A Novel Iterative Scheme and Its Application to Differential Equations

Abstract

The purpose of this paper is to employ an alternative approach to reconstruct the standard variational iteration algorithm II proposed by He, including Lagrange multiplier, and to give a simpler formulation of Adomian decomposition and modified Adomian decomposition method in terms of newly proposed variational iteration method-II (VIM). Through careful investigation of the earlier variational iteration algorithm and Adomian decomposition method, we find unnecessary calculations for Lagrange multiplier and also repeated calculations involved in each iteration, respectively. Several examples are given to verify the reliability and efficiency of the method.

1. Introduction

Over the last few decades several analytical/approximate methods have been developed to solve nonlinear ordinary and partial differential equations. For initial and boundary-value problems in ordinary and partial differential equations, some of these techniques include the perturbation method [1], the variational iteration method [2–4], the decomposition method [5–8], and the homotopy methods [9–11].

The Adomian decomposition method [12–16] for solving differential and integral equations, linear or nonlinear, has been the subject of extensive analytical and numerical studies. The method, well addressed in [12–16], has a significant advantage in which it provides the solution in a rapid convergent series with elegantly computable components. In recent years, a large amount of literature has been developed concerning the application of Adomian decomposition method in applied sciences. In addition, the method reveals the analytical structure of the solution which is absent in numerical solutions.

He’s variational iteration method [2–4] is based on a Lagrange multiplier technique developed by Inokuti et al. [17]. This method is, in fact, a modification of the general Lagrange multiplier method into an iteration method, which is called correction functional. The method has been shown to solve effectively, easily, and accurately a large class of nonlinear problems [18–23]. Generally, one or two iterations lead to high accurate solutions.

In the present study, we have linked up variational iteration method and Adomian decomposition method through Lagrange multiplier, which shows that VIM is another form of expressing ADM and vice versa. This study reveals that there is no need to integrate the differential equation again and again as we do in Adomian decomposition method. Advantage of new iterative scheme over the variational iteration method is that it avoids the unnecessary calculations and we can construct Lagrange multiplier very easily without construction of the correctional functional.

2. New Formulation for Adomian Decomposition Method and Variational Iteration Algorithm II

In order to elucidate the solution procedure, we consider the following 𝑛th order partial differential equation: 𝐿^𝑛𝑓(𝑥,𝑡)=𝑅𝑓(𝑥,𝑡)+𝑁𝑓(𝑥,𝑡)+𝑔(𝑥,𝑡),𝑡>0,𝑥∈𝐿, (1) where 𝐿^𝑛=𝜕^𝑛/𝜕𝑡^𝑛, 𝑛≥1, 𝑅 is a linear differential operator, 𝑁 is a nonlinear differential operator, 𝑅 and 𝑁 are free of partial derivative with respect to variable 𝑡, and 𝑔 is the source term. As we are familiar with the fact that in all kinds of iteration techniques, except the operator rest of the terms, are treated as a known function on the behalf of initial guess. In this present newly proposed idea, we have used the same concept. We have bound all terms in one function except operator. Consider 𝑔+𝑁𝑓+𝑅𝑓=𝐹𝑡,𝑥,𝑔,𝑓,𝜕𝑓,𝜕𝜕𝑥²𝑓𝜕𝑥²,…. (2) By incorporating (2) in (1), we get 𝐿^𝑛𝑓=𝐹𝑡,𝑥,𝑔,𝑓,𝜕𝑓,𝜕𝜕𝑥²𝑓𝜕𝑥²,…. (3) On integrating (3), we obtain 𝐿^(𝑛−1)𝑓=^𝑡₀𝐹𝜉,𝑥,𝑔,𝑓,𝜕𝑓,𝜕𝜕𝑥²𝑓𝜕𝑥²,…𝑑𝜉+𝑐₁(𝑥). (4) Again, by integrating (4), we have 𝐿^(𝑛−2)𝑓=^𝑡₀^𝜉₀𝐹𝜏,𝑥,𝑔,𝑓,𝜕𝑓,𝜕𝜕𝑥²𝑓𝜕𝑥²,…𝑑𝜏𝑑𝜉+𝑐₁(𝑥)𝑡+𝑐₂(𝑥), (5) since we know that multiple integral can be reduce to a single integral by using integral property. Hence, we can write (5) in the following form: 𝐿^(𝑛−2)𝑓=^𝑡₀(𝑡−𝜉)𝐹𝜉,𝑥,𝑔,𝑓,𝜕𝑓,𝜕𝜕𝑥²𝑓𝜕𝑥²,…𝑑𝜉+𝑐₁(𝑥)𝑡+𝑐₂(𝑥). (6) If we continue this process of integration, we can get final form as follows: 𝑓(𝑥,𝑡)=^𝑡₀(𝑡−𝜉)^𝑛−1()!𝐹𝑛−1𝑡,𝑥,𝑔,𝑓,𝜕𝑓,𝜕𝜕𝑥²𝑓𝜕𝑥²+𝑐,…𝑑𝜉₁(𝑥)𝑡^𝑛−1()!+𝑐𝑛−1²(𝑥)𝑡^𝑛−2(𝑛−2)!+⋯𝑐_𝑛(𝑥). (7) By writing the constant of integration in the form 𝑐_𝑘(𝑥)=(𝜕𝑓^𝑛−𝑘(𝑥,0⁺))/𝜕𝑡^𝑛−𝑘, 𝑘=1,…,𝑛 and substituting (2) in (7) then (7), we have 𝑓(𝑥,𝑡)=^𝑛−1_𝑘=0𝜕^𝑘𝑓𝑥,0⁺𝜕𝑡^𝑘𝑡^𝑘+𝑘!_𝑡0(𝑡−𝜉)^𝑛−1(𝑛−1)!𝑅𝑓+𝑁𝑓+𝑔𝑑𝜉. (8) In iteration form (8), it can be written as follows: 𝑓_𝑗+1(𝑥,𝑡)=𝑓₀(𝑥,𝑡)+^𝑡₀(𝑡−𝜉)^𝑛−1(𝑛−1)!𝑅𝑓_𝑗+𝑁𝑓_𝑗+𝑔𝑑𝜉,𝑗=0,1,2,…, (9) where 𝑓₀(𝑥,𝑡)=∑^𝑛−1_𝑘=0((𝜕^𝑘𝑓(𝑥,0⁺))/𝜕𝑡^𝑘)(𝑡^𝑘/𝑘!).

In (9), (𝑡−𝜉)^𝑛−1/(𝑛−1)! is Lagrange multiplier of He’s variational iteration method, denoted by 𝜆, if 𝑛 is an odd integer, and (9) can be written in standard variational iteration algorithm II [3] 𝑓_𝑗+1(𝑥,𝑡)=𝑓₀(𝑥,𝑡)+^𝑡₀𝜆𝑅𝑓_𝑗+𝑁𝑓_𝑗𝑓+𝑔𝑑𝜉,₀(𝑥,𝑡)=^𝑛−1_𝑘=0𝜕^𝑘𝑓𝑥,0⁺𝜕𝑡^𝑘𝑡^𝑘()𝑘!,𝜆=𝑡−𝜉^𝑛−1()!.𝑛−1 (10) Equation (10) is exactly the same as the standard He’s variational iteration algorithm II [3]. Here is a point to be noted, if we change our initial guess by adding source term in it, the resulting formulation will give the results obtained by well-known Adomian decomposition method by decomposing the nonlinear term in (10). Consider 𝑓_𝑗+1(𝑥,𝑡)=^𝑡₀𝜆𝑅𝑓_𝑗+𝑁𝑓_𝑗𝑓𝑑𝜉,₀()()()𝑥,𝑡=𝐻𝑥,𝑡,𝜆=𝑡−𝜉^𝑛−1()!,𝐻()=𝑛−1𝑥,𝑡^𝑛−1_𝑘=0𝜕𝑓𝑥,0⁺𝜕𝑡^𝑘𝑡^𝑘+𝑘!^𝑡₀𝜆𝑔(𝑥,𝜉)𝑑𝜉. (11) Equation (11) is an alternative approach of Adomian decomposition method, where 𝐻(𝑥,𝑡) is a term which arises from prescribed initial condition and source term. Furthermore, if we decompose the term 𝐻(𝑥,𝑡) in (11) and write the resulting equation in the form 𝑓₁(𝑥,𝑡)=𝐻₁(𝑥,𝑡)+^𝑡₀𝜆𝑅𝑓₀+𝑁𝑓₀𝐻()=𝑑𝜉,𝑥,𝑡^𝑛−1_𝑘=0𝜕𝑓𝑥,0⁺𝜕𝑡^𝑘𝑡^𝑘+𝑘!^𝑡₀()𝐻()𝜆𝑔𝑥,𝜉𝑑𝜉,𝑥,𝑡=𝐻₀(𝑥,𝑡)+𝐻₁()()𝑥,𝑡,𝜆=𝑡−𝜉^𝑛−1()!,𝑓𝑛−1₀(𝑥,𝑡)=𝐻₀(),𝑓𝑥,𝑡_𝑗+1(𝑥,𝑡)=^𝑡₀𝜆𝑅𝑓_𝑗+𝑁𝑓_𝑗𝑑𝜉,𝑗≥1, (12)(13) equation (12) is an alternative form of modified Adomian decomposition method.

3. Illustrative Examples

In order to illustrate the solution procedure, we consider the following examples for ordinary and partial differential equations.

Example 1. Consider the Blasius equation 𝑢^′′′(𝑥)+12𝑢(𝑥)𝑢^′′(𝑥)=0, (14) subject to the boundary conditions 𝑢(0)=0,𝑢^′(0)=1,𝑢^′⟶0,𝑥⟶∞. (15) To solve the above given problem, we consider an extra initial condition; that is, 𝑢^′′(0)=𝛼. In order to solve (14) with this extra initial condition, we follow the formulation given in (10). Consider 𝑢_𝑗+1(𝑥)=𝑢₀(𝑥)−^𝑥₀𝜆2𝑢_𝑗(𝜉)𝑢_𝑗^′′(𝜉)𝑢𝑑𝜉,₀(𝑥)=𝑢(0)+𝑥𝑢^′(0)+𝑥²𝑢2!^′′(𝑥)𝑥=𝑥+²𝛼,()2!𝜆=𝑥−𝜉².2! (16)(17) By using (16), we obtain the following successive approximations: 𝑢₁(𝑥)=𝑥+𝛼𝑥²2−𝛼𝑥⁴−𝛼48²𝑥⁵,𝑢240₂(𝑥)=𝑥+𝛼𝑥²2−𝛼𝑥⁴−𝛼48²𝑥⁵+240𝛼𝑥⁶+96011𝛼²𝑥⁷+2016011𝛼³𝑥⁸−𝛼161280²𝑥⁹−𝛼193536³𝑥¹⁰−𝛼518400⁴𝑥¹¹,⋮5702400 (18) Equation (18) is the exactly the same as obtained by using classical VIM in [20] and one can find the value of 𝛼 by using Padé approximant [21].

Example 2. Consider the nonhomogeneous wave equation 𝜕²𝑢(𝑥,𝑡)𝜕𝑡²=𝜕²𝑢(𝑥,𝑡)𝜕𝑥²+𝜂(𝑥,𝑡), (19) where 𝜂(𝑥,𝑡)=2𝑒^−𝜋𝑡sin𝜋𝑥, subject to the initial conditions 𝑢(𝑥,0)=sin𝜋𝑥,𝑢_𝑡(𝑥,0)=−𝜋sin𝜋𝑥, (20) whose exact solution is 𝑢(𝑥,𝑡)=𝑒^−𝜋𝑡sin𝜋𝑥. (21) To solve (19), we follow the formulation, given in (11). Consider 𝑢_𝑗+1(𝑥,𝑡)=^𝑡₀𝜆𝜕²𝑢_𝑗𝜕𝑥²𝑢𝑑𝜉,₀()()(),𝐻()+𝑥,𝑡=𝐻𝑥,𝑡,𝜆=𝑡−𝜉𝑥,𝑡=sin𝜋𝑥−𝑡𝜋sin𝜋𝑥_𝑡0(𝑡−𝜉)2𝜋²𝑒^−𝜋𝜉𝑢sin𝜋𝑥𝑑𝜉,₀(𝑥,𝑡)=𝐻(𝑥,𝑡)=−sin𝜋𝑥+𝑡𝜋sin𝜋𝑥+2𝑒^−𝜋𝑡𝑢sin𝜋𝑥_𝑗+1(𝑥,𝑡)=^𝑡₀()𝜕𝑡−𝜉²𝑢_𝑗𝜕𝑥²𝑢𝑑𝜉,₁()=𝜋𝑥,𝑡2−2𝜋𝑡+²𝑡²−𝜋2!³𝑡³3!sin𝜋𝑥−2𝑒^−𝜋𝑡𝑢sin𝜋𝑥,₂(𝑥,𝑡)=−2+2𝜋𝑡−𝜋²𝑡²+𝜋³𝑡³3−𝜋⁴𝑡⁴+𝜋4!⁵𝑡⁵5!sin𝜋𝑥−2𝑒^−𝜋𝑡𝑢sin𝜋𝑥,₃(𝑥,𝑡)=2−2𝜋𝑡+𝜋²𝑡²−𝜋³𝑡³3+𝜋⁴𝑡⁴3(4)−𝜋⁵𝑡⁵3(45)+𝜋)(⁶𝑡⁶−𝜋6!⁷𝑡⁷7!sin𝜋𝑥−2𝑒^−𝜋𝑡⋮sin𝜋𝑥, (22) Upon summing these iterations, we observe that 𝑢()=𝜋𝑥,𝑡1−𝜋𝑡+²𝑡²−𝜋2!³𝑡³+𝜋3!⁴𝑡⁴−𝜋4!⁵𝑡⁵+𝜋5!₆𝑡⁶−𝜋6!⁷𝑡⁷7!+⋯sin𝜋𝑥≈𝑒^−𝜋𝑡sin𝜋𝑥. (23) Solution (23) is exactly the same as obtained by using ADM in [22].

4. Conclusion

This paper helps us to gain insight into the idea of Adomian decomposition method and variational iteration method. By keeping in view both methods, we propose more simplified forms to calculate Lagrange multipliers. By introducing this Lagrange multiplier in ADM and VIM following the observations that have been made,(i)there is no need to do integration process again and again like we do in Adomian decomposition method and one can get the same results of Adomian method.(ii)It is easy to calculate the Lagrange multiplier of He’s variational iteration method.(iii)This new approach avoids the unnecessary calculations like we did in He’s variational iteration method and Adomian decomposition method.(iv)This study shows that VIM is another form of expressing ADM and vice versa.

So we can say that the present method is parallel form of ADM and can give good results of VIM with less effort.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Authors’ Contribution

The authors have made the same contribution. All authors read and approved the final paper.

Acknowledgments

The authors are grateful to the reviewers for their comments and useful suggestions and the third author was supported by Project no. FEKT-S-14-2200 of Faculty of Electrical Engineering and Communication, Brno University of Technology, Czech Republic.