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Fourier/Laplace transform for PDE

  1. May 10, 2006 #1
    hello
    i am trying to find the fundamental solution to
    [tex]\frac{\partial c}{\partial t} = D\frac{\partial ^2 c}{\partial x^2}[/tex]
    where c=c(x,t)
    with initial condition being [tex]c(x,0)=\delta (x)[/tex]
    where [tex]\delta (x)[/tex] is the dirac delta function.
    i have the solution and working written out in front of me.
    first off its got the laplace transform of [tex]\frac{\partial c}{\partial t}[/tex] as
    [tex] u\hat c (x,u) -c(x,0)[/tex]
    and the Fourier transform of [tex]\frac{\partial ^2 c}{\partial x^2}[/tex] as
    [tex]-q^2 \tilde c (q,t) [/tex]
    and then out of nowhere we get
    [tex]\hat c (q,u) = \frac{c(q,0)}{u+Dq^2}[/tex]
    once that bit of magic is done and a leap of faith is taken then i can see how the rest of it falls into place but can anyone explain the above steps to me?
     
  2. jcsd
  3. Jun 2, 2006 #2
    To get the fourier and laplace transform of the first derivative you put df/dx into the definitions, integrate by parts and use, in the fourier case, that f(x) must vanish as x goes to infinity in order for the the Fourier transform of f(x) to exist. And this you easily can generalize to higher derivatives.

    If you set c(q,0)=1 you get the Green function for the diffusion equation in the transformed space. The Green function is very useful for solving more complicated problems. A general solution to the diffusion equation is then integrals over the Green function G(r,r';t,t') times the source term, the boundary conditions, the initial condition, respectively.
     
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