In this paper, He’s variational iteration method is successfully employed to solve a nonlinear boundary value problem arising in the study of thin film flow of a third grade fluid down an inclined plane. For comparison, the same problem is solved by the Adomian decomposition method. The results show that the difference between the two solutions is negligible. The conclusion is that this technique may be considered an alternative and efficient method for finding approximate solutions of both linear and nonlinear boundary value problems. Furthermore, the variational iteration method has an advantage over the decomposition method in that it solves the nonlinear problems without using the Adomian polynomials.
Thin Film Flow; Third Grade Fluid; Nonlinear Boundary Value Problem; Variational Iteration Method; Adomian Decomposition Method1. Introduction
Recently, many approximate analytical and numerical methods have been suggested for solving linear and nonlinear boundary value problems arising in different branches of science and engineering. It is not difficult to solve a linear problem because of the availability of high performance digital computers, but finding solutions of nonlinear problems is still not easy. It is well known that getting an exact analytic solution of a given nonlinear problem is often more difficult compared to getting a numerical solution, despite the availability of supercomputers and software packages such as Maple, Mathe matica, Matlab etc, which provide an easy way to perform high quality symbolic computations. However, results obtained by numerical methods may give discon tinuous points of a curve when plotted; besides that complete physical understanding of a nonlinear problem is also difficult. If a nonlinear problem contains some sort of singularity or has multiple solutions then this also adds to the numerical difficulties. Though numerical and analytical solution methods have their limitations, at the same time they have their own advantages too. Therefore, we cannot neglect either of the two approaches but usually it is pleasing to solve a nonlinear problem analytically. In the recent decades, many different analytic methods have been introduced to solve the nonlinear problems, such as the homotopy analysis method (HAM) [1], the homotopy perturbation method (HPM) [2,3], the variational iteration method (VIM) [4,5], the Adomian decomposition method (ADM) [6,7], optimal homotopy asymptotic method (OHAM) [8,9]. In this study, we have applied the VIM and the ADM to find the approximate solutions of nonlinear and inhomogeneous differential equation governing the thin film flow of a third grade fluid down an inclined plane, and have made a graphical comparison of the numerical results from these two methods. Very recently, Mustafa Inc and Ebru Cavlak [10] have provided a comparative study of ADM and VIM in solving a new coupled MKdV system of equations. These methods generate the solution in a convergent series with components that are elegantly computed. Furthermore, these analytic methods avoid the complexities provided by other pure numerical methods [11,12]. The results reveal that the proposed methods provide an effective mathematical tool to handle a large class of linear and nonlinear differential equations.
2. Governing Equation
The thin film flow of a third grade fluid down an inclined plane of inclination is governed by the following nonlinear boundary value problem [13]
Introducing the parameters
the problem in Equations (2.1) and (2.2), after omitting asterisks, takes the following form
where is the dynamic viscosity, g is the gravity, is the fluid density and β > 0 is the material constant of a third grade fluid. We note that Equation (2.4) is a second order nonlinear and inhomogeneous differential equation with two boundary conditions; therefore, it is a well-posed problem.
Through integration of Equation (2.4) we have
where is a constant of integration. Employing the second condition of (2.5) in Equation (2.6), we obtain = 1. Thus, the system (2.4)-(2.5) can be written as
It should be noted that for, Equation (2.4) corresponds to that of Newtonian fluid whose exact solution subjected to the boundary conditions (2.5) is given by
In what follows, we will obtain the approximate analytic solutions of the nonlinear system (2.7)-(2.8) by using the VIM and the ADM techniques.
3. Solution by Variational Iteration Method
To illustrate the basic idea of He’s VIM, we consider the following nonlinear functional equation [4,5]
where is a linear operator, a nonlinear operator and an inhomogeneous term. Ji-Huan He has modified the general Lagrange multiplier method into an iteration method, which is called correction functional, in the following way [10]
where is a Lagrange multiplier that can be identified optimally via the variational theory [10]. The subscript denotes the approximation and is considered to be restricted variation, that is, The solution of the linear problem can be achieved in a single iteration step due to the exact identification of the Lagrange multiplier. This method requires the Lagrange multiplier be first determined optimally. The successive approximations of the solution can be readily obtained by using this determined Lagrange multiplier and any selective function Consequently, the solution is given by s. For the convergence criteria and error estimates of the VIM we refer the reader to [12,14].
According to the VIM, we can construct a correction functional of Equation (2.7) as follows
with We start with the initial guess in the above iteration formula and obtain the following approximate solutions:
In the solution (3.7) the terms involving the powers of gives the contribution of the non-Newtonian fluid. It is worth noting that by setting in the above approximations, we recover the exact solution for the case of Newtonian fluid. Thus, the first approximation of the nonlinear system (2.7)-(2.8) obtained by the VIM is identical with the exact solution of the corresponding linear problem. This shows that the VIM can be equally applied to linear equations.
The effects of the non-Newtonian parameter on the velocity given in (3.7) are plotted in Figure 1. It is shown that as we decrease the non-Newtonian parameter the solution converges to the Newtonian case.
4. Solution by Adomian Decomposition Method
A detailed description of the ADM is given in [6,7]. Here, we convey only the basic steps as a reminder. Writing Equation (2.7) in operator form, we obtain
where ,.
Here, is the highest order derivative which is assumed to be easily invertible, represents the nonlinear term and is the source term. According to the ADM, the solution can be expanded into the infinite series
where the components are usually determined recursively. The nonlinear term can be decomposed into infinite polynomials given by
where are the so-called Adomian polynomials of defined by
or equivalently,
It is well known that these polynomials can be constructed for all classes of nonlinearity according to the algorithm set by Adomian [15].
The general algorithm of this decomposition method for the nonlinear system (2.7)-(2.8) yields the recurrence relation,
where C2 is a constant of integration and can be found from the boundary condition (2.8). The first few terms of the Adomian polynomials for this problem are given by:
From these above results, we obtain the following components
In this manner, the rest of the terms in the decomposition series can be calculated. Summing up, we write the solution in the decomposition series form
This, after inserting the values of and from (4.9)-(4.14), becomes
which is the same as that obtained by the VIM. As before, setting in (4.15) one can recover the exact solution for the Newtonian fluid.
5. Conclusion
In this study, we have illustrated how the VIM and the ADM can be employed to obtain the approximate analytical solution of a nonlinear boundary value problem arising in the study of non-Newtonian fluid mechanics. The comparison between the fourth iteration solution of the VIM and five terms of the ADM is given in Figure 2. In fact, for an excellent agreement is observed.
Therefore, these methods are very powerful and efficient techniques for solving different kinds of linear and nonlinear problems arising in various fields of science and engineering. However, the VIM has an advantage over the ADM in that it solves the nonlinear problems without using the Adomian polynomials. Also, the use of the Lagrange multiplier reduces the successive use of the integral operator and it may be considered as an added advantage of this technique over the decomposition method.
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is governed by the following nonlinear boundary value problem [
is the dynamic viscosity, g is the gravity, 
is the fluid density and β > 0 is the material constant of a third grade fluid. We note that Equation (2.4) is a second order nonlinear and inhomogeneous differential equation with two boundary conditions; therefore, it is a well-posed problem.
is a constant of integration. Employing the second condition of (2.5) in Equation (2.6), we obtain 
= 1. Thus, the system (2.4)-(2.5) can be written as
, Equation (2.4) corresponds to that of Newtonian fluid whose exact solution subjected to the boundary conditions (2.5) is given by
is a linear operator, 
a nonlinear operator and 
an inhomogeneous term. Ji-Huan He has modified the general Lagrange multiplier method into an iteration method, which is called correction functional, in the following way [
is a Lagrange multiplier that can be identified optimally via the variational theory [
denotes the 
approximation and 
is considered to be restricted variation, that is, 
The solution of the linear problem can be achieved in a single iteration step due to the exact identification of the Lagrange multiplier. This method requires the Lagrange multiplier 
be first determined optimally. The successive approximations
of the solution 
can be readily obtained by using this determined Lagrange multiplier and any selective function 
Consequently, the solution is given by s. For the convergence criteria and error estimates of the VIM we refer the reader to [12,14].
We start with the initial guess 
in the above iteration formula and obtain the following approximate solutions:
gives the contribution of the non-Newtonian fluid. It is worth noting that by setting 
in the above approximations, we recover the exact solution for the case of Newtonian fluid. Thus, the first approximation of the nonlinear system (2.7)-(2.8) obtained by the VIM is identical with the exact solution of the corresponding linear problem. This shows that the VIM can be equally applied to linear equations.
on the velocity given in (3.7) are plotted in 
the solution converges to the Newtonian case.
,
.
is the highest order derivative which is assumed to be easily invertible, 
represents the nonlinear term and 
is the source term. According to the ADM, the solution 
can be expanded into the infinite series
are usually determined recursively. The nonlinear term 
can be decomposed into infinite polynomials given by
are the so-called Adomian polynomials of 
defined by
for this problem are given by:
and 
from (4.9)-(4.14), becomes
in (4.15) one can recover the exact solution for the Newtonian fluid.
an excellent agreement is observed.