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International Journal of Differential Equations
Volume 2012, Article ID 187902, 22 pages
Research Article

Application of Heat Balance Integral Methods to One-Dimensional Phase Change Problems

1MACSI, Department of Mathematics and Statistics, University of Limerick, Limerick, Ireland
2Centre de Recerca Matemàtica, Campus de Bellaterra, Edifici C, Bellaterra, 08193 Barcelona, Spain

Received 15 December 2011; Accepted 12 February 2012

Academic Editor: Ebrahim Momoniat

Copyright © 2012 S. L. Mitchell and T. G. Myers. 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.


We employ the Heat Balance Integral Method (HBIM) to solve a number of thermal and phase change problems which occur in a multitude of industrial contexts. As part of our analysis, we propose a new error measure for the HBIM that combines the least-squares error with a boundary immobilisation method. We describe how to determine this error for three basic thermal problems and show how it can be used to determine an optimal heat balance formulation. We then show how the HBIM, with the new error measure, may be used to approximate the solution of an aircraft deicing problem. Finally we apply the new method to two industrially important phase change problems.