a x
ð Þw x
ð Þ
d
2 u
dx 2 þ b x
ð Þw x
ð Þ
du
dx
þ c x
ð Þw x
ð Þu
¼
d
dx
p x
ð Þ
du
dx
!
þ cw x
ð Þu:
Rewriting this, we have
L x u ¼
1
w x
ð Þ
d
dx
p x
ð Þ
du
dx
!
þ cu w x
ð Þ > 0
½
Š ,
ð10:70Þ
where we have
p x
ð Þ ¼ a x
ð Þw x
ð Þ and
dp x
ð Þ
dx
¼ b x
ð Þw x
ð Þ:
ð10:71Þ
The latter equation of (10.71) corresponds to (10.63) if we assume w(x) 1. When
the differential operator L x is defined as (10.70), L x is said to be self-adjoint with
respect to a weight function of w(x).
Now we examine boundary functionals. The homogeneous adjoint boundary
functionals are described as follows:
B
{
1 u
ð Þ ¼ α 1 v
à r
ð Þ þ β 1
dv
Ã
dx
x¼r þ γ 1 v
à s
ð Þ þ δ 1
dv
Ã
dx
x¼s
¼ 0,
ð10:72Þ
B
{
2 u
ð Þ ¼ α 2 v
à r
ð Þ þ β 2
dv
Ã
dx
x¼r þ γ 2 v
à s
ð Þ þ δ 2
dv
Ã
dx
x¼s
¼ 0:
ð10:73Þ
In (10.3) putting α 1 ¼ 1 and β 1 ¼ γ 1 ¼ δ 1 ¼ 0, we have
B 1 u
ð Þ ¼ u r
ð Þ ¼ σ 1 :
ð10:74Þ
Also putting γ 2 ¼ 1 and α 2 ¼ β 2 ¼ δ 2 ¼ 0, we have
B 2 u
ð Þ ¼ u s
ð Þ ¼ σ 2 :
ð10:75Þ
Further putting
σ 1 ¼ σ 2 ¼ 0,
ð10:76Þ
we also get homogeneous BCs of
B 1 u
ð Þ ¼ B 2 u
ð Þ ¼ 0; i:e:, u r
ð Þ ¼ u s
ð Þ ¼ 0:
ð10:77Þ
For RHS of (10.69) to vanish, it suffices to define B
{
1 u
ð Þ and B
{
2 u
ð Þ such that
10.3 Second-Order Differential Operators
393