Hi guys, I have posted the thread before but this time I have come with PdF!!
As you know, Laplace inversion of A1*exp(x*s^.5)+A2*exp(-x*s^.5) in finite x domain which we can not deduce A1=0, is not possible with routine procedure. As you may see in attachment (PdF!!) an inversion is done which I can not understand how. Also the answer is presented in a series form.
The coefficients are not important at all, you may download the whole text freely by a simple google search of :
THERMO-HYDRO-MECFmCAL MODELS FOR
SATURATED
AND UNSATURATED POROUS MEDIA, YAFEI ZHOU,
Page 25, Eq 2-53
In the attachment you find the brief expression of results.