I need to solve this:
$u_t(x,t)-ku_{xx}(x,t)=xte^{-t^2}$
By using Duhamel's Principle and the Heat Kernel.
So far this is what I've done:
$u_t(x,t)-ku_{xx}(x,t)=xte^{-t^2}$
where u(x,0)=0
$u(x,t)= \int h(x,t-\sigma|\sigma)d\sigma$
is the solution to the IVP
where h solves
$h_t(x,t|\sigma)-kh_xx(x,t|\sigma)=0$ for all t>0
subject to $h(x,0|\sigma)=f(\sigma)$
where I believe $f(\sigma)=y\sigma e^{-\sigma^2}$
and then using the Heat Kernel Method
$h(x,t|\sigma)=\frac{1}{\sqrt{4 \pi k t}}\int y\sigma e^{-\sigma^2} $$e^\frac{-(x-y)^2}{4kt}dy$
where I substitute in z=x-y and therfroe dy=-dz to get
$h(x,t|\sigma)=\frac{1}{\sqrt{4 \pi k t}}\int (z+x)\sigma e^{-\sigma^2} $$e^\frac{-z^2}{4kt}dz$
But I'm not really sure where to go from here? I'd be grateful if anyone could suggest anything