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 [1–67]. He [20–35] 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 [1–8, 12–35, 41, 46–58, 63–65] 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 Adomian’s polynomials and correction functional of He’s VIM and solved quadratic Riccati differential and Klein-Gordon equations. This concept was subsequently exploited by Mohyud-Din and Noor [49, 54] for solving various singular and nonsingular initial and boundary value problems. Recently, Ghorbani and Saberi-Nadjafi [17] and Ghorbani [18] introduced He’s polynomials by splitting the nonlinear term and also proved that He’s polynomials are fully compatible with Adomian’s polynomials but are easier to calculate. More recently, Noor and Mohyud-Din coupled He’s polynomials and correction functional of the VIM and applied this reliable version (VIMHPS) to a number of physical problems; see [51–53]. The VIMHPS [51–53] has a very simple solution procedure and absorbs all of the positive features of He’s variational iteration (VIM) and homotopy perturbation (HPM) methods and is highly compatible with the diversity of the physical problems. Moreover, the inclusion of He’s polynomials 𝑝 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 He’s polynomials (VIMHPSs) [51–53] 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, He’s polynomials are easier to calculate as compared to Adomian’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 ill-conditioned 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 [51–53]. 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 He’s VIM, we consider the following general differential equation:𝐿𝑢+𝑁𝑢=𝑔(𝑥), (2.1) where 𝐿 is a linear operator, 𝑁 a nonlinear operator, and 𝑔(𝑥) is the inhomogeneous term. According to variational iteration method [1–8, 13, 14, 21–24, 26, 27, 41, 46, 51–54], we can construct a correction functional as follows:𝑢_𝑛+1(𝑥)=𝑢_𝑛(𝑥)+^𝑥₀𝜆(𝐿𝑢_𝑛(𝑠)+𝑁𝑢_𝑛(𝑠)−𝑔(𝑠))𝑑𝑠, (2.2) where 𝜆 is a Lagrange multiplier [21–24], which can be identified optimally via variational iteration method. The subscripts 𝑛 denote the nth approximation, and 𝑢_𝑛 is considered as a restricted variation. That is, 𝛿𝑢_𝑛=0; (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 [21–24]. In this method, it is required first to determine the Lagrange multiplier 𝜆 optimally. The successive approximation 𝑢_𝑛+1,𝑛≥0, of the solution 𝑢 will be readily obtained upon using the determined Lagrange multiplier and any selective function 𝑢₀; consequently, the solution is given by 𝑢=lim_𝑛→∞𝑢_𝑛. The convergence of variational iteration method has been discussed in [58].

3. Homotopy Perturbation Method (HPM) and He’s Polynomials

To explain He’s homotopy perturbation method, we consider a general equation of the type𝐿(𝑢)=0, (3.1) where 𝐿 is any integral or differential operator. We define a convex homotopy 𝐻(𝑢,𝑝) by𝐻(𝑢,𝑝)=(1−𝑝)𝐹(𝑢)+𝑝𝐿(𝑢), (3.2) where 𝐹(𝑢) is a functional operator with known solutions 𝑣₀, which can be obtained easily. It is clear that, for𝐻(𝑢,𝑝)=0, (3.3) we have𝐻(𝑢,0)=𝐹(𝑢),𝐻(𝑢,1)=𝐿(𝑢). (3.4) This shows that 𝐻(𝑢,𝑝) continuously traces an implicitly defined curve from a starting point 𝐻(𝑣₀,0) to a solution function 𝐻(𝑓,1). The embedding parameter monotonically increases from zero to unit as the trivial problem 𝐹(𝑢)=0 continuously deforms the original problem 𝐿(𝑢)=0. The embedding parameter 𝑝∈(0,1] can be considered as an expanding parameter [17, 18, 29–35]. The homotopy perturbation method uses the homotopy parameter 𝑝 as an expanding parameter [29–35] to obtain 𝑢=^∞_𝑖=0𝑝^𝑖𝑢_𝑖=𝑢₀+𝑝𝑢₁+𝑝²𝑢₂+𝑝³𝑢₃+⋯. (3.5) If 𝑝→1, then (3.5) corresponds to (3.2) and becomes the approximate solution of the form𝑓=lim_𝑝→1𝑢=^∞_𝑖=0𝑢_𝑖. (3.6) It is well known that series (3.6) is convergent for most of the cases and also the rate of convergence is dependent on 𝐿(𝑢); see [29–35]. We assume that (3.6) has a unique solution. The comparisons of like powers of 𝑝 give solutions of various orders. In sum, according to [17, 18], He’s HPM considers the nonlinear term 𝑁(𝑢) as 𝑁(𝑢)=^∞_𝑖=0𝑝^𝑖𝐻_𝑖=𝐻₀+𝑝𝐻₁+𝑝²𝐻₂+⋯, (3.7) where 𝐻_𝑛’s are the so-called He’s polynomials [17, 18], which can be calculated by using the formula𝐻_𝑛(𝑢₀,…,𝑢_𝑛1)=𝜕𝑛!𝜕𝑝^𝑛𝑛(𝑁(^𝑛_𝑖=0𝑝^𝑖𝑢_𝑖))_𝑝=0,𝑛=0,1,2,…. (3.8)

4. Adomian’s Decomposition Method (ADM)

Consider the differential equation [59–61]𝐿𝑢+𝑅𝑢+𝑁𝑢=𝑔, (4.1) where 𝐿 is the highest-order derivative which is assumed to be invertible, 𝑅 is a linear differential operator of order lesser than 𝐿, 𝑁𝑢 represents the nonlinear terms, and 𝑔 is the source term. Applying the inverse operator 𝐿⁻¹ to both sides of (4.1) and using the given conditions, we obtain𝑢=𝑓−𝐿⁻¹(𝑅𝑢)−𝐿⁻¹(𝑁𝑢), (4.2) where the function 𝑓 represents the terms arising from integrating the source term 𝑔 and by using the given conditions. Adomian’s decomposition method [59–61] defines the solution 𝑢(𝑥) by the series𝑢(𝑥)=^∞_𝑛=0𝑢_𝑛(𝑥), (4.3) where the components 𝑢_𝑛(𝑥) are usually determined recurrently by using the relation𝑢₀=𝑓,𝑢_𝑘+1=𝐿⁻¹(𝑅𝑢_𝑘)−𝐿⁻¹(𝑁𝑢_𝑘),𝑘≥0. (4.4) The nonlinear operator 𝐹(𝑢) can be decomposed into an infinite series of polynomials 𝐹(𝑢)=∑^∞_𝑛=0𝐴_𝑛, where 𝐴_𝑛 are the so-called Adomian’s polynomials that can be generated for various classes of nonlinearities according to the specific algorithm developed in [59–61] which yields 𝐴_𝑛1=(𝑑𝑛!)(^𝑛𝑑𝜆^𝑛)𝑁(^𝑛_𝑖=0(𝜆^𝑖𝑢_𝑖))_𝜆=0,𝑛=0,1,2,…. (4.5)

5. Variational Iteration Method Using He’s Polynomials (VIMHPs)

This modified version of variational iteration method [49–53] is obtained by the elegant coupling of correction functional (3.1) of variational iteration method (VIM) with He’s polynomials [17, 18] and is given by^∞_𝑛=0𝑝^(𝑛)𝑢_𝑛=𝑢₀(𝑥)+𝑝^𝑥₀𝜆(𝑠)(^∞_𝑛=0𝑝^(𝑛)𝐿(𝑢_𝑛)+^∞_𝑛=0𝑝^(𝑛)𝑁(𝑢_𝑛))𝑑𝑠−^𝑥₀𝜆(𝑠)𝑔(𝑠)𝑑𝑠. (5.1) Comparisons of like powers of 𝑝 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 one-dimensional Burger’s equation: 𝑢_𝑡+𝑢𝑢_𝑥𝑥−𝑣𝑢_𝑥𝑥=0, (6.1) with initial condition 𝑢(𝑥,0)=𝛼+𝛽+(𝛽−𝛼)exp(𝛾)1+exp(𝛾),𝑡≥0, (6.2) where 𝛾=(𝛼/𝑣)(𝑥−𝜆) and the parameters 𝛼,𝛽,𝜆 are the arbitrary constants. The correction functional is given by 𝑢_𝑛+1(𝑥,𝑡)=𝑢_𝑛(𝑥,𝑡)+^𝑡₀𝜆(𝑠)(𝜕𝑢_𝑛+𝑢𝜕𝑡^𝑛𝜕²𝑢_𝑛𝜕𝑥²𝜕−𝑣²𝑢_𝑛𝜕𝑥²)𝑑𝑠. (6.3) Making the above functional stationary, the Lagrange multiplier can be determined as 𝜆(𝑠)=−1, and we get the following iterative scheme: 𝑢_𝑛+1(𝑥,𝑡)=𝑢_𝑛(𝑥,𝑡)−^𝑡₀(𝜕𝑢_𝑛𝜕𝑡+𝑢_𝑛𝜕²𝑢_𝑛𝜕𝑥²𝜕−𝑣²𝑢_𝑛𝜕𝑥²)𝑑𝑠. (6.4) Applying the variational iteration method using He’s polynomials (VIMHPS), we get 𝑢₀+𝑝𝑢₁+𝑝²𝑢₂+⋯=(𝛼+𝛽+(𝛽−𝛼)exp(𝛾)1+exp(𝛾))−𝑝^𝑡₀(𝜕𝑢₀𝜕𝑡+𝑝𝜕𝑢₁𝜕𝑡+𝑝²𝜕𝑢₂𝜕𝑡+⋯)𝑑𝑠−𝑝^𝑡₀((𝑢₀+𝑝𝑢₁+𝑝²𝑢₂𝜕+⋯)(²𝑢₀𝜕𝑥²𝜕+𝑝²𝑢₁𝜕𝑥²𝜕+⋯)−𝑣(²𝑢₀𝜕𝑥²𝜕+𝑝²𝑢₁𝜕𝑥²+⋯))𝑑𝑠. (6.5) Comparing the coefficient of like powers of 𝑝, 𝑝⁽⁰⁾∶𝑢₀(𝑥,𝑡)=𝛼+𝛽+(𝛽−𝛼)exp(𝛾),𝑝1+exp(𝛾)₍₁₎∶𝑢₁(𝑥,𝑡)=𝛼+𝛽+(𝛽−𝛼)exp(𝛾)+1+exp(𝛾)2𝛼𝛽²exp(𝛾)𝑝𝑣(1+exp(𝛾))𝑡,₍₂₎∶𝑢₂(𝑥,𝑡)=(𝛼+𝛽+(𝛽−𝛼)exp(𝛾)1+exp(𝛾))+(2𝛼𝛽²exp(𝛾)𝛼𝑣(1+exp(𝛾)))𝑡+(³𝛽²exp(𝛾)(−1+exp(𝛾))𝑣²(1+exp(𝛾))³)𝑡³,⋮. (6.6) The series solution is given by 𝑢(𝑥,𝑡)=(𝛼+𝛽+(𝛽−𝛼)exp(𝛾)1+exp(𝛾))+(2𝛼𝛽²exp(𝛾)𝛼𝑣(1+exp(𝛾)))𝑡+(³𝛽²exp(𝛾)(−1+exp(𝛾))𝑣²(1+exp(𝛾))³)𝑡³𝛼+(⁴𝛽³exp(𝛾)(1−4exp(𝛾)+exp(𝛾)²)3𝑣³(1+exp(𝛾))⁴)𝑡⁴+⋯, (6.7) and in a closed-form by 𝑢(𝑥,𝑡)=(𝛼+𝛽+(𝛽−𝛼)exp(𝑥/𝑣)(𝑥−𝛽𝑡−𝜆)1+exp(𝑥/𝑣)(𝑥−𝛽𝑡−𝜆)). (6.8)
 Figure 1 depicts series solution at 𝛼=𝛽=𝜈=𝜆=1.

Figure 1 

(𝛼=𝛽=𝜈=𝜆=1).

Example 2. Consider the following homogeneous coupled Burger’s equation: 𝑢_𝑡−𝑢_𝑥𝑥−2𝑢𝑢_𝑥+(𝑢𝑣)_𝑥=0,𝑣_𝑡−𝑣_𝑥𝑥−2𝑣𝑣_𝑥+(𝑢𝑣)_𝑥=0, (6.9) with initial conditions 𝑢(𝑥,0)=sin𝑥,𝑣(𝑥,0)=sin𝑥. (6.10) The correction functional for the above coupled system is given by 𝑢_𝑛+1(𝑥,𝑡)=𝑢_𝑛(𝑥,𝑡)+^𝑡₀𝜆(𝑠)(𝜕𝑢_𝑛−𝜕𝜕𝑠²𝑢_𝑛𝜕𝑥²−2𝑢_𝑛(𝑢_𝑛)_𝑥+(𝑢_𝑛𝑣_𝑛)_𝑥𝑣)𝑑𝑠,_𝑛+1(𝑥,𝑡)=𝑣_𝑛(𝑥,𝑡)+^𝑡₀𝜆(𝑠)(𝜕𝑣_𝑛−𝜕𝜕𝑠²𝑣_𝑛𝜕𝑥²−2𝑣_𝑛(𝑣_𝑛)_𝑥+(𝑢_𝑛𝑣_𝑛)_𝑥)𝑑𝑠. (6.11) Making the above functional stationary, the Lagrange multiplier can be determined as 𝜆(𝑠)=−1, and we get 𝑢_𝑛+1(𝑥,𝑡)=𝑢_𝑛(𝑥,𝑡)−^𝑡₀(𝜕𝑢_𝑛−𝜕𝜕𝑠²𝑢_𝑛𝜕𝑥²−2𝑢_𝑛(𝑢_𝑛)_𝑥+(𝑢_𝑛𝑣_𝑛)_𝑥𝑣)𝑑𝑠,_𝑛+1(𝑥,𝑡)=𝑣_𝑛(𝑥,𝑡)−^𝑡₀(𝜕𝑣_𝑛−𝜕𝜕𝑠²𝑣_𝑛𝜕𝑥²−2𝑣_𝑛(𝑣_𝑛)_𝑥+(𝑢_𝑛𝑣_𝑛)_𝑥)𝑑𝑠. (6.12) Applying the variational iteration method using He’s polynomials (VIMHPS), we get 𝑢₀+𝑝𝑢₁+𝑝²𝑢₂+⋯=𝑢₀(𝑥,𝑡)+𝑝^𝑡₀𝜆(𝑠)((𝜕𝑢₀𝜕𝑡+𝑝𝜕𝑢₁𝜕𝜕𝑡+⋯)−(²𝑢₀𝜕𝑥²𝜕+𝑝²𝑢₁𝜕𝑥²+𝑝²𝜕²𝑢₂𝜕𝑥²+⋯))𝑑𝑠−𝑝^𝑡₀(2(𝑢₀+𝑝𝑢₁+⋯)(𝜕𝑢₀𝜕𝑥+𝑝𝜕𝑢₁𝜕𝑥+⋯)−((𝑢₀+𝑝𝑢₁+⋯)(𝑣₀+𝑝𝑣₁+⋯))_𝑥𝑣)𝑑𝑠,₀+𝑝𝑣₁+𝑝²𝑣₂+⋯=𝑣₀(𝑥,𝑡)+𝑝^𝑡₀𝜆(𝑠)((𝜕𝑣₀𝜕𝑡+𝑝𝜕𝑣₁𝜕𝜕𝑡+⋯)−(²𝑣₀𝜕𝑥²𝜕+𝑝²𝑣₁𝜕𝑥²+𝑝²𝜕²𝑣₂𝜕𝑥²+⋯))𝑑𝑠−𝑝^𝑡₀(2(𝑣₀+𝑝𝑣₁+⋯)(𝜕𝑣₀𝜕𝑥+𝑝𝜕𝑣₁𝜕𝑥+⋯)−((𝑢₀+𝑝𝑢₁+⋯)(𝑣₀+𝑝𝑣₁+⋯))_𝑥)𝑑𝑠. (6.13) Comparing the coefficient of like powers of 𝑝, 𝑝⁽⁰⁾∶𝑢₀𝑣(𝑥,𝑡)=sin𝑥,₀𝑝(𝑥,𝑡)=sin𝑥,₍₁₎∶𝑢₁𝑣(𝑥,𝑡)=sin𝑥−𝑡sin𝑥,₀𝑝(𝑥,𝑡)=sin𝑥−𝑡sin𝑥,₍₂₎∶𝑢₂𝑡(𝑥,𝑡)=sin𝑥−𝑡sin𝑥+²𝑣2!sin𝑥,₂𝑡(𝑥,𝑡)=sin𝑥−𝑡sin𝑥+²2!sin𝑥,⋮. (6.14) The series solutions are given by 𝑡𝑢(𝑥,𝑡)=sin𝑥(1−𝑡+²−𝑡2!³𝑡3!+−⁴𝑡4!+⋯),𝑣(𝑥,𝑡)=sin𝑥(1−𝑡+²−𝑡2!³𝑡3!+−⁴4!+⋯), (6.15) and the closed-form solutions are given as 𝑢(𝑥,𝑡)=exp(−𝑡)sin𝑥,𝑣(𝑥,𝑡)=exp(−𝑡)sin𝑥. (6.16)
 Figure 2 depicts the series solutions 𝑢(𝑥,𝑡) or 𝑣(𝑥,𝑡).

Figure 2 

Example 3. Consider the Thomas-Fermi equation 𝑦^′′𝑦(𝑥)=^3/2𝑥^1/2, (6.17) with boundary conditions 𝑦(0)=1,lim_𝑥→∞𝑦(𝑥)=0. (6.18) The correction functional is given as 𝑦_𝑛+1(𝑥)=𝑦₀(𝑥)+^𝑥₀𝑑𝜆(𝑠)(²𝑦_𝑛(𝑥,𝑠)𝑑𝑠²−𝑥^−1/2𝑦_𝑛^3/2)𝑑𝑠. (6.19) Making the correction functional stationary, the Lagrange multiplier can be identified as 𝜆(𝑠)=(𝑠−𝑥), and the following iterative scheme is obtained: 𝑦_𝑛+1(𝑥)=𝑦₀(𝑥)+^𝑥₀𝑑(𝑠−𝑥)(²𝑦_𝑛(𝑥,𝑠)𝑑𝑠²−𝑥^−1/2𝑦_𝑛^3/2)𝑑𝑠. (6.20) Applying the variational iteration method using He’s polynomials (VIMHPS), we get 𝑦₀+𝑦₁+𝑦₂+⋯=𝑦₀(𝑥)+𝑝^𝑥₀𝑑(𝑠−𝑥)((²𝑦₀𝑑𝑠²𝑑+𝑝²𝑦₁𝑑𝑠²+𝑝²𝑑²𝑦₂𝑑𝑠²+⋯)−(𝑥^−1/2(𝑦₀+𝑝𝑦₁+𝑝²𝑦₂+⋯)^3/2))𝑑𝑠. (6.21) Now, we apply a slight modification in the conventional initial value and take 𝑦₀(𝑥)=1, instead of 𝑦₀(𝑥)=1+𝐵𝑥, where 𝐵=𝑦^′(0). Comparing the coefficient of like powers of 𝑝, 𝑝⁽⁰⁾∶𝑦₀𝑝(𝑥)=1,₍₁₎∶𝑦₁4(𝑥)=1+𝐵𝑥+3𝑥^3/2,𝑝⁽²⁾∶𝑦₂4(𝑥)=1+𝐵𝑥+3𝑥^3/2+25𝐵𝑥^5/2+13𝑥³,𝑝⁽³⁾∶𝑦₃4(𝑥)=1+𝐵𝑥+3𝑥^3/2+25𝐵𝑥^5/2+13𝑥³+3𝐵70²𝑥^7/2+215𝐵𝑥⁴+2𝑥27^9/2+13𝑥³,⋮. (6.22) The series solution is given as 4𝑦(𝑥)=1+𝐵𝑥+𝐵𝑥+3𝑥^3/2+25𝐵𝑥^5/2+13𝑥³+3𝐵70²𝑥^7/2+215𝐵𝑥⁴+2𝑥27^9/2+3𝐵70²𝑥^7/2−1𝐵252³𝑥^9/2+1𝐵175²𝑥⁵+2𝑥27^9/2+3𝐵70²𝑥^7/2+1𝐵1056⁴𝑥^11/2+4𝐵1575³𝑥⁶+557𝐵100100²𝑥^13/2+4693𝐵𝑥⁷+101𝑥52650^15/2−3𝐵9152⁵𝑥^13/2−29𝐵24255⁴𝑥⁷−512𝐵351000³𝑥^15/2−46𝐵45045²𝑥⁸−1131178100𝐵𝑥^17/2+23𝑥473850⁹⋯,⋮. (6.23) Setting 𝑥^1/2=𝑡, the series solution is obtained as 𝑦(𝑡)=1+𝐵𝑡²+43𝑡³+25𝐵𝑡⁵+13𝑡⁶+3𝐵70²𝑡⁷+215𝐵𝑡⁸1+(−𝐵252³+227)𝑡⁹+1𝐵175²𝑡¹⁰1+(𝐵1056⁴+311485𝐵)𝑡¹¹4+(𝐵1575³+4405)𝑡¹²3+(−𝐵9152⁵+557𝐵100100²)𝑡¹³+(−29𝐵24255⁴+4693𝐵)𝑡¹⁴7+(𝐵499⁶−623𝐵351000³[phone])𝑡¹⁵+(68𝐵105105⁴−46𝐵45045²)𝑡¹⁶3+(−𝐵43520⁷+153173𝐵116424000⁴−1131178100𝐵)𝑡¹⁷+⋯. (6.24) The diagonal Padé approximants can be applied [53] in order to study the mathematical behavior of the potential 𝑦(𝑥) and to determine the initial slope of the potential 𝑦^′(0).

Example 4. Consider the following nonlinear third-order boundary layer problem which appears mostly in the mathematical modeling of physical phenomena in fluid mechanics: 𝑓^′′′(𝑥)+(𝑛−1)𝑓(𝑥)𝑓^′′(𝑥)−2𝑛(𝑓^′(𝑥))²=0,𝑛>0, (6.25) with boundary conditions 𝑓(0)=0,𝑓^′(0)=1,𝑓^′(∞)=0,𝑛>0. (6.26) The correction functional is given as 𝑓_𝑛+1(𝑥)=𝑓_𝑛(𝑥)+^𝑥₀𝜆(𝑠)(𝑓_𝑛^′′′(𝑥)+(𝑛−1)𝑓_𝑛(𝑥)𝑓_𝑛^′′(𝑥)−2(𝑓^′_𝑛(𝑥))²)=0,𝑛>0. (6.27) Making the correction functional stationary, the Lagrange multipliers can be identified as 𝜆(𝑠)=−(1/2!)(𝑠−𝑥)²; consequently 𝑓_𝑛+1(𝑥)=𝑓_𝑛(𝑥)−^𝑥₀12!(𝑠−𝑥)²(𝑓_𝑛^′′′(𝑥)+(𝑛−1)𝑓_𝑛(𝑥)𝑓_𝑛(𝑥)−2𝑛(𝑓^′_𝑛(𝑥))²)=0,𝑛>0. (6.28) Applying the variational iteration method using He’s polynomials (VIMHPS), we get 𝑓₀+𝑝𝑓₁+𝑝²𝑓₂1+⋯=𝑥+2𝛼𝑥²−𝑝^𝑥₀13!(𝑠−𝑥)²((𝑓₀^′′′+𝑝𝑓₁^′′′+⋯)+(𝑛−1)(𝑓₀+𝑝𝑓₁+⋯)(𝑓₀^′′+𝑝𝑓₁^′′+⋯))𝑑𝑥+2𝑛𝑝^𝑥₀13!(𝑠−𝑥)²(𝑓^′₀+𝑝𝑓₁+𝑝²𝑓₂+⋯)²𝑑𝑥, (6.29) where 𝑓^′′(0)=𝛼<0. Comparing the coefficient of like powers of 𝑝, 𝑝⁽⁰⁾∶𝑓₀𝑝(𝑥)=𝑥,₍₁₎∶𝑓₁1(𝑥)=𝑥+2𝛼𝑥²+13𝑥³,𝑝⁽²⁾∶𝑓₂1(𝑥)=𝑥+2𝛼𝑥²+13𝑥³+124𝛼(3𝑛+1)𝑥⁴+130𝑛(𝑛+1)𝑥⁵,𝑝⁽³⁾∶𝑓₃1(𝑥)=𝑥+2𝛼𝑥²+13𝑥³+124𝛼(3𝑛+1)𝑥⁴+130𝑛(𝑛+1)𝑥⁵+1𝛼120²(3𝑛+1)𝑥⁵+1720𝛼(19𝑛²+18𝑛+3)𝑥⁶+1315𝑛(2𝑛²+2𝑛+1)𝑥⁷,⋮. (6.30) The series solution is given as 𝑓(𝑥)=𝑥+𝛼𝑥²2+𝑛𝑥³31+(81𝑛𝛼+24𝛼)𝑥⁴1+(𝑛30²+140𝑛𝛼²+1𝛼120²+130𝑛)𝑥⁵+(19𝑛720²1𝛼+1240𝛼+40𝑛𝛼)𝑥⁶1+(120𝑛𝛼²+12315𝑛+𝑛315³+11𝛼5040²+3𝑛560²𝛼²+2𝑛315²)𝑥⁷+(11𝛼40320³+33𝑛4480²3𝛼+𝛼4480³𝑛²+2315760𝑛𝛼+2688𝛼+167𝑛40320³1𝛼+𝛼960³𝑛)𝑥⁸1+(3780𝑛+527𝑛362880³𝛼²+19𝑛11340³[phone]𝑛𝛼²+23𝑛8064²𝛼²+23𝑛22680²+13𝑛22680⁴+43𝛼120960²)𝑥⁹+⋯. (6.31)(1)(2)(3)(4)(5)(6)

Example 5. Consider the following nonlinear inhomogeneous Goursat problem: 𝑢_𝑥𝑡=−𝑢³+𝑥³+3𝑥²𝑡+3𝑥𝑡²+𝑡³,𝑢(𝑥,0)=𝑥,𝑢(0,𝑡)=𝑡,𝑢(0,0)=0. (6.32) The correction functional for the above problem is given by 𝑢_𝑛+1(𝑥,𝑡)=𝑢_𝑛(𝑥,𝑡)+^𝑡₀𝜕𝜆(𝑠)(²𝑢_𝑛+𝑢𝜕𝑥𝜕𝑠^3𝑛−𝑥³−3𝑥²𝑠−3𝑥𝑠²−𝑠³)𝑑𝑠,𝑛≥0. (6.33) Making the above correctional functional stationary, the Lagrange multiplier can be identified as 𝜆(𝑠)=−1; consequently 𝑢_𝑛+1(𝑥,𝑡)=𝑢_𝑛(𝑥,𝑡)−^𝑡₀(𝜕²𝑢_𝑛𝜕𝑥𝜕𝑠+𝑢³_𝑛−𝑥³−3𝑥²𝑠−3𝑥𝑠²−𝑠³)𝑑𝑠,𝑛≥0. (6.34) Applying the variational iteration method using He’s polynomials (5.1) (VIMHPS), we get 𝑢₀+𝑝𝑢₁+𝑝²𝑢₂+⋯=𝑢₀(𝑥,𝑡)−𝑝^𝑡₀𝜕((²𝑢₀𝜕𝜕𝑥𝜕𝑠+𝑝²𝑢₁𝜕𝑥𝜕𝑠+⋯)+(𝑢₀+𝑝𝑢₁+𝑝²𝑢₂+⋯)³−𝑥³−3𝑥²𝑠−3𝑥𝑠²−𝑠³)𝑑𝑠. (6.35) The initial approximation can be selected (by utilizing the boundary conditions) as 𝑢₀(𝑥,𝑡)=𝐴𝑥+𝐵𝑡, where 𝐴,𝐵 are constants and would be determined by using the boundary conditions; consequently 𝑢₀+𝑝𝑢₁+𝑝²𝑢₂+⋯=𝐴𝑥+𝐵𝑡−𝑝^𝑡₀𝜕((²𝑢₀𝜕𝜕𝑥𝜕𝑠+𝑝²𝑢₁𝜕𝑥𝜕𝑠+⋯)+(𝑢₀+𝑝𝑢₁+𝑝²𝑢₂+⋯)³−𝑥³−3𝑥²𝑠−3𝑥𝑠²−𝑠³)𝑑𝑠. (6.36) Comparing the coefficient of like powers of 𝑝, 𝑝⁽⁰⁾∶𝑢₀𝑝(𝑥,𝑡)=𝐴𝑥+𝐵𝑡,₍₁₎∶𝑢₁1(𝑥,𝑡)=𝐴𝑥+𝐵𝑡−4(𝐵³−1)𝑡⁴−(𝐴𝐵²−1)𝑥𝑡³−32(𝐴²𝐵−1)𝑥²𝑡²−𝐴³𝑥³𝑡+𝑥³𝑡,⋮. (6.37) Imposing the boundary conditions 𝑢(𝑥,0)=𝑥,𝑢(0,𝑡)=𝑡, to find the constants 𝐴 and 𝐵 will yield 𝐴=1, and 𝐵=1, consequently 𝑢(𝑥,𝑡)=𝑥+𝑡. (6.38)
 Figure 3 depicts the solution 𝑢(𝑥,𝑡).

Figure 3 

Example 6. Consider the following nonlinear differential equation which governs the unsteady flow of gas through a porous medium: 𝑦^′′(𝑥)+2𝑥𝑦1−𝛼𝑦^′(𝑥)=0,0<𝛼<1, (6.39) with the following typical boundary conditions imposed by the physical properties: 𝑦(0)=1,lim_𝑥→∞𝑦(𝑥)=0. (6.40) The correction functional is given as 𝑦_𝑛+1(𝑥)=𝑦_𝑛(𝑥)+^𝑥₀𝜆(𝑠)(𝑦^′′(𝑥)+2𝑠𝑦1−𝛼𝑦^′(𝑥))𝑑𝑠,0<𝛼<1. (6.41) Making the correction functional stationary, the Lagrange multipliers can be identified as 𝜆=𝑠−𝑥, and thus we get 𝑦_𝑛+1(𝑥)=𝑦_𝑛(𝑥)+^𝑥₀(𝑠−𝑥)(𝑦_𝑛^′′(𝑠)+2𝑠1−𝛼𝑦_𝑛𝑦(𝑠)^′𝑛(𝑠))𝑑𝑠,0<𝛼<1. (6.42) Applying the variational iteration method using He’s polynomials (VIMHPS), we get 𝑦₀+𝑝𝑦₁+𝑝²𝑦₂+⋯=𝑦₀+^𝑥₀(𝑠−𝑥)(𝑦₀^′′+𝑝𝑦₁^′′+𝑝²𝑦₂^′′+⋯)𝑑𝑠+2^𝑥₀(𝑠−𝑥)𝑥(1−𝛼(𝑦^′₀+𝑝𝑦₁+𝑝²𝑦₂+⋯))^−1/2(𝑦^′₀+𝑝𝑦^′₁+𝑝²𝑦^′₂+⋯)𝑑𝑠, (6.43) where 𝐴=𝑦^′(0). Comparing the coefficient of like powers of 𝑝, consequently, 𝑝⁽⁰⁾∶𝑦₀𝑝(𝑥)=1,₍₁₎∶𝑦₁𝑝(𝑥)=1+𝐴𝑥,₍₂₎∶𝑦₂𝐴(𝑥)=1+𝐴𝑥−3√𝑥1−𝛼³,𝑝⁽³⁾∶𝑦₃𝐴(𝑥)=1+𝐴𝑥−3√𝑥1−𝛼³−𝛼𝐴²12(1−𝛼)^3/2𝑥⁴+𝐴𝑥10(1−𝛼)⁵,⋮. (6.44) The series solution is given as 𝐴𝑦(𝑥)=1+𝐴𝑥−3√𝑥1−𝛼³−𝛼𝐴²12(1−𝛼)^3/2𝑥⁴𝐴+(−10(1−𝛼)3𝛼²𝐴³80(1−𝛼)^5/2)𝑥⁵+(𝛼𝐴²15(1−𝛼)²−𝛼³𝐴⁴48(1−𝛼)^7/2)𝑥⁶+⋯. (6.45)
Table 4 exhibits the initial slopes 𝐴=𝑦^′(0) for various values of 𝛼.
Table 5 exhibits the values of 𝑦(𝑥) for 𝛼=0.5 for 𝑥=0.1 to 1.0.

Table 1 

Padé approximants and initial slopes 𝑦^′(0) [56].

Table 2 

Numerical values for 𝛼=𝑓^′′(0) for 0<𝑛<1 by using diagonal Padé approximants [51].

Table 3 

Numerical values for 𝛼=𝑓^′′(0) for 𝑛>1 by using diagonal Padé approximants [51].

Table 4 

Table 5 

Example 7. Consider the following singularly perturbed sixth-order Boussinesq equation 𝑢_𝑡𝑡=𝑢_𝑥𝑥+(𝑝(𝑢))_𝑥𝑥+𝛼𝑢_{𝑥𝑥𝑥𝑥}+𝛽𝑢_{𝑥𝑥𝑥𝑥𝑥𝑥}. (6.46) Taking 𝛼=1,𝛽=0, and 𝑝(𝑢)=3𝑢², the model equation is given as 𝑢_𝑡𝑡=𝑢_𝑥𝑥+3(𝑢²)_𝑥𝑥+𝑢_{𝑥𝑥𝑥𝑥}, (6.47) with initial conditions 𝑢(𝑥,0)=2𝑎𝑘²𝑒^𝑘𝑥(1+𝑎𝑒^𝑘𝑥)²,𝑢_𝑡(𝑥,0)=2𝑎𝑘³1+𝑘²(1−𝑎𝑒^𝑘𝑥)𝑒^𝑘𝑥(1+𝑎𝑒^𝑘𝑥)³, (6.48) where a and k are arbitrary constants. The exact solution 𝑢(𝑥,𝑡) of the problem is given as 𝑢(𝑥,𝑡)=2𝑎𝑘²exp(𝑘𝑥+𝑘1+𝑘²𝑡)((1+𝑎exp(𝑘𝑥+𝑘1+𝑘²𝑡)))². (6.49) The correction functional is given by 𝑢_𝑛+1(𝑥,𝑡)=2𝑎𝑘²𝑒^𝑘𝑥(1+𝑎𝑒^𝑘𝑥)²+2𝑎𝑘³1+𝑘²(1−𝑎𝑒^𝑘𝑥)𝑒^𝑘𝑥(1+𝑎𝑒^𝑘𝑥)³𝑡+^𝑡₀𝜕𝜆(𝑠)(²𝑢_𝑛𝜕𝑡²−((𝑢_𝑛)_𝑥𝑥+3(𝑢_𝑛²)_𝑥𝑥+(𝑢_𝑛)_{𝑥𝑥𝑥𝑥}))𝑑𝑠. (6.50) Making the correction functional stationary, the Lagrange multiplier can easily be identified as 𝜆=𝑠−𝑥; consequently 𝑢_𝑛+1(𝑥,𝑡)=2𝑎𝑘²𝑒^𝑘𝑥(1+𝑎𝑒^𝑘𝑥)²+2𝑎𝑘³1+𝑘²(1−𝑎𝑒^𝑘𝑥)𝑒^𝑘𝑥(1+𝑎𝑒^𝑘𝑥)³𝑡+^𝑡₀𝜕(𝑠−𝑥)(²𝑢_𝑛𝜕𝑡²−((𝑢_𝑛)_𝑥𝑥+3(𝑢_𝑛²)_𝑥𝑥+(𝑢_𝑛)_{𝑥𝑥𝑥𝑥}))𝑑𝑠. (6.51) Applying the variational iteration method using He’s polynomials (VIMHPS), we get 𝑢₀+𝑝𝑢₁+𝑝²𝑢₂+𝑝³𝑢₃=+⋯2𝑎𝑘²𝑒^𝑘𝑥(1+𝑎𝑒^𝑘𝑥)²+2𝑎𝑘³1+𝑘²(1−𝑎𝑒^𝑘𝑥)𝑒^𝑘𝑥(1+𝑎𝑒^𝑘𝑥)³𝑡+𝑝^𝑡₀𝜕(𝑠−𝑥)((²𝑢₀𝜕𝑡²𝜕+𝑝²𝑢₁𝜕𝑡²+𝑝²𝜕²𝑢₂𝜕𝑡²𝜕+⋯)−(²𝑢₀𝜕𝑥²𝜕+𝑝²𝑢₁𝜕𝑥²+𝑝²𝜕²𝑢₂𝜕𝑥²+⋯))𝑑𝑠−𝑝^𝑡₀𝜕(𝑠−𝑥)((⁴𝑢₀𝜕𝑥⁴𝜕+𝑝⁴𝑢₁𝜕𝑥⁴+𝑝²𝜕⁴𝑢₂𝜕𝑥⁴𝜕+⋯)+3(²𝑢₀𝜕𝑥²𝜕+𝑝²𝑢₁𝜕𝑥²+𝑝²𝜕²𝑢₂𝜕𝑥²+⋯)²)𝑑𝑠. (6.52) Comparing the coefficients of like powers of 𝑝, 𝑝⁽⁰⁾∶𝑢₀(𝑥,𝑡)=2𝑒^𝑥(1+𝑒^𝑥)²𝑝₍₁₎,∶𝑢₁(𝑥,𝑡)=2𝑒^𝑥(1+𝑒^𝑥)²+2𝑎𝑘³1+𝑘²(1−𝑎𝑒^𝑘𝑥)𝑒^𝑘𝑥(1+𝑎𝑒^𝑘𝑥)³𝑡+2𝑒^𝑥(1−4𝑒^𝑥+𝑒^2𝑥)(1+𝑒^𝑥)⁴𝑡²𝑝₍₂₎,∶𝑢₂(𝑥,𝑡)=2𝑒^𝑥(1+𝑒^𝑥)²+2𝑎𝑘³1+𝑘²(1−𝑎𝑒^𝑘𝑥)𝑒^𝑘𝑥(1+𝑎𝑒^𝑘𝑥)³𝑡+2𝑒^𝑥(1−4𝑒^𝑥+𝑒^2𝑥)(1+𝑒^𝑥)⁴𝑡²−2√2𝑒^𝑥(−1+𝑒^𝑥)(1−10𝑒^𝑥+𝑒^2𝑥)3(1+𝑒^𝑥)⁵𝑡³+𝑒^𝑥(1−4𝑒^𝑥+𝑒^2𝑥)(1−44𝑒^𝑥+78𝑒^2𝑥−44𝑒^3𝑥+𝑒^4𝑥)3(1+𝑒^𝑥)⁸𝑡⁴,⋮. (6.53)(7)(8)(9)(10)(11)(12) The series solution is given as 𝑢(𝑥,𝑡)=2𝑒^𝑥(1+𝑒^𝑥)²+2𝑎𝑘³1+𝑘²(1−𝑎𝑒^𝑘𝑥)𝑒^𝑘𝑥(1+𝑎𝑒^𝑘𝑥)³𝑡+2𝑒^𝑥(1−4𝑒^𝑥+𝑒^2𝑥)(1+𝑒^𝑥)⁴𝑡²−2√2𝑒^𝑥(−1+𝑒^𝑥)(1−10𝑒^𝑥+𝑒^2𝑥)3(1+𝑒^𝑥)⁵𝑡³+𝑒^𝑥(1−4𝑒^𝑥+𝑒^2𝑥)(1−44𝑒^𝑥+78𝑒^2𝑥−44𝑒^3𝑥+𝑒^4𝑥)3(1+𝑒^𝑥)⁸𝑡⁴+8𝑒^2𝑥(1−10𝑒^𝑥+20𝑒^2𝑥−10𝑒^3𝑥+𝑒^4𝑥)(1+𝑒^𝑥)⁸𝑡⁴−√2𝑒^𝑥(−1+𝑒^𝑥)(1−56𝑒^𝑥+246𝑒^2𝑥−56𝑒^3𝑥+𝑒^4𝑥)15(1+𝑒^𝑥)⁷𝑡⁵+𝑒^𝑥(1−452𝑒^𝑥+19149𝑒^2𝑥−207936𝑒^3𝑥+807378𝑒^4𝑥−1256568𝑒^5𝑥)45(1+𝑒^𝑥)¹²𝑡⁶+𝑒^𝑥(807378𝑒^6𝑥−207936𝑒^7𝑥+19149𝑒^8𝑥−452𝑒^9𝑥+𝑒^10𝑥)45(1+𝑒^𝑥)¹²𝑡⁶+⋯. (6.54)
 Table 6 exhibits the absolute error between the exact and the series solutions. Higher accuracy can be obtained by introducing some more components of the series solution.
Figure 4 depicts the series solution 𝑢(𝑥,𝑡).

Table 6 

Error estimates.

Table 7 

Numerical results for Fisher’s equation.

Table 8 

Error estimates (𝛿=.01).

Figure 4 

Example 8. Consider Fisher’s equation of the following form: 𝑢_𝑡(𝑥,𝑡)−𝑢_𝑥𝑥(𝑥,𝑡)−𝑢(𝑥,𝑡)(1−𝑢(𝑥,𝑡))=0, (6.55) with initial conditions 𝑢(𝑥,0)=𝛽. (6.56) The correction functional is given by 𝑢_𝑛+1(𝑥,𝑡)=𝑢_𝑛(𝑥,𝑡)+^𝑡₀𝜆(𝜕𝑢_𝑛(𝑥,𝑠)−𝜕𝜕𝑠²𝑢_𝑛(𝑥,𝑠)𝜕𝑥²−𝑢_𝑛(𝑥,𝜏)(1−𝑢_𝑛(𝑥,𝑠)))𝑑𝑠. (6.57) Making the above correction functional stationary, the Lagrange multiplier can be identified as 𝜆(𝑠)=−1; consequently, we obtain the following iteration formula: 𝑢_𝑛+1(𝑥,𝑡)=𝑢_𝑛(𝑥,𝑡)−^𝑡₀(𝜕𝑢_𝑛(𝑥,𝑠)−𝜕𝜕𝑠²𝑢_𝑛(𝑥,𝑠)𝜕𝑥²−𝑢_𝑛(𝑥,𝑠)(1−𝑢_𝑛(𝑥,𝑠)))𝑑𝑠. (6.58) Applying the variational iteration method using He’s polynomials (5.1) (VIMHPS), we get 𝑢₀+𝑝𝑢₁+𝑝²𝑢₂+⋯=𝑢₀(𝑥,𝑡)−𝑝^𝑡₀((𝜕𝑢₀(𝑥,𝑠)𝜕𝑠+𝑝𝜕𝑢₁(𝑥,𝑠)𝜕𝑠+𝑝²𝜕𝑢₂(𝑥,𝑠)𝜕𝑠+⋯))𝑑𝑠+𝑝^𝑡₀𝜕((²𝑢₀(𝑥,𝑠)𝜕𝑥²𝜕+𝑝²𝑢₁(𝑥,𝑠)𝜕𝑥²+⋯)+((𝑢₀+𝑝𝑢₁+⋯)(1−(𝑢₀+𝑝𝑢₁+⋯))))𝑑𝑠. (6.59) Comparing the coefficient of like powers of 𝑝, following approximants are obtained: 𝑝⁽⁰⁾∶𝑢₀𝑝(𝑥,𝑡)=𝛽,₍₁₎∶𝑢₁𝑝(𝑥,𝑡)=𝛽+𝛽(1−𝛽)𝑡,₍₂₎∶𝑢₂𝑡(𝑥,𝑡)=𝛽+𝛽(1−𝛽)𝑡+²2!𝛽(1−3𝛽+2𝛽²𝑡)−³3𝛽²(−1+𝛽)²,𝑝⁽³⁾∶𝑢₃𝑡(𝑥,𝑡)=𝛽+𝛽(1−𝛽)𝑡+³3!𝛽(1−𝛽)(1−6𝛽+6𝛽²𝑡)+⁴3(−1+𝛽)²𝛽²−𝑡(−1+2𝛽)₅60(−1+𝛽)²𝛽²(3−20𝛽+20𝛽²𝑡)+⁶18(−1+𝛽)³𝛽³𝑡(−1+2𝛽)−⁷63(−1+𝛽)⁴𝛽⁴,⋮. (6.60) The series solution is given by 𝑡𝑢(𝑥,𝑡)=𝛽+𝛽(1−𝛽)𝑡+³3!𝛽(1−𝛽)(1−6𝛽+6𝛽²𝑡)+⁴3(−1+𝛽)²𝛽²−𝑡(−1+2𝛽)₅60(−1+𝛽)²𝛽²(3−20𝛽+20𝛽²𝑡)+⁶18(−1+𝛽)³𝛽³𝑡(−1+2𝛽)−⁷63(−1+𝛽)⁴𝛽⁴+⋯, (6.61) and in a closed-form by 𝑢(𝑥,𝑡)=𝛽exp𝑡1−𝛽+𝛽exp𝑡. (6.62)

Example 9. Consider the following seventh-order generalized KdV (SOG-KdV) equation: 𝑢_𝑡+𝑢𝑢_𝑥+𝑢_𝑥𝑥𝑥−𝑢_{𝑥𝑥𝑥𝑥𝑥}+𝜎𝑢_{𝑥𝑥𝑥𝑥𝑥𝑥𝑥}=0, (6.63) where 𝜎=𝛽𝛿/𝛾², with initial conditions 𝑢(𝑥,0)=𝑎₀+𝑎₆secℎ⁶(𝑘𝑥), (6.64) where 𝐾=5/√1538,𝑎₀=𝑐−18000/769²,𝑎₆=519750/769², and c is an arbitrary parameter. The correction functional is given as 𝑢_𝑛+1(𝑥,𝑡)=𝑢_𝑛+^𝑡₀𝜆(𝑠)(𝜕𝑢_𝑛+𝑢𝜕𝑠^𝑛𝜕𝑢_𝑛+𝜕𝜕𝑥³𝑢_𝑛𝜕𝑥³−𝜕⁵𝑢_𝑛𝜕𝑥⁵𝜕+𝜎⁷𝑢_𝑛𝜕𝑥⁷)𝑑𝑠. (6.65) Making the correction functional stationary, the Lagrange multiplier can be identified as 𝜆(𝑠)=−1; consequently 𝑢_𝑛+1(𝑥,𝑡)=𝑢_𝑛−^𝑡₀(𝜕𝑢_𝑛𝜕𝑠+𝑢_𝑛𝜕𝑢_𝑛+𝜕𝜕𝑥³𝑢_𝑛𝜕𝑥³−𝜕⁵𝑢_𝑛𝜕𝑥⁵𝜕+𝜎⁷𝑢_𝑛𝜕𝑥⁷)𝑑𝑠. (6.66) Applying the variational iteration method using He’s polynomials (VIMHPS), we get 𝑢₀+𝑝𝑢₁+𝑝²𝑢₂+⋯=𝑢₀−𝑝^𝑡₀((𝜕𝑢₀𝜕𝑠+𝑝𝜕𝑢₁𝜕𝑠+𝑝²𝜕𝑢₂𝜕𝑠+⋯)+(𝑢₀𝜕𝑢₀𝜕𝑥+𝑝𝑢₁𝜕𝑢₁𝜕𝑥+𝑝²𝑢₂𝜕𝑢₂𝜕𝑥+⋯))𝑑𝑠−𝑝^𝑡₀𝜕((³𝑢₀𝜕𝑥³𝜕+𝑝³𝑢₁𝜕𝑥³𝜕+⋯)−(⁵𝑢₀𝜕𝑥⁵𝜕+𝑝⁵𝑢₁𝜕𝑥⁵𝜕+⋯)+𝜎(⁷𝑢₀𝜕𝑥⁷𝜕+𝑝⁷𝑢₁𝜕𝑥⁷+⋯))𝑑𝑠. (6.67) Comparing the coefficient of like powers of 𝑝, 𝑝⁽⁰⁾∶𝑢₀(𝑥,𝑡)=𝑎₀+𝑎₆secℎ⁶𝑝(𝑘𝑥),₍₁₎∶𝑢₁(𝑥,𝑡)=𝑎₀+𝑎₆secℎ⁶(𝑘𝑥)+3𝑎₆𝑘𝑡16×{10𝑎₀+32𝑎₆−312𝑘²−26400𝑘⁴−9866112𝛿𝑘⁶+(15𝑎₀−256𝑘²−10480𝑘⁴+9932224𝛿𝑘⁶)cosh(2𝑘𝑥)+(6𝑎₀−8𝑘²+14624𝑘⁴−1443968𝛿𝑘⁶)cosh(4𝑘𝑥)+(𝑎₀+36𝑘²−1296𝑘⁴+46656𝛿𝑘⁶)cosh(6𝑘𝑥)}secℎ¹²(𝑘𝑥)tanh(𝑘𝑥),⋮. (6.68) The series solution is given by 𝑢(𝑥,𝑡)=𝑎₀+𝑎₆secℎ⁶(𝑘𝑥)+3𝑎₆𝑘𝑡16×{10𝑎₀+32𝑎₆−312𝑘²−26400𝑘⁴−9866112𝛿𝑘⁶+(15𝑎₀−256𝑘²−10480𝑘⁴+9932224𝛿𝑘⁶)cosh(2𝑘𝑥)+(6𝑎₀−8𝑘²+14624𝑘⁴−1443968𝛿𝑘⁶)cosh(4𝑘𝑥)+(𝑎₀+36𝑘²−1296𝑘⁴+46656𝛿𝑘⁶)cosh(6𝑘𝑥)}secℎ¹²(𝑘𝑥)tanh(𝑘𝑥)−3𝑎₆𝑘𝑡²4096×[1320𝑎²₀+6400𝑎₀𝑎₆+40960𝑎²₆−93120𝑎₀𝑘²−642560𝑎₆𝑘²+3592320𝑘⁴−718464𝑎₀𝑘⁴−223897600𝑎₆𝑘⁴[phone]𝑘⁶−1066859520𝑎₀𝛿𝑘⁶−371744399360𝑎₆𝛿𝑘⁶[phone]𝑘⁸[phone]𝛿𝑘¹⁰[phone]𝛿²𝑘¹²+(2079𝑎²₀+6912𝑎₀𝑎₆−36864𝑎²₆−124488𝑎₀𝑘²−291840𝑎₆𝑘²+3662064𝑘⁴−7324128𝑎₀𝑘⁴[phone]𝑎₆𝑘⁴[phone]𝑘⁶−643886208𝑎₀𝛿𝑘⁶[phone]𝑎₆𝛿𝑘⁶[phone]𝑘⁸[phone]𝛿𝑘⁸−115427212572672𝛿𝑘¹⁰−318167001264623616𝛿²𝑘¹²)cosh(2𝑘𝑥)+(924𝑎²₀−1536𝑎₀𝑎₆−14112𝑎₀𝑘²+592896𝑎₆𝑘²−1544256𝑘⁴+3088512𝑎₀𝑘⁴[phone]𝑎₆𝑘⁴−1003580928𝑘⁶[phone]𝑎₀𝛿𝑘⁶−173675003904𝑎₆𝛿𝑘⁶−180653147136𝑘⁸−361306294272𝛿𝑘⁸−159045011693568𝛿𝑘¹⁰+129148767835766784𝛿²𝑘¹²)cosh(4𝑘𝑥)+(77𝑎²₀−2816𝑎₀𝑎₆+36904𝑎₀𝑘²−2283568𝑘⁴+4567136𝑎₀𝛿𝑘⁶[phone]𝑘⁸[phone]𝛿𝑘⁸[phone]𝛿𝑘¹⁰−27618434663723008𝛿²𝑘¹²)cosh(6𝑘𝑥)−(168𝑎²₀+768𝑎₀𝑎₆−25536𝑎₀𝑘²+69120𝑎₆𝑘²+594048𝑘⁴−1188096𝑎₀𝑘⁴−8460288𝑎₆𝑘⁴−118291504𝑘⁶[phone]𝑎₀𝛿𝑘⁶[phone]𝑎₆𝛿𝑘⁶−35208886272𝑘⁸−70417772544𝛿𝑘⁸[phone]𝛿𝑘¹⁰−2901989324193792𝛿²𝑘¹²)cosh(8𝑘𝑥)−(105𝑎²₀−5880𝑎₀𝑘²−132720𝑘⁴−67643520𝑘⁶[phone]𝑎₀𝛿𝑘⁶[phone]𝑘⁸[phone]𝛿𝑘⁸−2194541905920𝛿𝑘¹⁰[phone]𝛿²𝑘¹²)cosh(10𝑘𝑥)−(28𝑎²₀+224𝑎₀𝑘²−53312𝑘⁴+106626𝑎₀𝑘⁴[phone]𝑘⁶−11178496𝑎₀𝛿𝑘⁶−436093852𝑘⁸−872187904𝛿𝑘⁸[phone]𝛿𝑘¹⁰−2068416315392𝛿²𝑘¹²)cosh(12𝑘𝑥)−(3𝑎²₀+216𝑎₀𝑘²+3888𝑘⁴−7776𝑎₀𝑘⁴−279936𝑘⁶+279936𝑎𝛿𝑘⁶+5038848𝑘⁸[phone]𝛿𝑘⁸−362797056𝛿𝑘¹⁰[phone]𝛿²𝑘¹²)cosh(14𝑘𝑥)]secℎ²⁰(𝑘𝑥)+⋯. (6.69)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27)(28)(29)(30)(31)(32)(33)(34)(35)(36)(37)(38)(39)(40)(41)(42)(43)(44)(45)(46)(47) The closed-form solution is given as 𝑢(𝑥,𝑡)=𝑎₀+𝑎₆secℎ⁶(𝑘(𝑥−𝑐𝑡)), (6.70) where 𝑘=5/√1538,𝑎₀=𝑐−18000/769²,𝑎₆=519750/769², and c is an arbitrary parameter.
Figure 5 depicts series solution at 𝛿=.01.

Figure 5 

7. Conclusion

In this paper, we applied variational iteration method using He’s polynomials (VIMHPS) for solving various initial and boundary value problems. The proposed modified method (VIMHPS) is employed without using linearization, discretization, transformation, or restrictive assumptions, absorb the positive features of the coupled techniques, and hence is very much compatible with the diversified and versatile nature of the physical problems. Moreover, the modification based upon He’s polynomials (VIMHPS) is easier to implement and is more user friendly as compared to the decomposition method where Adomian’s polynomials along with their complexities are used.

Acknowledgment

The authors are highly grateful to the referees for their constructive comments.