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Research Article
 

Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations



M. Kazemnia, S.A. Zahedi, M. Vaezi and N. Tolou
 
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ABSTRACT

This study has been devoted to investigate the semi-analytical solution of nonlinear differential equations with boundary value problems (BVPs). Modified variation iteration method has been utilized to solve some BVPs nonlinear differential equations. In this method, general lagrange multipliers have been introduced to construct correction functions for the problems. The multipliers can be identified optimally via the variational theory. The results have been compared with those of exact solutions and Homotopy Analysis Method (HAM). A clear conclusion can be drawn from the numerical results that proposed method provides excellent approximations to the solutions of this kind of problems in the terms of simplicity and accuracy, thus, it can be easily extended to other BVPs nonlinear differential equations and so can be found widely applicable in engineering sciences.

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  How to cite this article:

M. Kazemnia, S.A. Zahedi, M. Vaezi and N. Tolou, 2008. Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations. Journal of Applied Sciences, 8: 4192-4197.

DOI: 10.3923/jas.2008.4192.4197

URL: https://scialert.net/abstract/?doi=jas.2008.4192.4197
 

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; Al-Hayani and Casasús, 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 Exp-function 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:

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(1)

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:

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(2)

where, λ is a general Lagrangian multiplier which can be identified optimally via the variational theory. The subscript n indicates the nth approximation and Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations is considered as a restricted variation Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations.

Implementation of MVIM
Example one: Consider the following nonlinear fourth-order differential equation:

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(3)

Subject to the boundary conditions:

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(4)

Its correction variational functional can be expressed as follows:

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(5)

After some manipulations, the following stationary conditions have been obtained:

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(6)

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(7)

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(8)

The Lagrangian multipliers can therefore be identified as:

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(9)

And the variational iteration formula is obtained in the form:

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(10)

We are to start with the initial approximation of Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential EquationsSince, no initial approximation of Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations is available, we make one in the form of a polynomial as:

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(11)

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:

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(12)

For a special case, M = 2, Re = 0, α = 0.2 will be as follows:

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(13)

Incorporating the boundary conditions, Eq. 4, into y1(x), we have:

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(14)

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(15)

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(16)

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(17)

Solving the above system of equations simultaneously, we obtain:

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(18)

Therefore, we obtain the following first–order approximate solution for special case, M = 2, Re = 0, α = 0.2:

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(19)

In the same manner, the rest of the components of the iteration formula can be obtained.

Example two: Now consider another nonlinear fourth-order BVP (Liu and Wv, 2002):

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(20)

Subject to the boundary conditions:

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(21)

Its correction variational functional in can be expressed as:

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(22)

After some computations, we obtain the following stationary conditions:

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(23)

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(24)

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(25)

The Lagrangian multipliers can, therefore, be identified as:

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(26)

And the variational iteration formula is obtained in the form of

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(27)

Assuming initial approximation as following form:

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(28)

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:

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(29)

Incorporating the boundary conditions, Eq. 21, into y1(x), we have:

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(30)

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(31)

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(32)

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(33)

Solving the above system of equations simultaneously, we obtain:

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(34)

Therefore, we obtain the following first-order approximate solution, as follows:

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(35)

The exact solution is y(x) = x4 (Liu and Wv, 2002).

Example three: Consider the nonlinear boundary value problem as follows:

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(36)

Subject to the following conditions:

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(37)

Its correction variational functional can be expressed as:

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(38)

After some computations, we obtain the following stationary conditions:

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(39)

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(40)

The Lagrangian multipliers can, therefore, be identified as:

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(41)

And the variational iteration formula is obtained in the form of

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(42)

Now, we assume that the initial approximation has the form of

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(43)

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:

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(44)

Incorporating the boundary conditions, Eq. 37, into y1(x) and solving the equations, we obtain:

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(45)

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(46)

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(47)

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(48)

Solving the above system of equations simultaneously, we obtain:

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(49)

In the same manner, the rest of the components of the iteration formula can be obtained as:

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
(50)

The exact solution for this problem have already been obtained by Wazwaz (2001) as (1-x)ex.

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.

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
Fig. 1:
y (x) values versus x obtained from (a) MVIM and (b) HAM, in different values of M (Hayat et al., 2007)

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
Fig. 2:
Plot of the approximated y (x) versus x, this plot has excellent agreement with reference (Hayat et al., 2007)

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
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 Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations 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. 1-5 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.

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
Fig. 4:
The comparison between the Exact and MVIM solutions for second example

Image for - Assessment of Modified Variational Iteration Method in BVPs High-Order Differential Equations
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. 1-5 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.

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