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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- Sep 22nd 2009, 03:25 AMjin_nzzangFourier transforms
Use Fourier transforms to solve the ODE

-uxx + (a^2)u = δ(x)

could anyone please help me with this problem ? - Sep 22nd 2009, 04:21 AMCaptainBlack
Well you could start by taking the FT of this.

If we take the definition:

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

then we have:

$\displaystyle

(\mathcal{F}D^nf) (\omega)=(i\omega)^n F(\omega)

$

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

Then the equation becomes after taking the FT:

$\displaystyle

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

$

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

CB