Homework #3

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12 thoughts on “Homework #3

  1. For question 2 when you get to the point of integrating the side with the sum and alpha_k how do we know whether k = m or not?

    • I take it you’re ending up with something like:
      \sum_{k=1}^\infty \alpha_k\int_0^L \left(e^{i\frac{k\pi}{L}x}-e^{-i\frac{k\pi}{L}x}\right)\left(\overline{e^{i\frac{m\pi}{L}x}-e^{-i\frac{m\pi}{L}x}}\right)\,dx
      The sum is shorthand for :
      plug in k=1, calculate the integral, multiply by \alpha_1
      plug in k=2, calculate the integral, multiply by \alpha_2
      plug in k=3, …
      (and add up all the results)

      unless k=m, the integral is just going to give zero, so you the sum reduces to:
      0 + 0 + \dots + 0 + \alpha_m\,\cdot 2L + 0 + 0 + \dots

      Does that answer your question?

  2. I am having trouble with matching and solving alpha_k. As in I am left with the integral of u(x,0) * conjugate complex exponentials = alpha_m * 2 * L. Do we manually solve these integrals? I am not able to match what is done in class as a solution.

  3. Question from a student:

    In question 4a, If one of the constraint is a derivative of V with respect to x, I am not sure what to even do

    This is a separation of variables question, so you’ve made the guess v(x,t) = X(x)T(t). Plug this guess into the constraints.

    For example, v(2,t)=0.
    You plug in X(2)T(t)=0. You don’t want T(t)=0, because that makes v(x,t)=0, which is a solution, but not a helpful one. It must be then that X(2)=0.

    Do a similar thing for \partial_xv(0,t)=0, then start solving the ODE for X(x).

  4. Question from a student:

    In question 1a after I convert the sin + sin functions into exponential using Euler’s method, how do I proceed from there? I have looked at the slide mentioned but it only shows that if exponential with power of K’s multiplied by its conjugate(with power of M’s instead of K’s) is either 0 if K does not equal M and 2L if it is.

    Try going the other way around, and convert slide 7 into sine functions.
    You should get:
    \int_0^L\sin\left(\frac{k\pi}{L}x\right)\sin\left(\frac{m\pi}{L}x\right) = \left\{\begin{aligned}&0&&\text{ if }k\neq m\\&\frac{L}{2}&&\text{ if }k=m\end{aligned}\right.

  5. For Question 4c, what exactly they mean by the general solution for part b?
    The solution to the PDE with inhomogeneous constraints involves u_p(x) and u_h(x,t), and the latter requires to have f(x) from the data for the PDE, i.e. u(x,0) = f(x). How can we find the general solution if we are not given f(x)? Or does the question ask about some other solution?

  6. For the solution to 4b):

    We had a constraint that Up(2) = -3, but it seems if Up(x) = x +1, then Up(2) = 2 + 1 = 3 (not -3).
    Which is right? Just a little confused.
    Thank you.

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