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Fourier Series Solution of 1-D Heat Flow

  1. Mar 30, 2013 #1
    1. The problem statement, all variables and given/known data


    Length of rod = 1

    Initial Conditions: u(x,0)=sin(πx)

    Boundary conditions: u(0,t)=0 and u(1,t)=5.


    Alright I am supposed to find the temperature at all times, but I am curious about the setup of the problem itself.

    When x = 1, the boundary condition says that u = 5.

    When t = 0, the initial condition says that u = sin(x∏).

    So u(1,0) is supposed to equal what exactly? The boundary says it should be 5, but the initial condition says that sin(∏) = 0, so what would the answer be. I feel like this is contradictory but people are saying that it isn't and I am an idiot.

    I have asked this somewhere else and got responses just saying this was a stupid question.


    Thanks for any help
     
  2. jcsd
  3. Mar 30, 2013 #2

    micromass

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    You're right, it is contradictory. You should ask for clarification at your instructor.

    A possible way to interpret the problem is that the ##u(1,t)=5## condition only holds for large ##t## and not for all ##t##.
     
  4. Mar 30, 2013 #3
    You have no idea how much I appreciate this response. Thank you very much!
     
  5. Mar 30, 2013 #4

    LCKurtz

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    @curtis15: You didn't specify for what ##t## your boundary conditions ##u(0,t)=0,\ u(1,t)=5## apply. A reasonable interpretation would be for ##t>0##. Or you could think of the bar having the initial temperature distribution ##u(x,0) = \sin(\pi x)## suddenly inserted into a situation with those boundary conditions. I don't think there is anything contradictory here and working the problem should be straightforward.

    [Edit, added later:] Another reasonable interpretation is to assume the initial condition ##u(x,0) = \sin(\pi x)## holds for ##0 < x < 1##. That works even better intuitively.
     
    Last edited: Mar 30, 2013
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