10 03/27 F

  1. 3.30. In (a), use the property mentioned in class that the minimum of a harmonic function must occur on the boundary. In (b), try $1-y$ in the domain $\Omega$ given by $y\ge0$. In (c), consider the functions $s=v-u$ and $t=w-v$.

  2. 4.16. This is the heat equation equivalent of the uniqueness proof of the Poisson equation. You need to use method 2 of solved problem 4.2. Ignore the hint, which is wrong. Instead, you can assume that $\beta\ne0$, since $\beta=0$ has already been covered in 4.2. Use that to eliminate $\partial{v}/\partial{n}$.

  3. 4.17a. (Question (b) was done in class, and the stated condition that $F$ only needs to be continuous is not sufficient, but integrable and continuous would do it.)

  4. 4.18. (This is the basic solution for the temperature $u$ in a bar of length 1 where the ends of the bar are kept at zero temperature. Of course, the values of the constants $C_n$ will normally follow from some given initial temperature, and the number of terms in the sum $N$ will normally be infinite.)

  5. 4.19. Plane wave solutions are solutions that take the form (2) in solved problem 4.12, with $\vec\alpha$ a constant vector and $\mu$ a constant. This sort of solutions are a multi-dimensional generalization of the $f(x-at)$ moving “wave” solution of the one-dimensional wave equation. In fact, if you take $\vec\alpha$ to be a unit vector, it gives the oblique direction of propagation of the wave and $\mu$ gives the wave propagation speed. However, in this case you will see that the function $F$ cannot be an arbitrary function unless $b=0$. You may want to do the case $b=0$ separately. And also split up the cases for $\mu$.

  6. Solve the wave equation $u_{tt}=a^2u_{xx}$ using a Fourier transform in $x$. From it, show that the solution takes the form

    \begin{displaymath}
u(x,t)=f_1(x+at)+f_2(x-at)
\end{displaymath}

    Hint: stick with complex exponentials in the solution of the ODE.

  7. 5.27(a). Include a sketch of the characteristic lines. Is the solution you get valid everywhere?