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Math Help - Fourier transforms

  1. #1
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    Fourier transforms

    Use Fourier transforms to solve the ODE
    -uxx + (a^2)u = δ(x)

    could anyone please help me with this problem ?
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by jin_nzzang View Post
    Use Fourier transforms to solve the ODE
    -uxx + (a^2)u = δ(x)

    could anyone please help me with this problem ?
    Well you could start by taking the FT of this.

    If we take the definition:

    F(\omega)=\mathcal{F}f (\omega)=\frac{1}{\sqrt{2\pi}}\int_{-infty}^{\infty}f(x) e^{-i\omega t}\; dt

    then we have:

     <br />
(\mathcal{F}D^nf) (\omega)=(i\omega)^n F(\omega)<br />

    and \mathcal{F} \delta(\omega)=\frac{1}{\sqrt{2\pi}}

    Then the equation becomes after taking the FT:

     <br />
-(i\omega)^2U(\omega)+a^2U(\omega)=\frac{1}{\sqrt{2  \pi}}<br />

    Now solve this for U(\omega) and take the inverse FT

    CB
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