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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