INTRODUCTION
Boundary value problems (BVPs) in nonlinear differential equations have
been one of the major problems in engineering sciences. In the past several
decades, both mathematicians and physicists have made significant progress
in this direction (He, 2000; AlHayani and CasasÃºs, 2005; Liu,
2004). Since these equations are nonlinear, thus do not have precise analytical
solutions. On the other hand, solving these nonlinear equations analytically
may guide authors to know the described some physical process deeply and
sometimes leads them to know some facts which are not simply understood
through common observations. As a result, these equations have to be solved
using other methods. Many different methods have been presented recently;
for instance, the homotopy analysis method (Khatami et al., 2008;
Abbasbandy, 2006), the Variational Iteration Method (VIM) (Khatami
et al., 2008; Ganji et al., 2006; Sadighi and Ganji, 2007;
He, 1999, 2002, 2007), modified variational method (Abassy et al.,
2007; Tari, 2007), the Adomian`s Decomposition Method (ADM) (Adomian,
1994; Wazwaz, 2007), Homotopy perturbation method (He, 2003, 2005, 2006;
Tolou et al., 2008; Ghasemi, 2007) and Expfunction method (Mahmoudi
et al., 2008).
This study deals with analytical solution of some BVPs nonlinear equation
by the means of modified variational iteration method. Three examples
are presented to assess the benefits of method. In the first example,
the steady flow of a second grade fluid in a porous channel is considered
and solved using MVIM and its results are compared with those of homotopy
analysis method (HAM) (Hayat et al., 2007). The last two examples
are nonlinear equations which are solved by MVIM and the results are compared
with the obtained results from the exact solutions from Liu and Wv (2002)
and Wazwaz (2001).
MATERIALS AND METHODS
Fundamental of He`s variational iteration method: To clarify the
basic ideas of VIM, we consider the following differential equation:
where, L is a linear operator, N is a nonlinear operator and g (t) an
inhomogeneous term. According to VIM, we can write down a correction functional
as follows:
where, Î» is a general Lagrangian multiplier which can be
identified optimally via the variational theory. The subscript n indicates
the nth approximation and
is considered as a restricted variation .
Implementation of MVIM
Example one: Consider the following nonlinear fourthorder
differential equation:
Subject to the boundary conditions:
Its correction variational functional can be expressed as follows:
After some manipulations, the following stationary conditions have been
obtained:
The Lagrangian multipliers can therefore be identified as:
And the variational iteration formula is obtained in the form:
We are to start with the initial approximation of Since,
no initial approximation of
is available, we make one in the form of a polynomial as:
This depends on the order of differentiation and a, b, c and d are unknown
constants to be later determined.
Using the above iteration formula (10), we can directly obtain other
components as:
For a special case, M = 2, Re = 0, Î± = 0.2 will be as follows:
Incorporating the boundary conditions, Eq. 4, into y_{1}(x),
we have:
Solving the above system of equations simultaneously, we obtain:
Therefore, we obtain the following firstâ€“order approximate solution
for special case, M = 2, Re = 0, Î± = 0.2:
In the same manner, the rest of the components of the iteration formula
can be obtained.
Example two: Now consider another nonlinear fourthorder BVP (Liu
and Wv, 2002):
Subject to the boundary conditions:
Its correction variational functional in can be expressed as:
After some computations, we obtain the following stationary conditions:
The Lagrangian multipliers can, therefore, be identified as:
And the variational iteration formula is obtained in the form of
Assuming initial approximation as following form:
where, a, b, c and d are unknown constants to be further determined.
Using the above iteration formula (27), we can directly obtain the other
components as follows:
Incorporating the boundary conditions, Eq. 21, into
y_{1}(x), we have:
Solving the above system of equations simultaneously, we obtain:
Therefore, we obtain the following firstorder approximate solution,
as follows:
The exact solution is y(x) = x^{4} (Liu and Wv, 2002).
Example three: Consider the nonlinear boundary value problem as
follows:
Subject to the following conditions:
Its correction variational functional can be expressed as:
After some computations, we obtain the following stationary conditions:
The Lagrangian multipliers can, therefore, be identified as:
And the variational iteration formula is obtained in the form of
Now, we assume that the initial approximation has the form of
where ,a, b, c and d are unknown constants to be further determined.
Using the above iteration formula (42), we can directly obtain the other
components as follows:
Incorporating the boundary conditions, Eq. 37, into
y_{1}(x) and solving the equations, we obtain:
Solving the above system of equations simultaneously, we obtain:
In the same manner, the rest of the components of the iteration formula
can be obtained as:
The exact solution for this problem have already been obtained by Wazwaz
(2001) as (1x)e^{x}.
RESULTS AND DISCUSSION
In this survey, some BVPs type nonlinear equations are solved by using MVIM.
Figure 1a and b show the y(x) obtained from MVIM and HAM versus
x respectively. In comparison to those of HAM (Fig. 1b) (Hayat
et al., 2007) this results illustrates high accuracy in different values
of M.

Fig. 1: 
y (x) values versus x obtained from (a) MVIM and (b)
HAM, in different values of M (Hayat et al., 2007) 

Fig. 2: 
Plot of the approximated y (x) versus x, this plot has
excellent agreement with reference (Hayat et al., 2007) 

Fig. 3: 
Plot of the approximated y (x) versus x, this plot has
excellent agreement with reference (Hayat et al., 2007) 
Figure 2 and 3 show the results of
y (x) and versus x for different values of Re that are in good agreement with those of
HAM (Hayat et al., 2007). The results of example two from proposed solution
and exact solution (Liu and Wv, 2002) are prorated in Fig. 4 which are is in excellent agreement. Finally the results of example three from
MVIM and exact solution (Wazwaz, 2001) shown in Fig. 5 which
are in good agreement. The result shown in Fig. 15 indicates that the MVIM experiences a high accuracy. In addition, in comparison
with conventional method, a considerable reduction of the volume of the calculation
can be seen in MVIM.

Fig. 4: 
The comparison between the Exact and MVIM solutions
for second example 

Fig. 5: 
The comparison between the Exact and MVIM solutions
for third example 
CONCLUSION
In this research, we studied the application of the modified variation
iteration method (MVIM) to nonlinear and linear integral equations and
the Blasius problem. It was founded that the approximations obtained by
MVIM are valid when compared with the exact solutions.
The Fig. 15 clearly show that the
results by MVIM are in excellent agreement with the exact solutions. MVIM
provides highly accurate numerical solutions in comparison with other
methods and it is expected here that this powerful mathematical tool can
solve a large class of nonlinear differential systems, especially nonlinear
integral systems and equations used in engineering and physics.