Transform expression involving Erfc back and forth with Laplace transform and its inversion

 Transform expression involving Erfc back and forth with Laplace transform and its inversion

It’s a problem I encountered when answering this question and I think it’s worth starting a new question for it. Consider the following expressions:

expr = Erfc[x/(2 Sqrt[t])] - E^(t + x) Erfc[Sqrt[t] + x/(2 Sqrt[t])];
texpr = E^(-Sqrt[s] x)/(s + s^(3/2));

texpr is the Laplace transform of expr when x > 0. This can be verified numerically:

With[{pre = 32}, 
 Block[{x = RandomReal[{0, 100}, WorkingPrecision -> pre], 
        s = RandomReal[{0, 100}, WorkingPrecision -> pre]}, 
  N[{texpr, NIntegrate[expr Exp[-s t], {t, 0, Infinity}, WorkingPrecision -> pre]}, 
    pre/2 // Floor]]]

But LaplaceTransform and InverseLaplaceTransform can’t handle them well:

LaplaceTransform[expr, t, s]
(* Partly unevaluated *)
InverseLaplaceTransform[texpr, s, t]
(* Unevaluated *)

My question is, can we transform expr to texpr, and texpr to expr with some extra coding?

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