Environmental Engineering Reference
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find exact solutions for some special cases. We apply the perturbation method here
mainly for one-dimensional problems. A similar approach can be followed for two-
and three-dimensional problems.
6.9.1 Introduction
The perturbation method aims to find approximate solutions of mathematical prob-
lems involving a small parameter
; for example: algebraic equations, initial-value
problems of ODE and PDE. Let
P
ε
be the problem involving such a small parameter
ε
ε
0.
Once the exact solution of
P
0
is available, we can obtain an approximate solution
of
P
ε
by expanding the solution of
P
ε
and
P
0
be the corresponding problem when
ε
=
in terms of power series of
ε
and keeping the
0
term in the expansion is the exact solution of
P
0
.
Therefore, the perturbation method is a method of obtaining an approximate so-
lution of
P
ε
that is based on the exact solution of
P
0
and corrected by a few terms of
the power function of
first few terms of the series. The
ε
ε
.
In the case
B
2
=
ε
, consider the one-dimensional version of PDS (6.164)
:
u
t
/
τ
0
+
A
2
R
1
u
tt
=
Δ
u
+
ε
u
txx
,
×
(
0
,
+
∞
)
,
P
ε
(6.167)
u
(
x
,
0
)=
0
,
u
t
(
x
,
0
)=
ψ
(
x
)
,
0
<
ε
1
.
The solution of
P
0
is available in Section 5.2,
I
0
b
2
τ
0
x
+
At
x
1
t
2
A
e
−
2
u
(
x
,
t
)=
2
(
At
)
−
(
ξ
−
x
)
ψ
(
ξ
)
d
ξ
−
At
I
0
b
u
2
τ
0
At
−
1
t
2
A
e
−
2
=
2
(
At
)
−
ψ
(
u
+
x
)
d
u
,
(6.168)
At
where
b
0
.
The perturbation method for
P
ε
=
1
/
2
A
τ
is to correct the
u
(
x
,
t
)
in Eq. (6.168) by a poly-
nomial of power terms of
to obtain an approximate analytical solution of (6.167).
We focus our discussion only on regular perturbation. When
ε
is a polynomial
of
x
, in particular, the perturbation method can lead to the exact solution of
P
ε
.Note
that elementary functions can normally be approximated by Taylor polynomials. It
is thus very useful to discuss solutions of PDS (6.167) with a polynomial
ψ
(
x
)
ψ
(
x
)
,i.e.
N
n
=
0
a
n
x
n
ψ
(
x
)=
P
N
(
x
)=
.
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