B ψ
ð Þ ¼ 2ψ a
ð Þ À ψ b
ð Þ ¼ 0,
ð10:45Þ
B
0
φ
ð Þ ¼
1
2
φ a
ð Þ À φ b
ð Þ ¼ 0:
ð10:46Þ
The boundary functional B
0 (φ) is said to be adjoint to B(ψ). The two boundary
functionals are admittedly different. If, however, we set ψ(b) ¼ ψ(a), then we should
have φ(b) ¼ φ(a) for the surface term to vanish. That is
B ψ
ð Þ ¼ ψ a
ð Þ À ψ b
ð Þ ¼ 0,
ð10:47Þ
B
0
φ
ð Þ ¼ φ a
ð Þ À φ b
ð Þ ¼ 0:
ð10:48Þ
Thus, the two functionals are identical and ψ and φ satisfy homogeneous BCs with
respect to these functionals.
As discussed above, a FOLDE is characterized by its differential operator as well
as a BC (or boundary functional). This is similarly the case with SOLDEs as well.
Example 10.3 Next, let us consider a following differential operator:
L x ¼
1
i
d
dx
:
ð10:49Þ
As in the case of Example 10.2, we have
Z b
a
dxφ
à 1
i
d
dx
ψ
À
Z b
a
dx
1
i
d
dx
φ
! Ã
ψ ¼
1
i
φ
Ã
ψ
½
Š
b
a :
ð10:50Þ
Also rewriting (10.50) using an inner product notation, we get
φjL x ψ
h
iÀ L x φjψ
h
i¼
1
i
φ
Ã
ψ
½
Š
b
a ,
ð10:51Þ
Apart from the factor
1
i , RHS of (10.51) are again given by
φ
à b
ð Þψ b
ð Þ À φ
à a
ð Þψ a
ð Þ:
Repeating a discussion similar to Example 10.2, when the surface term vanishes, we
get
φjL x ψ
h
i¼ L x φjψ
h
i:
ð10:52Þ
Comparing (10.43) and (10.52), we have
388
10 Introductory Green’s Functions