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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