HallsofIvy, that is not difficult like you see! by semigroupe theory the expression e^{tL}f is the solution of the differential equation on Banach space : u'+Lu=0, u(0)=f, so for example for the laplacian,
e^{t\Delta}f is the solution of the Heat equation (for Dirichlet or Neumann...
It solve by which space you choose?
if continuous functions ok no problem
but if you solve on other spaces like sobolev spaces, in this case your method will not be avalable