Environmental Engineering Reference
In-Depth Information
Appendix A
Special Functions
In solving PDS, separation of variables sometimes leads to some special kinds of
linear ordinary differential equations. Two typical examples are the Bessel equation
and the Legendre equation. Particular solutions of these equations are called special
functions . They play the same role as the orthogonal set of trigonometric functions
in Fourier series and serve as the function bases for expanding solutions of PDS.
The series of function terms so obtained is a generalized Fourier series.
A.1 Bessel and Legendre Equations
Consider a mixed problem of two-dimensional heat-conduction equations
a 2
x 2
y 2
R 2
=
,
+
<
,
<
,
u t
Δ
u
0
t
(
,
,
)= ϕ (
,
) ,
u
x
y
0
x
y
u
| x 2 + y 2 = R 2
=
0
.
Let u
=
U
(
x
,
y
)
T
(
t
)
. Substituting it into the equation yields, with
λ
as the separa-
tion constant,
Δ
U
+ λ
U
=
0
,
U
| x 2
=
0
.
(A.1)
y 2
R 2
+
=
T + λ
T
=
0
.
(A.2)
If
λ =
0, U
(
x
,
y
)
0. Therefore
λ =
0byEq.(A.1).ByEq.(A.2),wehave T
(
t
)=
c e λ t .Since T
0. The equation in (A.1) is called
the Helmholtz equation . In a polar coordinate system, Eq. (A.1) reads
(
t
)
must be bounded, we obtain
λ >
2 U
2 U
∂θ
1
r
U
r 2
1
r 2 +
r +
2 + λ
U
=
0
,
0
<
r
<
R
,
(A.3)
U
| r = R =
0
,
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