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Heat Equation, 1-D Rectangular Case.

Consider the following boundary-value problem for temperature in a 1-D body in rectangular coordinates:

\begin{eqnarray*}
\alpha \frac{\partial ^{2}T}{\partial x^{2}}+\frac{\alpha }{k}...
... n_{i}}\right\vert
_{x_{i}}+h_{i}T(x_{i},t) &=&f_{i}(t);\;i=1,2.
\end{eqnarray*}

Note that $n_{i}$ is the outward normal on boundary $i$. The convective (type 3) boundary conditions are specified on boundaries $i=1$ and $i=2$. Boundary conditions of type 1 or 2 are also included by this relationship by taking $k_{i}=0$ or $h_{i}=0$, respectively, on boundaries $i=1$ or $i=2$.

The Green's Function Solution Equation for temperature $T(x,t)$ is given by:

\begin{eqnarray*}
T(x,t) &=&\int_{x^{\prime }}G(x,t \left\vert  x^{\prime },0\...
...{x^{\prime }=x_{i}}%
\right] \; \mbox{(for b.c. of type 1 only)}
\end{eqnarray*}

The spatial integrals should be evaluated over the whole body, for example, on $(0<x^{\prime }<L)$ for a plate, or over $(0<x^{\prime }<\infty )$ for a semi-infinte body. The sumations in the boundary condition terms represent at most two boundaries.

The same GF appears in each integral term, evaluated at the source location $%
(x^{\prime },\tau )$ appropriate for that integral term. For example, in the initial-condition integral the GF is evaluated at $\tau =0$; in a boundary-condition integral the GF is evaluated at $x^{\prime }=x_{i}$,



2004-01-31