My Question:
How do I complete the inverse Fourier Transform of:
$\displaystyle \int_{-\infty}^\infty F(\omega)e^{-k\omega^2t}e^{-\gamma t}e^{-i\omega x}\,d\omega$
I cant figure out quite how to use the convolution theorem/table of transforms here.
The Problem:
Solve:
$\displaystyle \frac{\partial u}{\partial t} = k\frac{\partial^2 u}{\partial x^2}-\gamma u, -\infty \lt x \lt \infty$
$\displaystyle u(x,0)=f(x)$
What I have done so far:
$\displaystyle \mathcal{F}\left[\frac{\partial u}{\partial t}\right] = k\mathcal{F}\left[\frac{\partial^2 u}{\partial x^2}\right]-c\mathcal{F}\left[\gamma u\right]$
...
$\displaystyle \frac{dU}{dt}=-k\omega^2U-\gamma U$
$\displaystyle \implies U(\omega,t)=C(\omega)e^{-k\omega^2t}e^{-\gamma t}$
$\displaystyle u(x,0)=f(x)\implies U(\omega,t)=F(\omega)e^{-k\omega^2t}e^{-\gamma t}$
...
$\displaystyle u(x,t)=\mathcal{F}^{-1}[U(\omega,t)]$
$\displaystyle = \int_{-\infty}^\infty F(\omega)e^{-k\omega^2t}e^{-\gamma t}e^{-i\omega x}\,d\omega$ (Stuck here, when trying to do the inverse....)
I cant figure out if any of these apply? Also is Gamma just a constant here (the Euler-Mascheroni constant?)
https://en.wikipedia.org/wiki/Fourier_transform#Tables_of_important_Fourier_transforms