## HW help, PDE solving with Fourier Transforms

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### HW help, PDE solving with Fourier Transforms

Right so I've got the heat equation [imath]u_t = u_xx[/imath], and some initial condition [imath]u(x,0) = f(x)[/imath].

Now according to my notes, my solution should be:
$\mathcal{F}^1 \left( F(k)e^{ikx +k^2 t} \right)$
(where [imath]F(k)= \mathcal{F}(f(x))[/imath])

But with the initial condition [imath]f(x) = 1 \, \mbox{if} \, x \in (-1,2) , \, 0 \, \mbox{o/w}[/imath], [imath]F(k)= \tfrac{i}{k}(e^{2ik} - e^{ik})[/imath], I wind up with the godawful integral:

$\frac{i}{\pi} \int_{-\infty}^{\infty} \!{\frac {{{\rm e}^{ik \left( x-2 \right) + \left( {k}^{2}-ik \right) t}}+{{\rm e}^{ik \left( x+1 \right) + \left( {k}^{2}-ik \right) t}}}{k}}{dk}$

Can anyone tell me where I've gone wrong please?

william
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### Re: HW help, PDE solving with Fourier Transforms

The Fourier transform of a uniform distribution is going to look like $F(k)=\frac{e^{ix_1k}-e^{ix_2k}}{k}$ so your integral looks about right.
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