Analytic Pade-like approximations of exp(–sqrt(s)) for simulations of diffusion processes in the semi-infinite geometry

Abstract

Detailed transient behaviors of mass and heat transfer processes are required to solve partial differential equations. When those partial differential equations are coupled, they are still difficult to solve in time domain. For linear mass and heat transfer processes, their Laplace-domain solutions are obtainable and, when they are approximated by rational polynomials in the Laplace variable s, the problems can be transformed to a set of ordinary differential equations solved easily in time domain for various initial conditions. In this approximation, the conventional Pade method based on the Tayler series expansion of the Laplace-domain solutions has been well developed and effective. However, for some mass and heat transfer processes in the semi-infinite geometry, the Pade approximation is not applicable because the Laplace-domain solutions involving exp(–sqrt(s)) are not analytic at s = 0. Here, for such processes, analytical methods to approximate exp(–sqrt(s)) by rational polynomials are proposed. First it is expanded in series in terms of cosh–1(2ksqrt(s)) which converges fast. This series, when truncated, is analytic at s = 0 and its Pade approximations are available. The proposed method enables partial differential equations be replaced to a set of ordinary differential equations, reducing computations considerably for coupled partial differential equations. Performances of the proposed method are illustrated with several realistic mass and heat transfer processes.