Fourier transforms

**jin_nzzang** · September 22nd 2009, 03:25 AM

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

could anyone please help me with this problem ?

**CaptainBlack** · September 22nd 2009, 04:21 AM

Originally Posted by jin_nzzang

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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