I am trying to solve the heat equation with both conduction and advective terms:

alpha*(d^2)T/(d^2)z-v*dT/dz-dT/dt (1)

where v=u*Cw/Cv, alpha is a constant. According to the literature the temperature frequency cariation solution of equation 1 is:

That=T0*exp(gamma*z)*exp(i*omega*t) (2)

where T0 and omega are constants. To get the actual temperature (T) value I am trying to use the Gaver-Stehfest numerical inversion. However, the values I get are not realistic, while my input parameters are fine. What am I doing wrong?

You need to use multi-precision toolbox to enhance results. Very good results for one staircase simulation with a big L. Beware when you simulate multiple staircase.