Heat Equation, General Case

Consider the following general boundary-value problem with vector coordinate $\mathbf{r}$ :

$$\begin{eqnarray*} \alpha \nabla ^{2}T+\frac{\alpha }{k}g(\mathbf{r},t)-\alpha m^... ...ac{\partial T}{\partial t}\right\vert _{\mathbf{r}_{i}};\;i=1,2. \end{eqnarray*}$$

The general boundary condition represents five different boundary conditions (type 1 through 5) by suitable choice of boundary parameters $k_{i}=0$ or $k$; $% h_{i}=0$ or $h$; $(\rho cb)_{i}=0$ or nonzero. $$Here $(\rho cb)_{i}$ 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 $T(\mathbf{r},t)$ is given by:

$$\begin{eqnarray*} &&T(\mathbf{r},t)=\int_{R}G(\mathbf{r},t \left\vert \mathbf... ...{r}_{i}}\right] ds_{j}^{\prime } \; \mbox{(b.c. of type 1 only)} \end{eqnarray*}$$

This equation applies to any orthogonal coordinate system if the correct form for differential area $ds_{i}$ and differential volume $dv$ are used. See the table below for $ds_{i}$ and $dv$ for several body shapes in the rectangular, cylindrical, and spherical coordinate systems. Spatial derivative $\partial /\partial n_{i}$ denotes differentiation along the outward normal on surface $S_{i}$, $i=1,2,\ldots ,s$where $s$ represents the number of boundary conditions. The number of boundary conditions $s$ only includes conditions at ``real'' boundaries; the number of ``real'' boundaries does not include the boundary at $x\rightarrow \infty$ for a semi-infinite body, for example.

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

Body shape	coordinates	$ds_{i}^{\prime }$	$dv^{\prime }$
plate	$x$	$^{\ast }1$	$dx^{\prime }$
rectangle	$x,y$	$dx^{\prime }\mbox{ or }dy^{\prime }$	$dx^{\prime }dy^{\prime }$
parallelpiped	$x,y,z$	$\begin{array}{l} dx^{\prime }dy^{\prime }\mbox{ or }dx^{\prime }dz^{\prime } \\ \mbox{ or }dy^{\prime }dz^{\prime } \end{array}$	$dx^{\prime }dy^{\prime }dz^{\prime }$
infinite cylinder	$r$	$2\pi r^{\prime }$	$2\pi r^{\prime }dr^{\prime}$
thin shell	$\phi$	$^{\ast }\delta \mbox{ (shell thickness)}$	$\delta ad\phi ^{\prime }$
finite cylinder	$r,z$	$^{\ast }2\pi r_{i}dz^{\prime }\mbox{ or }2\pi r^{\prime }dr^{\prime }$	$2\pi r^{\prime }dr^{\prime }dz^{\prime }$
wedge	$r,\phi$	$dr^{\prime }\mbox{ or }r_{i}d\phi ^{\prime }$	$r^{\prime }dr^{\prime }d\phi ^{\prime }$
sphere	$r$	$^{\ast }4\pi r_{i}^{2}\mbox{ }$	$4\pi (r^{\prime })^{2}dr^{\prime }$
$\ast \mbox{no integral on }S_{i}$