Review Article  Open Access
Variational Iteration Method for Initial and Boundary Value Problems Using He's Polynomials
Abstract
This paper outlines a detailed study of the coupling of He's polynomials with correction functional of variational iteration method (VIM) for solving various initial and boundary value problems. The elegant coupling gives rise to the modified versions of VIM which is very efficient in solving nonlinear problems of diversified nature. It is observed that the variational iteration method using He's polynomials (VIMHP) is very efficient, easier to implements, and more user friendly. Several examples are given to reconfirm the efficiency of the proposed VIMHP.
1. Introduction
With the rapid development of nonlinear sciences, many analytical and numerical techniques have been developed by various scientists. Most of the developed techniques have their limitations like limited convergence, divergent results, linearization, discretization unrealistic assumptions, and noncompatibility with the physical problems [1â€“67]. He [20â€“35] developed the variational iteration (VIM) and homotopy perturbation methods (HPM) which proved to be fully synchronized with the versatile nature of the physical problems showing the intellect of the author; see [1â€“8, 12â€“35, 41, 46â€“58, 63â€“65] and the references therein. In order to improve the efficiency of these algorithms several modifications have been introduced by different researchers for time and again. Abbasbandy [1, 2] made the elegant coupling of Adomianâ€™s polynomials and correction functional of Heâ€™s VIM and solved quadratic Riccati differential and KleinGordon equations. This concept was subsequently exploited by MohyudDin and Noor [49, 54] for solving various singular and nonsingular initial and boundary value problems. Recently, Ghorbani and SaberiNadjafi [17] and Ghorbani [18] introduced Heâ€™s polynomials by splitting the nonlinear term and also proved that Heâ€™s polynomials are fully compatible with Adomianâ€™s polynomials but are easier to calculate. More recently, Noor and MohyudDin coupled Heâ€™s polynomials and correction functional of the VIM and applied this reliable version (VIMHPS) to a number of physical problems; see [51â€“53]. The VIMHPS [51â€“53] has a very simple solution procedure and absorbs all of the positive features of Heâ€™s variational iteration (VIM) and homotopy perturbation (HPM) methods and is highly compatible with the diversity of the physical problems. Moreover, the inclusion of Heâ€™s polynomials in the correction functional enhances its capability to deal with the physical nature of the problems and makes this version closer to the versatility of the physical nature of the problems. The basic motivation of the present study is the implementation of VIMHPS for solving various initial and boundary value problems of diversified physical nature. Several examples are given for the comparison and to measure the efficiency of these couplings. It is to be highlighted that the variational iteration method using Heâ€™s polynomials (VIMHPSs) [51â€“53] has certain advantages as compared to the decomposition method. Firstly, the use of Lagrange multiplier reduces the successive applications of the integral operator and hence minimizes the computational work to a tangible level while still maintaining a very high level of accuracy. Moreover, Heâ€™s polynomials are easier to calculate as compared to Adomianâ€™s polynomials and this gives it a clear edge over the traditional decomposition method. The VIMHPS is also independent of the small parameter assumption (which is either not there in the physical problems or difficult to locate) and hence is more convenient to apply as compare to the traditional perturbation method. It is worth mentioning that the VIMHPS is applied without any discretization, restrictive assumption or transformation and is free from round off errors. We apply the proposed VIMHPS for all the nonlinear terms in the problem without discretizing either by finite difference or spline techniques at the nodes, involving laborious calculations coupled with a strong possibility of the illconditioned resultant equations which is a complicated problem to solve. Moreover, unlike the method of separation of variables that requires initial and boundary conditions, the VIMHPS provides the solution by using the initial conditions only; see [51â€“53]. It is to be highlighted that new developments of variational iteration and homotopy perturbation methods are available in [8, 12, 40]. Moreover, the convergence of variational iteration method has been discussed in [58]. It is worth mentioning that the examples which have been discussed in this paper are of utmost importance in applied, engineering, and nonlinear sciences.
2. Variational Iteration Method (VIM)
To illustrate the basic concept of Heâ€™s VIM, we consider the following general differential equation: where is a linear operator, a nonlinear operator, and is the inhomogeneous term. According to variational iteration method [1â€“8, 13, 14, 21â€“24, 26, 27, 41, 46, 51â€“54], we can construct a correction functional as follows: where is a Lagrange multiplier [21â€“24], which can be identified optimally via variational iteration method. The subscripts denote the nth approximation, and is considered as a restricted variation. That is, (2.2) is called a correction functional. The solution of the linear problems can be solved in a single iteration step due to the exact identification of the Lagrange multiplier. The principles of variational iteration method and its applicability for various kinds of differential equations are given in [21â€“24]. In this method, it is required first to determine the Lagrange multiplier optimally. The successive approximation of the solution will be readily obtained upon using the determined Lagrange multiplier and any selective function ; consequently, the solution is given by The convergence of variational iteration method has been discussed in [58].
3. Homotopy Perturbation Method (HPM) and Heâ€™s Polynomials
To explain Heâ€™s homotopy perturbation method, we consider a general equation of the type where is any integral or differential operator. We define a convex homotopy by where is a functional operator with known solutions , which can be obtained easily. It is clear that, for we have This shows that continuously traces an implicitly defined curve from a starting point to a solution function . The embedding parameter monotonically increases from zero to unit as the trivial problem continuously deforms the original problem The embedding parameter can be considered as an expanding parameter [17, 18, 29â€“35]. The homotopy perturbation method uses the homotopy parameter as an expanding parameter [29â€“35] to obtain If , then (3.5) corresponds to (3.2) and becomes the approximate solution of the form It is well known that series (3.6) is convergent for most of the cases and also the rate of convergence is dependent on ; see [29â€“35]. We assume that (3.6) has a unique solution. The comparisons of like powers of give solutions of various orders. In sum, according to [17, 18], Heâ€™s HPM considers the nonlinear term as where â€™s are the socalled Heâ€™s polynomials [17, 18], which can be calculated by using the formula
4. Adomianâ€™s Decomposition Method (ADM)
Consider the differential equation [59â€“61] where is the highestorder derivative which is assumed to be invertible, is a linear differential operator of order lesser than , represents the nonlinear terms, and is the source term. Applying the inverse operator to both sides of (4.1) and using the given conditions, we obtain where the function represents the terms arising from integrating the source term and by using the given conditions. Adomianâ€™s decomposition method [59â€“61] defines the solution by the series where the components are usually determined recurrently by using the relation The nonlinear operator can be decomposed into an infinite series of polynomials where are the socalled Adomianâ€™s polynomials that can be generated for various classes of nonlinearities according to the specific algorithm developed in [59â€“61] which yields
5. Variational Iteration Method Using Heâ€™s Polynomials (VIMHPs)
This modified version of variational iteration method [49â€“53] is obtained by the elegant coupling of correction functional (3.1) of variational iteration method (VIM) with Heâ€™s polynomials [17, 18] and is given by Comparisons of like powers of give solutions of various orders.
6. Numerical Applications
In this section, we apply the VIMHPS for solving various initial and boundary value problems.
Example 1. Consider the following onedimensional Burgerâ€™s equation:
with initial condition
where and the parameters are the arbitrary constants. The correction functional is given by
Making the above functional stationary, the Lagrange multiplier can be determined as and we get the following iterative scheme:
Applying the variational iteration method using Heâ€™s polynomials (VIMHPS), we get
Comparing the coefficient of like powers of ,
The series solution is given by
and in a closedform by
â€‰Figure 1 depicts series solution at .
Example 2. Consider the following homogeneous coupled Burgerâ€™s equation:
with initial conditions
The correction functional for the above coupled system is given by
Making the above functional stationary, the Lagrange multiplier can be determined as and we get
Applying the variational iteration method using Heâ€™s polynomials (VIMHPS), we get
Comparing the coefficient of like powers of ,
The series solutions are given by
and the closedform solutions are given as
â€‰Figure 2 depicts the series solutions or
Example 3. Consider the ThomasFermi equation with boundary conditions The correction functional is given as Making the correction functional stationary, the Lagrange multiplier can be identified as and the following iterative scheme is obtained: Applying the variational iteration method using Heâ€™s polynomials (VIMHPS), we get Now, we apply a slight modification in the conventional initial value and take instead of where Comparing the coefficient of like powers of , The series solution is given as Setting the series solution is obtained as The diagonal PadÃ© approximants can be applied [53] in order to study the mathematical behavior of the potential and to determine the initial slope of the potential .
Example 4. Consider the following nonlinear thirdorder boundary layer problem which appears mostly in the mathematical modeling of physical phenomena in fluid mechanics: with boundary conditions The correction functional is given as Making the correction functional stationary, the Lagrange multipliers can be identified as ; consequently Applying the variational iteration method using Heâ€™s polynomials (VIMHPS), we get where Comparing the coefficient of like powers of , The series solution is given as
Example 5. Consider the following nonlinear inhomogeneous Goursat problem:
The correction functional for the above problem is given by
Making the above correctional functional stationary, the Lagrange multiplier can be identified as ; consequently
Applying the variational iteration method using Heâ€™s polynomials (5.1) (VIMHPS), we get
The initial approximation can be selected (by utilizing the boundary conditions) as where are constants and would be determined by using the boundary conditions; consequently
Comparing the coefficient of like powers of ,
Imposing the boundary conditions to find the constants and will yield and consequently
â€‰Figure 3 depicts the solution
Example 6. Consider the following nonlinear differential equation which governs the unsteady flow of gas through a porous medium:
with the following typical boundary conditions imposed by the physical properties:
The correction functional is given as
Making the correction functional stationary, the Lagrange multipliers can be identified as and thus we get
Applying the variational iteration method using Heâ€™s polynomials (VIMHPS), we get
where . Comparing the coefficient of like powers of , consequently,
The series solution is given as
Table 4 exhibits the initial slopes for various values of .
Table 5 exhibits the values of for for to .


