Mathematical Problems in Engineering
Volume 2010 (2010), Article ID 421657, 24 pages
Research Article

Approximate Solution of the Nonlinear Heat Conduction Equation in a Semi-Infinite Domain

1Department of Mathematics and Statistics, University of Vermont, Burlington, VT 05401, USA
2Department of Mechanical Engineering, The University of Texas at San Antonio, San Antonio, TX 78239, USA

Received 31 December 2009; Revised 28 June 2010; Accepted 30 June 2010

Academic Editor: Mehrdad Massoudi

Copyright © 2010 Jun Yu 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.


We use an approximation method to study the solution to a nonlinear heat conduction equation in a semi-infinite domain. By expanding an energy density function (defined as the internal energy per unit volume) as a Taylor polynomial in a spatial domain, we reduce the partial differential equation to a set of first-order ordinary differential equations in time. We describe a systematic approach to derive approximate solutions using Taylor polynomials of a different degree. For a special case, we derive an analytical solution and compare it with the result of a self-similar analysis. A comparison with the numerically integrated results demonstrates good accuracy of our approximate solutions. We also show that our approximation method can be applied to cases where boundary energy density and the corresponding effective conductivity are more general than those that are suitable for the self-similar method. Propagation of nonlinear heat waves is studied for different boundary energy density and the conductivity functions.