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Use the Fourier transform directly to solve the heat equation

  1. Apr 12, 2014 #1
    1. The problem statement, all variables and given/known data

    Use the Fourier transform directly to solve the heat equation with a convection term
    [tex]u_t =ku_{xx} +\mu u_x,\quad −infty<x<\infty,\: u(x,0)=\phi(x),
    assuming that u is bounded and k > 0.

    2. Relevant equations

    fourier transform
    inverse fourier transform
    convolution thm

    3. The attempt at a solution

    taking the FT of both sides i get
    [tex]U_t=-k w^2U-iw\mu U[/tex]
    I solved the ode and got
    [tex]U(w)=e^{(\mu i w- w^2k)t}[/tex]
    but now I am a bit confused on the next step, is this where I want to get my initial condition involved, or do I want to try and get it back as u(x,t) using inverse FT. I can see that my solution is a gaussian multiplied by another function of F, so I think I might be able to use convolution thm?
  2. jcsd
  3. Apr 13, 2014 #2


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    Don't you mean ##U(\omega,0) = \Phi(\omega,0)##?

    You left out the arbitrary constant when you solved for ##U(\omega,t)##. You should have ##U(\omega,t) = A(\omega) e^{(i\mu\omega-k\omega^2)t}.##
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