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# Unit Impulse Response: General Homogeneous Boundary Conditions

From the viewpoint of technique, the Green's function most easily constructed is the one satisfying the separated boundary conditions. This Green's function is

It satisfies
 0 0

and

The graph of such a unit impulse response is depicted in Figure 4.4. Such a Green's function is obviously the simplest to construct: find any solution to the homogeneous problem for the left hand interval, then find any solution for the right hand interval, and for all intent and purposes one is done. The only remaining question is: What is the Green's function if the homogeneous boundary conditions are different?

The answer is illustrated by the following problem:

Given:
The above Green's function .
Find:
(a) The Green's function which satisfies the initial conditions''

and

(b) The Green's function, say , which is adjoint to .