On Monday July 6th, the nanoHUB will be intermittently unavailable due to scheduled maintenance. We apologize for any inconvenience this may cause. close


Support Options

Submit a Support Ticket


1-D Green's Functions For Heat Conduction Between Semi-infinite Slabs With Perfect and Imperfect Boundary Contact

By Donald E. Amos

Sandia National Laboratories, Retired



Published on


This document presents two derivations for 1-D Green's functions for semi-infinite slabs in contact along the boundary x=0. The case of imperfect contact with a heat transfer coefficient h is derived and the case of perfect contact is obtained by taking h to infinity. The two dimensional case with source point (x',y') is reduced to the one dimensional case by applying a constant source in the y' direction. Because the two-dimensional source solutions have complex representations, we get 1-D complex representations also. However, these complex, 1-D forms can also be reduced to all real, closed forms which agree with a direct attack using the 1-D equations.

The case of perfect insulation on x=0 is also computed by taking h to zero. The result is 'the method of images' solution in the source region and zero in the other region.

The convolution for a continuous (constant) source in time is carried out to produce a point source solution with continuous heat generation.



[1] Abramowitz S, Stegun IA (1965) Handbook of Mathematical Functions, AMS 55, Dover Publications Inc., New York, 1046pp

[2] Amos DE (2012) Handbook of Integrals Related to Heat Conduction and Diffusion, http://nanohub.org/resources/13874

[3] Amos DE, Beck JV, de Monte F (2011) Transient Heat Conduction in Adjacent Quadrants Separated by a Thermal Resistance, http://nanohub.org/resources/12465

[4] Amos DE (2012) Transient Heat Conduction in Adjacent Materials Heated on Part of the Common Boundary, http://nanohub.org/resources/12390

[5] Amos, DE (2012), Green's Functions For Heat Conduction in Adjacent Materials, http://nanohub.org/resources/12856

[6] Amos, DE (2012), Theory of Heat Conduction for Two-region Problems Using Green's Functions, http://nanohub.org/resources/13671

[7] Cole DC, Beck JV, Haji-Sheikh A, Litkouhi B (2010) Heat Conduction Using Green's Functions, 2nd Ed., CRC Press, 643p.

[8] Carslaw HS, Jaeger JC (1948) Conduction of Heat in Solids, Oxford Univ Press, London, 386pp

Cite this work

Researchers should cite this work as follows:

  • Donald E. Amos (2013), "1-D Green's Functions For Heat Conduction Between Semi-infinite Slabs With Perfect and Imperfect Boundary Contact," http://nanohub.org/resources/15237.

    BibTex | EndNote


nanoHUB.org, a resource for nanoscience and nanotechnology, is supported by the National Science Foundation and other funding agencies. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.