14 04/24 F

  1. Solve 7.26, 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 $s$.

  2. 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.

  3. Derive the (real) Fourier series for the $2\pi$-periodic function $f(\theta)$ that satisfies

    \begin{displaymath}
f = 0 \mbox{ for } \vert\theta\vert < {\textstyle\frac{1}{2...
...{ for } {\textstyle\frac{1}{2}}\pi < \vert\theta\vert \le \pi
\end{displaymath}

    If possible, check your answer versus 24.17 in the math handbook.

  4. 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 $2\pi$.

  5. 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. Are there points where the Fourier series does not converge? If so, give the $\theta$ values for which it does not converge.
    3. If it converges everywhere, the maximum difference between the Fourier series and the given function should obviously become zero. You might want to plot the function for $n_{{\rm {max}}}=100$ using 1800 equally spaced plot points before you answer that. Explain.