more than one linear coordinate transformation can turn the equation into the Laplace equation?

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. Show that the Laplace equation
   
   $$u_{xx} + u_{yy} = 0 \qquad 0 \le x \le 1 \quad 0 \le y \le T$$
   
   
   is improperly posed for the initial/boundary value problem
   
   $$\mbox{BC:}\quad u(0,y) = u(1,y) = 0 \qquad \mbox{IC:}\quad u(x,0) = u_0(x), u_y(x,0) = 0$$
   
   
   because the solution at $y=T$ can be arbitrarily much larger than the given initial condition $u_0(x)$. To do so, assume that $u_0(x)=\varepsilon\sin(n{\pi}x)$, where $\varepsilon$ is a small number. The solution is of the form $u=\sin(n{\pi}x)f(y)$ where $f(y)$ can be found from substitution into the Laplace equation and initial conditions. Show that the solution at $y=T$ can be any amount of times larger than $\varepsilon$, the magnitude of the initial condition. For example, show that the solution at $y=T$ can be a billion times larger than the initial condition at $y=0$.

8. Repeat the argument to show that the wave equation does not have a problem with the above initial value problem.