Continued Fraction Method for Approximation of Heat Conduction Dynamics in a Semi-Infinite Slab

Jietae Lee,Dong Hyun Kim

doi:10.4236/am.2014.57100

Abstract

Heat conduction dynamics are described by partial differential equations. Their approximations with a set of finite number of ordinary differential equations are often required for simpler computations and analyses. Rational approximations of the Laplace solutions such as the Pade approximation can be used for this purpose. For some heat conduction problems appearing in a semi-infinite slab, however, such rational approximations are not easy to obtain because the Laplace solutions are not analytic at the origin. In this article, a continued fraction method has been proposed to obtain rational approximations of such heat conduction dynamics in a semi-infinite slab.

Highlights

Partial differential equations describing dynamics of diffusional processes, when coupled with other differential equations, are difficult to simulate and analyze. To overcome such difficulties of partial differential equations, they are often approximated by a set of ordinary differential equations
The Pade approximation of Laplace solutions of the partial differential equations can be used for this purpose [1]-[4]
It is well-known that the Pade approximations are obtained through the continued fraction expansions [4]-[6] and routines for the Pade approximation and the continued fraction expansion are provided in the Maple package [7]

Summary

Introduction

Partial differential equations describing dynamics of diffusional processes, when coupled with other differential equations, are difficult to simulate and analyze. The Pade approximation of Laplace solutions of the partial differential equations can be used for this purpose [1]-[4]. For some diffusional problems appear in a semi-infinite slab, such Pade approximations do not exist because the Laplace solutions are not analytic at the origin. For the approximation of Equation (5), the Pade approximation method is often used, but it cannot be applied ( ) to our transfer function of G= (s) exp −x s because the transfer function is not analytic at the origin. ( ) cosh s terms in Equation (8), we can obtain a continued fraction expansion in s as ( ) exp= − s h= 0 0.98718 25.833s 387.52s.

Simulations

Conclusion