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A STUDY OF SIMPLIFIED SHALLOW WATER WAVES: ASSESSMENT OF ADOMIAN’S DECOMPOSITION METHOD
FOR THE ANALYTICAL SOLUTION
Mehdi Safari
A Study of Simplified Shallow Water Waves: Assessment of Adomian’s Decomposition Method for the
Analytical Solution
Mehdi Safari, ms_safari2005@yahoo.com
Department of Mechanical Engineering, Aligoodarz Branch, Islamic Azad University, P. O. Box 159, Aligoodarz, Iran.
Corresponding author: Tel/Fax: +98 861 3672399
ABSTRACT
In this paper, we consider two model equations for shallow water waves. Shallow water waves were introduced as a model equation which reduces to
the KdV equation in the long small amplitude limit. Large classes of linear and nonlinear differential equations, both ordinary as well as partial, can be
solved by the ADM.The decomposition method provides an effective procedure for analytical solution of a wide and general class of dynamical
systems representing real physical problems.This method efficiently works for initial- value or boundary-value problems and for linear or nonlinear,
ordinary or partial differential equations and even for stochastic systems. Moreover, we have the advantage of a single global method for solving
ordinary or partial differential equations as well as many types of other equations. We use Adomian’s decomposition method (ADM) to solve them.
The results show that Adomian's decomposition method is a powerful method for solving these equations and the obtained solutions are shown
graphically.
Keywords: Adomian’s decomposition method; Shallow water wave equation
INTRODUCTION
Clarkson et.al (Clarkson, Mansfield, 1994) investigated the generalized
short water wave (GSWW) equation
u
t
uxxt
uut
ux utdx
ux
0,
x
(1)
where and are non-zero constants.
Ablowitz et. al. (Ablowitz et al., 1974) studied the specific case
where Eq. (1) is reduced to
4 and 2
t
uxxt
4uut
2u
x utdx
ux
0,
x
u (2)
This equation was introduced as a model equation which reduces to the
KdV equation in the long small amplitude limit (Ablowitz et al., 1974,
Hirota, Satsuma, 1976). However, Hirota et.al. (Hirota, Satsuma, 1976)
examined the model equation for shallow water waves
t
uxxt
3uut
3u
x utdx
ux
0,
x
u (3)
obtained by substituting 3 in (1).
Equation (2) can be transformed to the bilinear forms
D
( D D D
D
1
) Dt
( D
3
D
)
f . f
0,
2
3
x t t x x
s x
(4)
where s is an auxiliary variable, and f satisfies the bilinear equation
D ( D
D
) f . f
0,
3
x s x
(5)
However, Eq.(3) can be transformed to the bilinear form
D ( D D D
D
) f . f
0,
2
x t t x x
(6)
and the solution of the equation is
u x,
t)
2(ln f ) ,
(7)
(
xx
where f(x, t) is given by the perturbation expansion
n1
n
f ( x,
t)
1
f ( x,
t),
(8)
n
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