12 04/11 F

1. In problem 7.26, show that if we define a new unknown $v$ by setting
   
   $$u=e^{\alpha x + \beta t}v$$
   
   
   the PDE for $v$ simplifies into the normal heat equation $v_t=\kappa v_{xx}$ for suitable values of the constants $\alpha$ and $\beta$. Find those values. Show that the initial condition for $v$ is the same as the one for $u$, but that the boundary condition becomes $v(0,t)=\bar f(t)$ where $\bar f$ is a different function than $f$. Identify what $\bar f$ is.

2. Solve the problem for $v$ using Laplace transformation in time. Write the solution in terms of $u$ and the given function $f$, of course. Note that the answer in the book is wrong.

3. Now solve 7.26 directly, by Laplace transforming the problem as given in time. This is a good way to practice back transform methods. Note that one factor in $\widehat u$ is a simpler function at a shifted value of coordinate $s$.

4. Solve 7.35 by Laplace transform in time. Clean up completely; only the given function may be in your answer, no Heaviside functions or other weird stuff. There is a minor error in the book’s answer.

5. Replot the solution to 7.27, summing only one, two, and three eigenfunctions and examine how the solution approaches the exact one if you sum more and more terms. Comment in particularly on the slope of the profiles at time $t=0$ and $t=1$ at the boundaries. What should the slope be at those times? What is it?

6. Derive the (real) Fourier series for the $2\pi$-periodic function $f(\theta)$ with values
   
   $$f = 0 \quad \vert\theta\vert<0.5\pi \qquad f = 1 \quad 0.5\pi<\vert\theta\vert<\pi$$
   
   
   If possible, check your answer versus 24.17 in the math handbook.

7. Plot the Fourier series for increasing number of terms, $n_{{\rm {max}}}=1,3,7,15$ using 360 equally spaced plot points in the range from 0 to $2\pi$.

8. Based on these plots, discuss whether the Fourier series really converges to the given function for high enough number of terms:
   1. When does an actual jump first show up?
   2. The original function is undefined when $\theta={\textstyle\frac{1}{2}}\pi$ and ${\textstyle\frac{3}{2}}\pi$. What happens with the Fourier series at those locations?
   3. Are there points where the Fourier series does not converge? If so, give the $\theta$ values for which it does not converge.

9. Supposing that the Fourier series converges everywhere, and it does, the difference between the converged function and the nonconverged function should become vanishingly small, not? In particular the maximum difference between the Fourier series and the converged function should become zero. Or does it? If not, explain why not. In particular, plot the function for $n_{{\rm {max}}}=100$ using 1800 equally spaced plot points to explain your reasoning.