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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:

Note that is the outward normal on boundary . The convective (type 3) boundary conditions are specified on boundaries and . Boundary conditions of type 1 or 2 are also included by this relationship by taking or , respectively, on boundaries or .

The Green's Function Solution Equation for temperature is given by:

The spatial integrals should be evaluated over the whole body, for example, on for a plate, or over 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 appropriate for that integral term. For example, in the initial-condition integral the GF is evaluated at ; in a boundary-condition integral the GF is evaluated at ,

2004-01-31