In finishing the evaluation of a partial differential equation, I've arrived at a stage of a convultion integral I'm stuck at.
I have to evaluate the following integral $\frac{a}{\sqrt{\pi}}\int^{t}_0 \frac{k}{\sqrt{t-\tau}}\exp({\frac{−x^2}{4a^2 (t-\tau)}})\,d\tau$
where $a>0, t>0, 0<x<\infty, k:=constant$
given that the solution will be $k[2a\sqrt{\frac{t}{\pi}}\exp({\frac{−x^2}{4a^2 t}})-xerfc(\frac{x}{2a\sqrt{t}})]$
and I'm suggested to use the change of variable
$z=\frac{x}{2a\sqrt{t-\tau}}$
Directly using the substitution hasn't been fruitful so far. Any suggestions?
