Green's Function for Heat Conduction in a Semi-infinite Medium X50 with a Type 5 Boundary Condition

By Donald E. Amos

Sandia National Laboratories, Retired


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The Green's function is the principal tool in construction of the general solution to the classical heat conduction problem. The solution is presented in terms of the internal heat generation, initial temperature and integrals which reflect the physical influence of the boundary. In the current literature ( ) the common boundary conditions are presented as Types 1,2,3,4, and 5 ranging from a specified temperature (Type 1) to the most general form (Type 5) where input energy (flux), heat loss to the surroundings, heat storage on a boundary layer and conduction into the material are considered. Since the driving energy for the Green's function is internal, the homogeneous form of the boundary condition is used to define the Green's function. The thrust of this work is to derive the Green's function, labeled X50, for a semi-infinite slab with Type 5 boundary condition at x=0.

In a special case where there is no loss or gain of energy to or from the surroundings (Type 4 condition), a closed form is derived in terms of the co-error function. This Green's function is applied to a system where the initial boundary temperature is higher than the connected medium. The medium acts as a heat sink and the solution tracks the cooling of the system.


This work was supported by NSF Award 1250625, Exact Analytical Conduction Toolbox, administered by the University of Nebraska, Kevin Cole, Director


  1. Cole, KD, Beck, JV, et. al. (2010), Heat Conduction Using Green's Functions, 2nd Ed., CRC Press Boca Raton, 643pp
  2. Amos, DE (2014) Theory of Heat Conduction with a Type 5 Boundary Condition,
  3. Amos, DE, (2014) Green's Function for Heat Conduction in a Slab X55 with Type 5 Boundary Conditions,
  4. Amos, DE, (2014) Green's Function for Heat Conduction in a Hollow Cylinder R55 with Type 5 Boundary Conditions,
  5. Amos, DE,(2012) Handbook of Integrals related to Heat Conduction and Diffusion,

Cite this work

Researchers should cite this work as follows:

  • Donald E. Amos (2014), "Green's Function for Heat Conduction in a Semi-infinite Medium X50 with a Type 5 Boundary Condition,"

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  1. Type5 BC
  2. Semi-infite Slab
  3. heat conduction heat transfer
  4. Green\'s function