Vol. 2, No. 4, 2009

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Newton's law of heating and the heat equation

Mark Gockenbach and Kristin Schmidtke

Vol. 2 (2009), No. 4, 419–437
Abstract

Newton’s law of heating models the average temperature in an object by a simple ordinary differential equation, while the heat equation is a partial differential equation that models the temperature as a function of both space and time. A series solution of the heat equation, in the case of a spherical body, shows that Newton’s law gives an accurate approximation to the average temperature if the body is not too large and it conducts heat much faster than it gains heat from the surroundings. Finite element simulation confirms and extends the analysis.

Keywords
heat equation, Newton's law of heating, finite elements, Bessel functions
Mathematical Subject Classification 2000
Primary: 35K05
Milestones
Received: 25 November 2008
Revised: 15 June 2009
Accepted: 13 July 2009
Published: 28 October 2009

Communicated by Suzanne Lenhart
Authors
Mark Gockenbach
Department of Mathematical Sciences
Michigan Technological University
1400 Townsend Drive
Houghton, MI 49931-1295
United States
Kristin Schmidtke
Department of Mathematical Sciences
Michigan Technological University
1400 Townsend Drive
Houghton, MI 49931-1295
United States