Environmental Engineering Reference
In-Depth Information
Equation (D.2) is called the
regular S-L equation
if conditions (D.3) and (D.4)
are satisfied. The problem of finding its nontrivial solutions under boundary con-
ditions (D.5) and (D.6) is called the
regular S-L problem
. It can be shown that the
problem has solutions only for certain values of parameter
.The
λ
k
are called the
eigenvalues
. The corresponding solutions are called the
eigenfunc-
tions
of the problem.
λ
=
λ
k
,
k
=
1
,
2
, ···
D.2 The Lagrange Equality and Self-Conjugate
Boundary-Value Problems
C
2
(
)
(
)
∈
[
,
]
Suppose that
u
x
,
v
x
a
b
. Integration by parts leads to
b
b
−
vqu
d
x
pu
)
+
vLu
d
x
=
v
(
a
a
b
)
v
pu
+
vqu
d
x
b
a
+
u
(
=
−
v
(
x
)
p
(
x
)
x
a
b
)
)
u
(
b
a
+
v
(
=
−
p
(
x
x
)
v
(
x
)
−
u
(
x
)
x
uLv
d
x
.
a
Thus
b
a
(
)
u
(
)
b
a
.
v
(
vLu
−
uLv
)
d
x
=
−
p
(
x
x
)
v
(
x
)
−
u
(
x
)
x
(D.7)
This is called the
Lagrange equality
.
If
u
(
x
)
and
v
(
x
)
satisfy boundary conditions (D.5) and (D.6), when
α
2
=
0and
β
2
=
0 the right-hand side of Eq. (D.7) becomes
)
u
(
)
+
)
u
(
)
v
(
v
(
−
p
(
b
b
)
v
(
b
)
−
u
(
b
)
b
p
(
a
a
)
v
(
a
)
−
u
(
a
)
a
−
β
1
)+
β
1
=
−
(
)
(
)
(
(
)
(
)
p
b
β
2
u
b
v
b
β
2
u
b
v
b
−
α
1
)+
α
1
+
p
(
a
)
α
2
u
(
a
)
v
(
a
α
2
u
(
a
)
v
(
a
)
=
0
.
Thus
b
a
(
vLu
−
uLv
)
d
x
=
0
.
(D.8)
This is called the
self-conjugate relation
. It can be shown that Eq. (D.8) is also valid
when one or all of
β
2
are vanished. A boundary-value problem is called the
self-conjugate boundary-value problem
if it satisfies the self-conjugate relation.
α
2
and
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