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?