F x, y
ð Þ
u 2 x
ð Þu 1 y
ð Þ À u 1 x
ð Þu 2 y
ð Þ
a y
ð ÞW u 1 y
ð Þ, u 2 y
ð Þ
ð
Þ
,
ð10:173Þ
we have
G x, y
ð Þ ¼ F x, y
ð ÞΘ x, y
ð Þ:
ð10:174Þ
Notice that
Θ x, y
ð Þ 6 ¼ Θ y, x
ð Þ and G x, y
ð Þ 6 ¼ G y, x
ð Þ:
ð10:175Þ
It is therefore obvious that the differential operator is not Hermitian.
10.6.3 Estimation of Surface Terms
To include the surface term of the inhomogeneous case, we use (10.62).
Z s
r
dx v
à L x u
ð ÞÀ L x
{ v
Â
à à u
h
i
¼ av
à du
dx
À u
d av
Ã
ð Þ
dx
þ buv
Ã
! s
r
:
ð10:62Þ
As before, we set r ¼ 0 in (10.62). Also, we classify (10.62) into two cases according
as s > 0 or s < 0.
(i) Case I (x > 0, y > 0): Equation (10.62) reads as
Z 1
0
dx v
à L x u
ð ÞÀ L x
{ v
Â
à à u
h
i
¼ av
à du
dx
À u
d av
Ã
ð Þ
dx
þ buv
Ã
! 1
0
:
ð10:176Þ
This time, inserting
g x, y
ð Þ ¼ G
à y, x
ð Þ ¼ G y, x
ð Þ
ð10:177Þ
into v of (10.176) and arranging terms, we have
u y
ð Þ ¼
Z 1
0
dxG y, x
ð Þd x
ð Þ
À a x
ð ÞG y,x
ð Þ
du x
ð Þ
dx
À u x
ð Þ
da x
ð Þ
dx
G y, x
ð ÞÀu x
ð Þa x
ð Þ
∂G y,x
ð Þ
∂x
þ b x
ð Þu x
ð ÞG y,x
ð Þ
! 1
x¼0
:
ð10:178Þ
414
10 Introductory Green’s Functions