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Math Help - Heat Equation Derivation

  1. #1
    Super Member Deadstar's Avatar
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    Heat Equation Derivation

    Solve the heat equation u_t = u_{xx} with the boundary conditions u(0, t) = u(\pi, t) = 0 and the following initial condition: u(x, 0) = T.

    Now, the question itself isn't a problem, its the actual derivation of the heat equation that is (its not asked in the question but we're told that in an exam it would be).

    So I'm looking to derive this. u(x, t) = \sum_1^\infty b_n e^{-n^2t} \sin(nx) where bn = \frac{2}{\pi} \int_0^{\pi} u(x, 0) \sin(nx) dx.

    So i start by doing seperation of variables to get T'X = X''T, ... , X'' = \lambda X and T' = \lambda T.

    Now at this point I think I should use eigenfunctions then superposition to solve it but i cant see where the e^{-n^2 t} or the b_n come from?

    Any help please?
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  2. #2
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    Opalg's Avatar
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    Quote Originally Posted by Deadstar View Post
    Solve the heat equation u_t = u_{xx} with the boundary conditions u(0, t) = u(\pi, t) = 0 and the following initial condition: u(x, 0) = T.

    Now, the question itself isn't a problem, its the actual derivation of the heat equation that is (its not asked in the question but we're told that in an exam it would be).

    So I'm looking to derive this. u(x, t) = \sum_1^\infty b_n e^{-n^2t} \sin(nx) where bn = \frac{2}{\pi} \int_0^{\pi} u(x, 0) \sin(nx) dx.

    So i start by doing separation of variables to get T'X = X''T, ... , X'' = \lambda X and T' = \lambda T.

    Now at this point I think I should use eigenfunctions then superposition to solve it but i cant see where the e^{-n^2 t} or the b_n come from?

    Any help please?
    Start with the x equation X'' = \lambda X. This has the solution X = A\cos(\omega x) + B\sin(\omega x), where \omega^2=-\lambda. The boundary conditions X(0) = X(\pi) = 0 tell you that A=0 and \omega = n (an integer). Thus \lambda = -n^2.

    The t equation then becomes T' = -n^2T, which has the solution T(t) = b_ne^{-n^2 t} (where b_n is a constant). You then have to use Fourier theory to get the right vaues of the b_ns to satisfy the initial condition when the eigenfunction solutions are superimposed.
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