Approximate solution of the nonlinear heat conduction equation in a semi-infinite domain
type: Item
instance of: scholarly article
title: Approximate solution of the nonlinear heat conduction equation in a semi-infinite domain
zbMATH Open document ID: 1203.80026
DOI: 10.1155/2010/421657
author: Jun Yu
author: Yi Yang
author: Antonio Campo
published in: Mathematical Problems in Engineering
publication date: 2010-09-30T00:00:00Z
full work available at URL: https://eudml.org/doc/228563
review text: This paper develops an approximation method for solving a nonlinear heat conduction problem in a semi-infinite domain. Based on equivalent integral formulation of the considered problem, the partial differential equation is reduced to a set of first-order ordinary differential equations in time by means of the expansion of an energy density function (defined as the internal energy per unit volume) as a Taylor polynomial in a spatial domain. A systematic approach to derive approximate solutions using Taylor polynomials of a different degree is described. For a special case, an analytical solution is derived and compared with the result of a self-similar analysis. The comparison with both self-similar and numerical integrated solutions confirms the accuracy of the approximate results and shows the suitability and efficiency of the approximation method for more general and realistic cases since it assumes no restriction on the form of boundary energy density and heat conductivity. The consistent improvement of the quadratic approximation over the linear one indicates that the method can be systematically extended to achieve higher order of accuracy. Propagation of nonlinear heat waves is studied for different boundary energy density and the conductivity functions. The idea in the approximation method is quite general, as it can be applied to any nonlinear physical processes in which the solution exhibits wave front behavior. This makes the approximation methodology potentially useful in many application fields, including fluid dynamics, combustion, and environmental and material sciences.
Mathematics Subject Classification ID: 80M25
Mathematics Subject Classification ID: 35K05
Mathematics Subject Classification ID: 80A20
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 35Q79
Mathematics Subject Classification ID: 35K86
zbMATH DE Number: 5793747
zbMATH Keywords: heat transfer
zbMATH Keywords: nonlinear heat conduction
zbMATH Keywords: temperature
zbMATH Keywords: approximation method
zbMATH Keywords: Taylor polynomial
Wikidata QID: Q58653084
MaRDI profile type: Publication
OpenAlex ID: W2065997824
cites work: Effect of Radiation on Shock Wave Behavior
cites work: Self-similar expansion of dense matter due to heat transfer by nonlinear conduction
description: scientific article; zbMATH DE number 5793747