There are several difficulties in constructing a Greens function for a given differential equation. First of all, the inverse does not necessarily exist. This shortcoming can be redeemed in many cases by some transformation. Even if the inverse exists, its construction is usually more difficult than finding solutions by using a properly chosen conventional trick. However, if the inverse exists, then the problem is equivalent to the task of finding or constructing the kernel, the Green's function of the integral equation (See e.g. G(x, t) in eq. [2.14]). There is more than one way to construct the Green's function associated with a differential equation. Each method may have corresponding physical meaning. Here we presented the one which is the easiest and the most revealing in terms of using the adjointness we have already introduced in our paper. Here we just further refine the basic concepts.
Definition: L* differential operator is formally adjoint to L if L and L* are associated with the following equation
∫ f*(x) Lg(x) dx = [...] + ∫ f(x) L*g(x) dx (A.1)
Definition: L is formally self-adjoint if L = L*.
Definition: L* differential operator is adjoint to L if the associated differential equations of L has
homogeneous boundary conditions, that if the eq. (A.1) takes the simple form
∫ f*(x) Lg(x) dx = ∫ f(x) L*g(x) dx. (A.2)
or using the inner-product notation
(f, g) = ∫ f(x)*g(x)dx (A.3)
eq. (A.2) takes the form
(Lf,g) = (f, L*g). (A.4)
The key step to achieve the adjointness is to recognize the importance of elimination of the
boundary terms in eq. (A.1). The very same idea leads us to the Green's function ( Greenberg, 1971; pp. 22-26). If we find a G function for a given g, for which
L*(G) = δ(x′-x) (A.5)
G(a, x) = G(x′, b) = 0 (A.6)