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## Heat Equation, General Case

Consider the following general boundary-value problem with vector coordinate : The general boundary condition represents five different boundary conditions (type 1 through 5) by suitable choice of boundary parameters or ; or ; or nonzero. Here represents properties of a high conductivity surface film (density, specific heat, thickness) which is thin enough that there is a negligible temperature gradient across the film and negligible heat flux parallel to the surface inside the film.

The Green's Function Solution Equation for temperature is given by: This equation applies to any orthogonal coordinate system if the correct form for differential area and differential volume are used. See the table below for and for several body shapes in the rectangular, cylindrical, and spherical coordinate systems. Spatial derivative denotes differentiation along the outward normal on surface , where represents the number of boundary conditions. The number of boundary conditions only includes conditions at real'' boundaries; the number of real'' boundaries does not include the boundary at for a semi-infinite body, for example.

Table 1. Differential area and volume for the GF Solution Equation.

 Body shape coordinates  plate   rectangle   parallelpiped   infinite cylinder   thin shell   finite cylinder   wedge   sphere       Next: Laplace and Helmholtz Equation Up: HEAT EQUATION (TRANSIENT CONDUCTION) Previous: Heat Equation, 1-D Rectangular
2004-01-31