1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

So here is what i have for a solution to the heat eqn.

  1. Jan 20, 2014 #1


    User Avatar
    Gold Member

    1. The problem statement, all variables and given/known data
    i am solving the heat equation and so far i know what i have is correct. basically, i am down to this [tex]\sum_{n=0}^{\infty}A_n\cos(\frac{n\pi x}{L})=273+96(2L-4x)[/tex] where all i need is to solve for [itex]A_n[/itex]

    3. The attempt at a solution
    i was thinking about taking advantage of the orthogonality of the cosine function and multiplying both sides by [itex]\cos(\frac{m\pi x}{L})[/itex] and then integrate over the interval [itex][0,L][/itex]. my question is, if [itex]m\neq n[/itex] then i can move this cosine into the sum, integrate term wise, yet the left side equals zero ([itex]m\neq n[/itex]). Thus, [itex]m = n[/itex], and then if i multiply both sides by [itex]\cos(\frac{n\pi x}{L})[/itex] i cannot put this cosine term inside the sum, and thus i have lost the idea of how to solve for [itex]A_n[/itex]. any help/advice is awesome!

    for what it's worth, this is not a class i am in, i'm just doing the problem for fun. thanks for your help!!
  2. jcsd
  3. Jan 20, 2014 #2


    User Avatar
    Homework Helper

    This is the standard procedure.

    No. The left hand side is
    \int_0^L \cos(m\pi x/L) \sum_{n=0}^\infty A_n \cos(n \pi x /L)\,dx
    = \sum_{n= 0}^\infty \left(A_n \int_0^L \cos(m \pi x/L) \cos(n\pi x/L)\,dx\right) \\
    = A_m \int_0^L \cos^2(m \pi x/L)\,dx
    Remember that [itex]n[/itex] varies in the summation, but [itex]m[/itex] is fixed. At some point [itex]n[/itex] must take the value [itex]m[/itex], and this is the only term in the sum which doesn't vanish on integration.
    Last edited: Jan 20, 2014
  4. Jan 20, 2014 #3


    User Avatar
    Gold Member

  5. Jan 20, 2014 #4


    User Avatar
    Gold Member

    i have another post in the pde/ode theory part on a similar topic. if youre not too busy, perhaps you could take a look?
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted