Asymptotic calculation of Laplace Inverse in analytical solutions of groundwater problems

Abstract

The Laplace transform with respect to time, t, is normally used in finding analytical solutions for transient groundwater problems. The behavior of a function at large or small t is known to correspond to that of its Laplace transform counterpart at small or large p, respectively; p is the Laplace transform parameter of t. This condition is generally translated as t being inversely related to p and vice versa. By this relationship many asymptotic solutions for large or small t have been determined from the Laplace domain solutions valid only for small or large p. However, an example is given here which shows this kind of asymptotic calculation may fail to yield correct asymptotic solutions. Hence, the asymptotic calculation must be exercised with care. To deal with this possible failure, the Tauberian theorem is offered to evaluate the asymptotic behavior of functions from their Laplace transform counterparts.