Green's Functions for Diffusion

The Green's function is the response to a delta function source with homogeneous boundary conditions. The delta function models a source that is instantaneously pulsed in time and infinitely concentrated in space. The differential equation governing the one-dimensional diffusion Green's function is
The delta source is applied at time at location . The general homogeneous boundary conditions are
For an insulated boundary is obtained. In the limit as , a zero boundary is recovered. Causality in time requires that no response is felt prior to application of a source. Therefore, the initial condition is
for .
The Green's function represents the most basic and fundamental response to any system of differential equations. It can be used to construct the solution to any linear problem subject to arbitrary volumetric sources, boundary conditions, and initial conditions by integrating the Green's function over the appropriate times and locations.


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The solution for the Green's function on a finite domain with general boundary conditions is constructed from solutions for an infinite domain using the method of images. In order to see the influence of boundary conditions, place the source point near a boundary and adjust the boundary parameter, .
Snapshot 1: the source point is inside the region, far away from boundaries
Snapshot 2: the source point is near an insulated boundary ()
Snapshot 3: the source point is near a zero potential boundary ( large)
= Green's function (1/)
= position (m)
= time (s)
= source location (m)
= source time (s)
= diffusivity (/s)
, = boundary coefficients (1/m)
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