A Comparative Study of Variational Iteration Method and He-Laplace Method

In this paper, variational iteration method and He-Laplace method are used to solve the nonlinear ordinary and partial differential equations. Laplace transformation with the homotopy perturbation method is called He-Laplace method. A comparison is made among variational iteration method and He-Laplace. It is shown that, in He-Laplace method, the nonlinear terms of differential equation can be easily handled by the use of He’s polynomials and provides better results.

In this paper, the main objective is to introduce a comparative study to nonlinear ordinary differential equation and partial differential equations by using variational iteration method and He-Laplace method.
It is worth mentioning that He-Laplace method is an elegant combination of the Laplace transformation, the homotopy perturbation method and He's polynomials.The use of He's polynomials in the nonlinear term was first introduced by Ghorbani [50].The proposed algorithm provides the solution in a rapid convergent series which may lead to the solution in a closed form.This paper contains basic idea of homotopy perturbation method in Section 2, variational iteration method in Section 3, Laplace homotopy perturbation method in Section 4 and conclusions in Section 5 respectively.

Basic Idea of Homotopy Perturbation
Method and He-Laplace Method

Homotopy Perturbation Method
Consider the following nonlinear differential equation with the boundary conditions of , 0, y B y r n where A, B,

 
f r and are a general differential operator, a boundary operator, a known analytic function and the boundary of the domain , respectively.

 
The operator A can generally be divided into a linear part L and a nonlinear part N. Equation (1) may therefore be written as: By the homotopy technique, we construct a homotopy or where is an embedding parameter, while 0 is an initial approximation of Equation ( 1), which satisfies the boundary conditions.Obviously, from Equatons (4) and ( 5), we will have: The changing process of p from zero to unity is just that of v(r, p) from y 0 to y(r).In topology, this is called deformation, while     0 L v L y 

and    
A v f r  are called homotopy.If the embedding parameter p is considered as a small parameter, applying the classical perturbation technique, we can assume that the solution of Equations ( 4) and ( 5) can be written as a power series in : p Setting in Equation ( 8), we have The combination of the perturbation method and the homotopy method is called the HPM, which eliminates the drawbacks of the traditional perturbation methods while keeping all its advantages.The series (9) is convergent for most cases.However, the convergent rate depends on the nonlinear operator   A v .Moreover, He [51] made the following suggestions: 1) The second derivative of

 
N v with respect to must be small because the parameter may be relatively large, i.e. .

He-Laplace Method
Consider the following nonlinear differential equation (IVP): where 1 By using linearity of Laplace transformation, the result is Applying the formula on Laplace transform, we obtain Using initial conditions in Equation ( 14), we have or Taking inverse Laplace transform, we have where   F x represents the term arising from the source term and the prescribed initial conditions.Now, we apply homotopy perturbation method [51], where the term are to recursively calculated and the nonlinear term n y   f y can be decomposed as   must be smaller than one so that the series converges.
for some He's polynomial n H (see [50,52]) that are given by Substituting Equations ( 18) and ( 19) in ( 17), we get ) which is the coupling of the Laplace transformation and the homotopy perturbation method using He's polynomials.Comparing the coefficient of like powers of p, the following approximations are obtained: : with the following conditions: .
Equation ( 22) can be written as By applying the Laplace transform to both sides of Equation ( 24) subject to the initial condition, we have The inverse of the Laplace transform implies that Now, we apply the homotopy perturbation method, we have where   n H u are He's polynomials.The first few components of He's polynomials are given by Comparing the coefficient of like powers of p, we have So that the solution   , u x y is given by   0 1 2 3 which is the exact solution of the problem.Example 2.2.Consider the following non-homogeneous nonlinear PDE [53]: with the following condition: By applying the Laplace transform method subject to the initial condition, we have The inverse of the Laplace transform implies that Now, we apply the homotopy perturbation method, we have x t H u u u Comparing the coefficient of like powers of p, we have , 4 Proceeding in a similar manner, we have  

Variational Iteration Method (VIM)
To illustrate the basic concept of the technique, we consider the following general differential equation where L is a linear operator, N is a nonlinear operator and g(x) is the forcing term.According to VIM, we can construct a correct functional as follows where  is a Lagrange multiplier.The subscripts n denote the nth approximation, n u is considered as a restricted variation i.e.
  If 0 is an initial approximation or trial-function then we can write down following expression for correction: where the last term of right is called "correction", n  is a general Lagrange multiplier.The above functional is called correction functional, the Lagrange multiplier in the functional should be such chosen that its correction solution is superior to its initial approximation (trialfunction) and is the best within the flexibility of the trialfunction, accordingly we can identified the multiplier by variational theory [54,55].Making the above correction functional stationary with y(0) = 1 so that, we can obtain following stationary conditions:   The Lagrange multiplier, therefore, can be identified as follows: To simplify the multiplier, we approximate Equation (44) as follows: Substituting Equation (45) in Equation ( 41) yields following variational iteration formula We start with by above iteration formula, we can obtain following results, , while its exact one is y(0.4) = 0.6667, the 0.17% accuracy is remarkably good in view of the crudeness of its initial approximation.The process can, in principle, be continued as far as desired, however, the resulting integrals quickly become very cumbersome, so some simplification in the process of identification of Lagrange multiplier will be discussed at below: .6678 y We re-consider the correction functional Equation ( 41) as follows: Where the nonlinear term 2 n y is considered as nonvariational variation or constrained variation [54], i.e.
The Lagrange multiplier, therefore, can be readily identified and the following variation iteration formula can be obtained: Putting n = 0, 1, in Equation ( 50), we can obtain fol- tional iteration technique mentioned above ca : ation can be obtained, which converges to its exact solution, a little more slowly due to the approximate identification of the Lagrange multiplier.

Remark
The varia n be readily extended to partial differential equations (PDEs).Here the author will illustrate its process.
Example 3.2.Consider the following equation [53]  which has the exact solution ate solution ca   u x xy a   ., an approxi According to Adomian [56] m n be obtained [57].
It is obvious that the approximation does not satisfy its boundary conditions.In 1995, Liu [57] proposes a modified Adomian's method called weighted residual decomposition method, with such method, he obtained following approximation: which satisfies all its boundary conditions and has more of Equation ( 51) is higher accuracy than Adomian's.In 1978, Inokuti et al. [55] proposed a general Lagrange multiplier method to solve nonlinear mathematical physics which was first applied to quantum mechanics.In this method, a more accurate solution, depending upon its trial-function can be obtained for some special points, but not an approximate analytical one.J. H.He [53] tries to solve it by variational iteration method as follows: Supposing the initial approximation 0 u , its correction variational functional in x-direction and -direction can be expressed respectively as follows: where is a nonvariational variation.Their stationary conditio are written down respectively as follows The Lagrange multipliers can be easily identified: The iteration formulae in x-direction and y-directions can be, therefore, expressed respectively as follows To ensure the approximations satisfy the boundary conditions at x = 1 and y = 1, we modify the variational iteration formulae in x-direction and y-direction lows: Now we start with an arbitrary initial approximation: x, where A and B are constants to be detere variational iteration formula in x-direction (59), we have By imposing the boundary conditions at x = 0 and x = 0 and B = a − 1/30, thus we have By (61), we have which is an exact solution.The approxima be ction.sider the following nonlinear PDE [53]:   , 0 0 u x  Its t-direction correction functional can be constructed as In which is nonvariational variation.The multiplier can be tified and its v nal iteration form ˆn u iden ariatio ula t-direction can be obtained We start with initial approximation by above ite u 0 ration form la, we can obtain successively its approximation: which is the same as Adomian's [56,58].

Comparision of Variational Iterational Method and He-Laplace Method
He's polynomials.The first few components of He's polynomials are given by

3 . 1 .
method, it is required first to determine the Lagrange multiplier  optimally.The successive approximation 1 n of the solution u will be readily obtained upon using the determined Lagrange multiplier and any selective function , consequently, the solution is given by , consider the following examples: Example Consider the following first order nonlinear differential equation[53]:

Example 4 . 1 .
Consider the following first order nonlin ear differential equation[53]:By applying the aforesaid method subject to the in condition, we have He's polynomials.The first few ponents of He's polynomials are given by com- t