Can anyone help me with this IVP heat equation problem? I have
$u_t-u_{xx}=g(x,t)$
where $x \in \mathbb{R}$, $t>0$, $u(x,0)=0$
So i've found by taking a Fourier transformation that
$\hat{u_t}(w,t)=-w^2\hat{u}(w,t)+\hat{g}(w,t)$
I understand this is a standard technique. The homogenous solution of this PDE is then
$\hat{u}=A(w)e^{-w^2t}$
Can anyone help me solve the non-homogenous form so I can show it must be the result of a convolution and hence find an integral equation for $u$ by inversion.
If I didn't explain well enough part d on page 81 of this set of notes explains the homogeneous case http://www.maths.ox.ac.uk/system/files/coursematerial/2011/979/36/DEnotes-fin.pdf
EDIT: So I have found
$\hat{u}=e^{-w^2t} \Big[ \int_0^t e^{w^2s}\hat{g}(w,s) ds +A(w)\Big]$
And as $u(x,0)=0 \Rightarrow \hat{u}(w,0)=0$ then $A(w)=0$ and
$\hat{u}=e^{-w^2t}\int_0^t e^{w^2s}\hat{g}(w,s) ds$