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

$$G(x;\xi)=\frac{-1}{c} u_1(x_<) u_2(x_>)~~.$$

Figure 4.4: Response to a unit impulse applied at $x=\xi$ .

It satisfies

$$\alpha G(a;\xi)+\alpha ' G'(a;\xi) =$$ 0

$$\beta G(b;\xi) +\beta ' G'(b;\xi) =$$ 0

and

$$LG(x;\xi) = -\delta(x-\xi)~~, ~~~\textrm{where}~~ L=\frac{d}{dx}p(x)\frac{d}{dx} +\gamma (x)~~.$$

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?

Figure 4.5: Unit impulse response of a system satisfying Dirichlet and Neumann conditions at $x=a$ . Boundary conditions imposed at the same point are called initial conditions. If these two conditions are imposed at the starting point (and $x$ is ``time''), then the response is called causal or retarded. If the two conditions were imposed at a later point, then the response would be called acausal or advanced.

The answer is illustrated by the following problem:

Given:

The above Green's function $G(x;\xi )$ .

Find:

(a) The Green's function $G^R(x;\xi)$ which satisfies the ``initial conditions''

$$G^R(a;\xi)=0$$

and

$$\left.\frac{dG^R(x;\xi)}{dx} \right\vert^{x=a} =0$$

(b) The Green's function, say $G^A$ , which is adjoint to $G^R$ .

(c) The adjoint boundary conditions.

(d) A qualitative graph of $G^A$ .

Solution: Use $L(G^R -G)=0$ and the theorem of Section 4.2.

Remark: If $x$ is the time, then $G^R$ would be the so-called causal or retarded Green's funtion depicted in Figure 4.5, while $G^A$ would be the so-called acausal or advanced Green's function.

The general philosophy is this: If it is too difficult to find the Green's function for a desired set of boundary conditions, consider alternative boundary conditions for which the Green's function can readily be found. The desired Green's function is obtained by adding that solution to the homogeneous differential equation which guarantees that the desired boundary conditions are fullfilled.