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A STUDY OF SIMPLIFIED SHALLOW WATER WAVES: ASSESSMENT OF ADOMIAN’S DECOMPOSITION METHOD
FOR THE ANALYTICAL SOLUTION
Mehdi Safari
The operating with the operator
1
0
(()()())
1
L on both sides of Eq. (18) we have
u f L g t R u F u , (20)
where f
0
is the solution of homogeneous equation
Lu 0 , (21)
involving the constants of integration. The integration constants
involved in the solution of homogeneous equation ( 21) are to be
determined by the initial or boundary condition according as the
problem is initial-value problem or boundary-value problem.
The ADM assumes that the unknown function u ( x ,) t can be
expressed by an infinite series of the form
u ( x ,)( t ,) u
n
x t , (22)
n 0
and the nonlinear operator F () u can be decomposed by an infinite
series of polynomials given by
F () u An
, (23)
n 0
where u
n
( x ,) t will be determined recurrently, and An
are the socalled
polynomials of u
0, u1,..., u
n
defined by
n
1 d
An F () , u 0,1, n2...
n
n ! d
i
i
n 0
0
(24)
ADM IMPLEMENT FOR FIRST MODEL OF SHALLOW
WATER WAVE EQUATION
We first consider the application of ADM to first model of shallow
water wave equation. If Eq. (2) is dealt with this method, it is formed as
L u L u 4uL u 2L u L udx L u,
(25)
t
where
xxt
t
3
L t
, L x
, L xxt
,
2
t x x
t
If the invertible operator
L
1
t
x
t
x
t
dt
0
x
(26)
is applied to Eq. 25, then
L L u L
1
t
t
1
t
( L
is obtained. By this
xxt
u(
x,
t)
u(
x,0)
L
u 4uL u 2L u
1
t
( L
xxt
t
x
x
u 4uL u 2L u
t
x
L udx L u),
x
t
x
L udx L u),
t
x
(27)
(28)
is found. Here the main point is that the solution of the decomposition
method is in the form of
u ( x,
t)
un
( x,
t)
, (29)
n0
Substituting from Eq. 29 in 28, we find
n0
L ( , ) 4 ( , ) ( , )
1
0
0
0
( , ) ( ,0)
xxt
un
x t
un
x t Lt
un
x t
n
n
n
u
n
x t u x Lt
, (30)
x
2 ( , )
( , )
( , )
Lx
un
x t
Lt
un
x t dx
Lx
un
x t
n0
n0
n0
is found.
According to Eq.19 approximate solution can be obtained as follows:
2 1 c 1
( c 1)sech
x
2 c
u0
( x,
t)
,
2c
1 c 1
c 1
( c 1)sinh
x
2
t
1(
, )
c c
x t
,
3 1 c 1
2c
cosh
x
2
c
(31)
u (32)
t
(33)
u2( x,
t)
( Lxxtu1
4u1Lt
u1
2Lxu1
Lt
u1dx
Lxu1
) dt,
0
Thus the approximate solution for first model of shallow water wave
equation is obtained as
u x,
t)
u ( x,
t)
u ( x,
t)
u ( x,
) , (34)
(
0 1
2
t
The terms u0 ( x,
t),
u1(
x,
t),
u2
( x,
t)
in Eq.34, obtained from
Eqs.31, 32, 33. In Fig.1 the first model of shallow water wave equation
with the first initial condition (31) of Eq. (2) when c=2 has been shown.
ADM IMPLEMENT FOR SECOND MODEL OF SHALLOW
WATER WAVE EQUATION
Now we consider the application of ADM to second model of shallow
water wave equation. If Eq. (3) is dealt with this method, it is formed as
x
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