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A STUDY OF SIMPLIFIED SHALLOW WATER WAVES: ASSESSMENT OF ADOMIAN’S DECOMPOSITION METHOD
FOR THE ANALYTICAL SOLUTION
Mehdi Safari
L u L u 3uL u 3L u L udx L u,
(35)
t
where
xxt
t
3
L t
, L x
, L xxt
,
2
t x x
t
x
x
t
x
(36)
The terms u0 ( x,
t),
u1(
x,
t),
u2
( x,
t)
in Eq.44, obtained from
Eqs.41, 42, 43. If we assume c=2 then by drawing 3-D figures of ADM
solutions. In Fig.2 the second model of shallow water wave equation
with the first initial condition (31) of Eq. (2) when c=2 has been shown.
Fig.1. For the first model of shallow water wave equation with the first
initial condition (31) of Eq. (2), ADM result for u ( x,
t)
, when c=2.
If the invertible operator
L L u L
1
t
t
1
t
( L
xxt
L
1
t
t
dt
u 3uL u 3L u
t
0
x
is applied to Eq. 45, then
x
L udx L u),
t
x
(37)
is obtained. By this
u(
x,
t)
u(
x,0)
L
1
t
( L
xxt
u 3uL u 3L u
t
x
x
L udx L u),
t
x
(38)
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)
, (39)
n0
Substituting from Eq. 49 in 48, we find
Fig.2. For the second model of shallow water wave equation with the
first initial condition (31) of Eq. (3), ADM result for u ( x,
t)
, when
c=2.
n0
L ( , ) 3 ( , ) ( , )
1
0
0
0
( , ) ( ,0)
xxtun
x t un
x t Lt
un
x t
n
n
n
u
n
x t u x Lt
, (40)
x
3 ( , ) ( , )
( , )
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
2ccosh
x
2
c
(41)
u (42)
t
(43)
u2( x,
t)
( Lxxtu1
3u1L tu1
3Lxu1
Lt
u1dx
Lxu1
) dt,
0
Thus the approximate solution for second model of shallow water wave
equation is obtained as
u( x,
t)
u0 ( x,
t)
u1(
x,
t)
u2
( x,
t)
, (44)
x
CONCLUSION
In this paper, Adomian’s decomposition method has been successfully
applied to find the solution of two model equations for shallow water
waves. The obtained results were showed graphically it is proved that
Adomian's decomposition method is a powerful method for solving
these equations. In our work; we used the Maple Package to calculate
the functions obtained from the Adomian’s decomposition method.
AcademyPublish.org – Journal of Engineering and Technology Vol.2, No.2 19