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Journal of Applied Mathematics
Volume 2014, Article ID 391606, 6 pages
http://dx.doi.org/10.1155/2014/391606
Research Article

Approximate Analytic Solutions for the Two-Phase Stefan Problem Using the Adomian Decomposition Method

1College of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi 030024, China
2College of Computer Science and Technology, Taiyuan University of Technology, Taiyuan, Shanxi 030024, China
3Department of Mathematics, Xin Zhou Teachers University, Xinzhou, Shanxi 034000, China

Received 22 January 2014; Accepted 3 June 2014; Published 18 June 2014

Academic Editor: Abdel-Maksoud A. Soliman

Copyright © 2014 Xiao-Ying Qin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

An Adomian decomposition method (ADM) is applied to solve a two-phase Stefan problem that describes the pure metal solidification process. In contrast to traditional analytical methods, ADM avoids complex mathematical derivations and does not require coordinate transformation for elimination of the unknown moving boundary. Based on polynomial approximations for some known and unknown boundary functions, approximate analytic solutions for the model with undetermined coefficients are obtained using ADM. Substitution of these expressions into other equations and boundary conditions of the model generates some function identities with the undetermined coefficients. By determining these coefficients, approximate analytic solutions for the model are obtained. A concrete example of the solution shows that this method can easily be implemented in MATLAB and has a fast convergence rate. This is an efficient method for finding approximate analytic solutions for the Stefan and the inverse Stefan problems.