An efficient method for asymptotic solution to some linear PDEs having arbitrary time-varying type I boundary conditions

Abstract

An asymptotic technique is presented for a class of initial-boundary value problems (IBVP) having an arbitrary time-varying boundary condition. This class of IBVP is traditionally solved using the Laplace transform, meaning that governing equation and boundary condition (BC) are solved jointly in Laplace space. This is inconvenient for many applications, particularly inverse methods that require solution for large numbers of different BC, as any change in the BC means that an entirely new problem has to be solved. In this paper, the Weeks method for asymptotic inversion of the Laplace transform, along with some useful properties of the Laguerre functions, are combined in order to circumvent this problem. It is shown how, once a single Green’s function IBVP has been solved asymptotically by the Weeks method, it is possible to compute a solution for any other BC by algebraic manipulation alone. Efficient numerical implementation is discussed, and the method is used to solve a real contaminant transport problem from the literature. It is seen that computational performance is superior to a direct approach that requires multiple inversions.