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Solving PDE heat problem with FFCT

  1. Aug 28, 2017 #1
    1. The problem statement, all variables and given/known data
    solve the following heat problem using FFCT:
    A metal bar of length L, is at constant temperature of ## U_0 ## , at ##t=0## the end ##x=L## is suddenly given the constant temperature of ##U_1## and the end x=0 is insulated. Assuming that the surface of the bar is insulated, find the temperature at any point x of the bar at any time ##t>0## , assume ##k=1##

    X5usGki.png
    2. Relevant equations
    heat eq.
    ## \frac {\partial^2 u} {\partial x^2} = \frac 1 k \frac {\partial u} {\partial t} ##
    with the following additional equations:
    GGjbJ4J.jpg

    3. The attempt at a solution
    my attempt goes like this:
    $$ \frac {\partial^2 u} {\partial x^2} = \frac 1 k \frac {\partial u} {\partial t} $$
    $$ \mathcal{F}_{fc} \left[ \frac {\partial u} {\partial t} \right] = \mathcal{F}_{fc} \frac {\partial^2 u} {\partial x^2} $$
    $$ \frac {dU} {dt} = {-\left( \frac {{n} {\pi}} L \right)}ˆ{2} * F(x,t) + \left( {-1} \right)ˆn \frac {\partial{f(L,t)}} {\partial x} - \frac {\partial{f(0,t)}} {\partial x} $$
    $$ \frac {dU} {dt} = - \left( \frac {{n} {\pi}} L \right)ˆ(2) * F(x,t) + \left( {-1} \right)ˆn \frac {\partial{f(L,t)}} {\partial x} $$

    and i dont know how to continue...
     
    Last edited: Aug 28, 2017
  2. jcsd
  3. Aug 29, 2017 #2

    RUber

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    @Aows, I have not worked much with the FFCT, but it seems like the method is much like that of other transforms.
    After you have rewritten the derivatives, you should separate the variables, setting F(x,t) = X(x)T(t) and use standard ODE methods to solve for T(t).
    What is unclear to me it that your problem does not explicitly give
    ## \frac{\partial f}{\partial x} (L,t)##
    It feels like the sudden change from ##U_0 ## to ##U_1## represents a discontinuity. Have you dealt with similar problems before? I know in Laplace transforms, there is a standard method for handling things that switch on. I am unfamiliar with the analogous method for the Cosine transform.
     
  4. Aug 29, 2017 #3
    Dr.RUber @RUber ,
    1. separate the variables, you mean in my last step? if so can you tell me how ?
    2. for the ## \frac{\partial f}{\partial x} (L,t) ## , this is my problem too, do you think that some information should be given or all the infos are available now ?
     
  5. Aug 29, 2017 #4

    RUber

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    To separate the variables, you assume that your function ##F## is a product of two functions, one dependent only on x, ##X(x)## and one dependent only on t ##T(t)##.
    If you let ## T(0) = 1##, then your initial conditions will fully describe ##X(x)##.
    In the case that ## \frac{\partial f}{\partial x} (L,t) = 0## then you have a simple differential equation to solve for each n.

    I am not sure using the Cosine series is the best choice, since the data given are Dirichlet. The source I saw online said that Sine series are more appropriate for Dirichlet data. Are you able to use the sine transform instead?
     
  6. Aug 29, 2017 #5
    Dr. @RUber ,
    how to apply the separation of variable after my last step??

    and regarding the using FFCT instead of FFST is because the problem says in the end ## x=0 ## is insulated which means that du/dx=0,
     
  7. Aug 29, 2017 #6

    RUber

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    Oh, I see. I read that information as saying that ##f(0,t) = U_0, \quad f(L,t) = U_1##. In this case, you seem to have mixed boundary conditions then.
    ##\frac{\partial u}{\partial x} (0,t) = 0, u(L,t) = U_1 ##
    To use the separation of variables after your last step, you will find an appropriate function of time that would satisfy your differential equation.
     
  8. Aug 29, 2017 #7
    Dr. @RUber , can you elaborate more please?

    many thanks to all of your contributions...
     
  9. Sep 2, 2017 #8
    any updates related to this problem ?
    thank you
     
  10. Sep 3, 2017 #9

    TSny

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    Are you required to solve it using the FFCT? Like @RUber said, this doesn't seem too appropriate because of the mixed boundary conditions. ( I think I see a trick way of solving it using the FFST.)

    Have you tried RUber's suggestion of approaching it using the standard separation of variables technique? What differential equations do you get for ##X(x)## and ##T(t)##?
     
  11. Sep 3, 2017 #10
    Hello Mr. @TSny ,
    yes am required to use the FFCT to solve this problem. it is said that if there is an insulated side $dp/dx = 0 $ , then we are required to use the FFCT as a method of solution.
    can you help with how to solve it using FFCT? @TSny
    here is the solution of the problem using laplace transform:

    F2dduhK.jpg

    nqwOE4R.jpg
     
  12. Sep 3, 2017 #11

    TSny

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    The LT solution looks good except for a couple of minor errors that cancel each other:

    In equation (6), there should be a minus sign in front of C2.

    cosh is defined with a positive sign in the numerator, not a negative sign.

    Using the FFCT, I don't see how to get past the snag of having U specified at x = L rather than having Ux specified at x = L.
     
  13. Sep 3, 2017 #12
    thanks indeed for your notes on the problem, I also noticed those errors too.
    regarding FFCT solution, this is my problem too, @TSny do you think that there are some missing information (i mean more should be given) or those info are enough but i need to learn more ??
     
  14. Sep 3, 2017 #13

    TSny

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    I don't think there is any missing information. The problem is well-posed.

    Note that the FFCT over the interval ##0<x<L## leads to an expansion of ##U## in terms of the set of functions ##\cos \left(\frac{n \pi x}{L} \right)##. But, the LT solution is in terms of the set ##\cos \left(\frac{(2n-1) \pi x}{2L} \right)##. The members of the second set are not contained in the first set. So, I don't see how the FFCT is going to lead to the solution.
     
  15. Sep 3, 2017 #14
    Mr. @TSny , what do you mean by this ((The members of the second set are not contained in the first set.)),
    in my last reply, i meant: in order to use FFCT, do you think there are some more infos should be given ?
     
  16. Sep 3, 2017 #15

    TSny

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    For example, in the set ##\cos \left(\frac{(2n-1) \pi x}{2L} \right)##, you have ##\cos \left(\frac{3 \pi x}{2L} \right)## when ##n = 2##. But this function is not contained in the set ##\cos \left(\frac{n \pi x}{L} \right)## for any value of the integer ##n##.

    I think there might be a "trick" way to solve the problem using the FFCT. Consider a related problem where you have a rod of length 2L which is insulated at both ends as well as the sides. Suppose the initial temperature distribution is ##U(x, 0) = U_0## for ##0 < x < L## and ##U(x, 0) = -U_0## for ##L < x <2L##. Also, suppose that ##U(L, 0) = 0##. You can solve this using the FFCT since the boundary condition at both ends is ##U_x = 0##. I think the solution for this problem relates in a simple way to the solution for the original problem.

    upload_2017-9-3_12-58-37.png

    Here, the "original problem" is your problem with the right end set at 0 rather than ##U_1## for convenience. But, that's easily taken care of.

    I have not actually carried out the calculation, but I think it should work.
     
    Last edited: Sep 3, 2017
  17. Sep 3, 2017 #16
    Mr. @TSny , that's exactly my problem, if i substitute ##U_1## , i don't know what to do after that ??
     
  18. Sep 3, 2017 #17

    Orodruin

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    I suggest first getting rid of the inhomogeneous boundary condition at x = L by shifting the solution.

    Once you have done that, you can solve the problem using the base functions given in #13 or make an odd extension around x=L to end up with a domain 0<x<2L with homogeneous Neumann conditions at both boundaries - a problem you can directly apply FFCT on.

    Edit: oops, that is exactly what #19 said. Should reload before posting after some time ...
     
  19. Sep 3, 2017 #18
    thanks indeed for your contribution Mr. @Orodruin , can you write the full answer in details ?
    exams are on the doors and you know there are a lot of subjects along with it.
    your help will be highly appreciated.
     
  20. Sep 3, 2017 #19

    Orodruin

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    No, this would be against the forum rules and posting the solution would just generate a warning for me. You need to solve the problem yourself. Feel free to ask further questions about things you still find unclear.
     
  21. Sep 3, 2017 #20
    appreciate your kind reply a lot,
    but i really don't know how to apply your idea of shifting the boundary condition after substituting the ##U_1## . @Orodruin
     
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