5 02/13 F

1. 2.19b, h. Show a picture of the different regions.

2. 2.20.

3. 2.21, in three spatial dimensions, and time as appropriate.

4. 4.20. A number $T$ is rational if it can be written as the ratio of a pair of integers, e.g. 1.5 = 3/2 = 6/4 = 9/6 = .... It is irrational if it cannot, like $\sqrt{2}$. Near any rational number, irrational numbers can be found infinitely closely nearby, and vice versa. For example, the value $\pi$ to one billion digits, as found on the internet, is the rational number $31415927\ldots/10000000\ldots$; $\pi$ itself is not rational. The wave equation problem when $T=\pi$ has no nonzero solutions, but when $T=\pi$ to 1 billion digits has infinitely many of them. Obviously, in physics it is impossible to determine the final time to infinitely many digits, so there is no physically meaningful solution to the stated problem.

   For nonzero solutions, try $u=\sin(n\pi x)\sin(n\pi t)$, which satisfies the wave equation and the boundary conditions at $x=0$ and $x=1$ and the initial condition at $t=0$. See when it satisfies the end condition at $t=T$.

   The wave equation needs two initial conditions at $t=0$, not one condition at $t=0$ and one at $t=T$.

5. Show that the Laplace equation
   
   $$u_{xx} + u_{yy} = 0 \qquad 0 \le x \le 1 \quad 0 \le y \le T$$
   
   
   with the same boundary conditions, (replacing $t$ by $y$), does not have the same problem. Hint: assume solutions of the form $u=\sin(n\pi x)f(y)$ and plug into the Laplace equation and $u(x,0)=0$ to figure out what $f(y)$ is. (You may want to recall the graphs of the hyperbolic functions.)

6. 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 $u_0(x)=C\sin(n{\pi}x)$, in which case 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.

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