Inverse Laplace Transform of a product of exponential functions

[metdave]

I am reviewing some material on Laplace Transforms, specifically in the context of solving PDEs, and have a question.

Suppose I have an Inverse Laplace Transform of the form u(s,t)=e^((as^2+bs)t) where a,b<0. How can I invert this with respect to s, giving a function u(x,t)? Would the inverse transform simply be a convolution?

Thanks!
David

 

[JJacquelin]

Hi !
I fear that this may not be possible.
Try to find the Inverse Laplace transform of exp(-s²) for example.