A novel iterative scheme and its application to differential equations

Research Article A Novel Iterative Scheme and Its Application to Differential Equations Yasir Khan,1F. Naeem,2and Zdenjk Šmarda3 1Department of Mathematics, Zhejiang University, Hangzhou 310027, China 2Modern Textile Institute, Donghua University, Shanghai 200051, China 3Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technicka8,61600Brno,CzechRepublic Correspondence should be addressed to Yasir Khan; [email protected]m Received 18 December 2013; Accepted 18 February 2014; Published 16 March 2014 AcademicEditors:H.JafariandC.M.Khalique Copyright © 2014 Yasir Khan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 boundaryvalue 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]forsolving 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]isbasedona 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 Lagrangemultiplier,whichshowsthatVIMisanotherformof 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: 𝐿𝑛𝑓(𝑥,𝑡)=𝑅𝑓(𝑥,𝑡)+𝑁𝑓(𝑥,𝑡)+𝑔(𝑥,𝑡),𝑡>0,𝑥∈𝐿, (1) Hindawi Publishing Corporation e Scientific World Journal Volume 2014, Article ID 605376, 4 pages http://dx.doi.org/10.1155/2014/605376 2The Scientific World Journal where 𝐿𝑛=𝜕𝑛/𝜕𝑡𝑛,𝑛≥1,𝑅is a linear differential operator, 𝑁 is a nonlinear differential operator, 𝑅and 𝑁are free of partial derivative with respect to variable 𝑡,and𝑔is 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 presentnewlyproposedidea,wehaveusedthesameconcept. We have bound all terms in one function except operator. Consider 𝑔+𝑁𝑓+𝑅𝑓=𝐹(𝑡,𝑥,𝑔,𝑓,𝜕𝑓 𝜕𝑥,𝜕2𝑓 𝜕𝑥2,...).(2) By incorporating (2)in(1), we get 𝐿𝑛𝑓=𝐹(𝑡,𝑥,𝑔,𝑓,𝜕𝑓 𝜕𝑥,𝜕2𝑓 𝜕𝑥2,...).(3) On integrating (3), we obtain 𝐿(𝑛−1)𝑓=∫𝑡 0𝐹(𝜉,𝑥,𝑔,𝑓,𝜕𝑓 𝜕𝑥,𝜕2𝑓 𝜕𝑥2,...)𝑑𝜉+𝑐1(𝑥).(4) Again, by integrating (4), we have 𝐿(𝑛−2)𝑓=∫𝑡 0∫𝜉 0𝐹(𝜏,𝑥,𝑔,𝑓,𝜕𝑓 𝜕𝑥,𝜕2𝑓 𝜕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=∫𝑡 0(𝑡−𝜉)𝐹(𝜉,𝑥,𝑔,𝑓,𝜕𝑓 𝜕𝑥,𝜕2𝑓 𝜕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)=∫𝑡 0(𝑡−𝜉)𝑛−1 (𝑛−1)!𝐹(𝑡,𝑥,𝑔,𝑓,𝜕𝑓 𝜕𝑥,𝜕2𝑓 𝜕𝑥2,...)𝑑𝜉 +𝑐1(𝑥)𝑡𝑛−1 (𝑛−1)!+𝑐2(𝑥)𝑡𝑛−2 (𝑛−2)!+⋅⋅⋅𝑐𝑛(𝑥).(7) By writing the constant of integration in the form 𝑐𝑘(𝑥) = (𝜕𝑓𝑛−𝑘(𝑥,0+))/𝜕𝑡𝑛−𝑘,𝑘=1,...,𝑛and substituting (2)in(7) then (7), we have 𝑓(𝑥,𝑡)=𝑛−1 ∑ 𝑘=0𝜕𝑘𝑓(𝑥,0+) 𝜕𝑡𝑘𝑡𝑘 𝑘! +∫𝑡 0(𝑡−𝜉)𝑛−1 (𝑛−1)!(𝑅𝑓+𝑁𝑓+𝑔)𝑑𝜉. (8) In iteration form (8),itcanbewrittenasfollows: 𝑓𝑗+1 (𝑥,𝑡)=𝑓0(𝑥,𝑡)+∫𝑡 0(𝑡−𝜉)𝑛−1 (𝑛−1)!(𝑅𝑓𝑗+𝑁𝑓𝑗+𝑔)𝑑𝜉, 𝑗=0,1,2,...,(9) where 𝑓0(𝑥,𝑡)=∑𝑛−1 𝑘=0((𝜕𝑘𝑓(𝑥,0+))/𝜕𝑡𝑘)(𝑡𝑘/𝑘!). In (9), (𝑡−𝜉)𝑛−1/(𝑛−1)!is Lagrange multiplier of He’s variational iteration method, denoted by 𝜆,if𝑛is an odd integer, and (9) can be written in standard variational iteration algorithm II [3] 𝑓𝑗+1 (𝑥,𝑡)=𝑓0(𝑥,𝑡)+∫𝑡 0𝜆(𝑅𝑓𝑗+𝑁𝑓𝑗+𝑔)𝑑𝜉, 𝑓0(𝑥,𝑡)=𝑛−1 ∑ 𝑘=0𝜕𝑘𝑓(𝑥,0+) 𝜕𝑡𝑘𝑡𝑘 𝑘!,𝜆= (𝑡−𝜉)𝑛−1 (𝑛−1)!.(10) Equation (10)isexactlythesameasthestandardHe’svariational 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 wellknown Adomian decomposition method by decomposing the nonlinear term in (10). Consider 𝑓𝑗+1 (𝑥,𝑡)=∫𝑡 0𝜆(𝑅𝑓𝑗+𝑁𝑓𝑗)𝑑𝜉, 𝑓0(𝑥,𝑡)=𝐻(𝑥,𝑡),𝜆= (𝑡−𝜉)𝑛−1 (𝑛−1)!, 𝐻(𝑥,𝑡)=𝑛−1 ∑ 𝑘=0𝜕𝑓(𝑥,0+) 𝜕𝑡𝑘𝑡𝑘 𝑘!+∫𝑡 0𝜆𝑔(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)andwritetheresulting equation in the form 𝑓1(𝑥,𝑡)=𝐻1(𝑥,𝑡)+∫𝑡 0𝜆(𝑅𝑓0+𝑁𝑓0)𝑑𝜉, 𝐻(𝑥,𝑡)=𝑛−1 ∑ 𝑘=0𝜕𝑓(𝑥,0+) 𝜕𝑡𝑘𝑡𝑘 𝑘!+∫𝑡 0𝜆𝑔(𝑥,𝜉)𝑑𝜉, 𝐻(𝑥,𝑡)=𝐻0(𝑥,𝑡)+𝐻1(𝑥,𝑡),𝜆= (𝑡−𝜉)𝑛−1 (𝑛−1)!, 𝑓0(𝑥,𝑡)=𝐻0(𝑥,𝑡), (12) 𝑓𝑗+1 (𝑥,𝑡)=∫𝑡 0𝜆(𝑅𝑓𝑗+𝑁𝑓𝑗)𝑑𝜉, 𝑗≥1, (13) equation (12) is an alternative form of modified Adomian decomposition method. The Scientific World Journal 3 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)+1 2𝑢(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.Inordertosolve(14)withthis extra initial condition, we follow the formulation given in (10). Consider 𝑢𝑗+1 (𝑥)=𝑢0(𝑥)−∫𝑥 0𝜆 2(𝑢𝑗(𝜉)𝑢󸀠󸀠 𝑗(𝜉))𝑑𝜉, 𝑢0(𝑥)=𝑢(0)+𝑥𝑢󸀠(0)+𝑥2 2!𝑢󸀠󸀠 (𝑥)=𝑥+𝑥2𝛼 2! ,(16) 𝜆=(𝑥−𝜉)2 2! .(17) By using (16), we obtain the following successive approximations: 𝑢1(𝑥)=𝑥+𝛼𝑥2 2−𝛼𝑥4 48 −𝛼2𝑥5 240, 𝑢2(𝑥)=𝑥+𝛼𝑥2 2−𝛼𝑥4 48 −𝛼2𝑥5 240 +𝛼𝑥6 960+11𝛼2𝑥7 20160 +11𝛼3𝑥8 161280−𝛼2𝑥9 193536−𝛼3𝑥10 518400−𝛼4𝑥11 5702400, . . . (18) Equation (18)istheexactlythesameasobtainedbyusing classical VIM in [20]andonecanfindthevalueof𝛼by using Pad´ eapproximant[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−𝜋𝑡 sin 𝜋𝑥,subjecttotheinitialconditions 𝑢(𝑥,0)=sin 𝜋𝑥, 𝑢𝑡(𝑥,0)=−𝜋sin 𝜋𝑥, (20) whose exact solution is 𝑢(𝑥,𝑡)=𝑒−𝜋𝑡 sin 𝜋𝑥. (21) To solve (19), we follow the formulation, given in (11). Consider 𝑢𝑗+1 (𝑥,𝑡)=∫𝑡 0𝜆(𝜕2𝑢𝑗 𝜕𝑥2)𝑑𝜉, 𝑢0(𝑥,𝑡)=𝐻(𝑥,𝑡),𝜆= (𝑡−𝜉), 𝐻(𝑥,𝑡)=sin 𝜋𝑥−𝑡𝜋sin 𝜋𝑥 +∫𝑡 0(𝑡−𝜉)(2𝜋2𝑒−𝜋𝜉 sin 𝜋𝑥)𝑑𝜉, 𝑢0(𝑥,𝑡)=𝐻(𝑥,𝑡)=−sin 𝜋𝑥+𝑡𝜋sin 𝜋𝑥 +2𝑒−𝜋𝑡 sin 𝜋𝑥 𝑢𝑗+1 (𝑥,𝑡)=∫𝑡 0(𝑡−𝜉)(𝜕2𝑢𝑗 𝜕𝑥2)𝑑𝜉, 𝑢1(𝑥,𝑡)=(2−2𝜋𝑡+𝜋2𝑡2 2! −𝜋3𝑡3 3! )sin 𝜋𝑥−2𝑒−𝜋𝑡 sin 𝜋𝑥, 𝑢2(𝑥,𝑡)=(−2+2𝜋𝑡−𝜋2𝑡2+𝜋3𝑡3 3−𝜋4𝑡4 4! +𝜋5𝑡5 5! )sin 𝜋𝑥 −2𝑒−𝜋𝑡 sin 𝜋𝑥, 𝑢3(𝑥,𝑡)=(2−2𝜋𝑡+𝜋2𝑡2−𝜋3𝑡3 3+𝜋4𝑡4 3(4) −𝜋5𝑡5 3(4)(5)+𝜋6𝑡6 6! −𝜋7𝑡7 7! )sin 𝜋𝑥 −2𝑒−𝜋𝑡 sin 𝜋𝑥, . . . (22) Upon summing these iterations, we observe that 𝑢(𝑥,𝑡)=(1−𝜋𝑡+𝜋2𝑡2 2! −𝜋3𝑡3 3! +𝜋4𝑡4 4! −𝜋5𝑡5 5! +𝜋6𝑡6 6! −𝜋7𝑡7 7! +⋅⋅⋅)sin 𝜋𝑥≈𝑒−𝜋𝑡 sin 𝜋𝑥. (23) Solution (23) is exactly the same as obtained by using ADM in [22]. 4. Conclusion ThispaperhelpsustogaininsightintotheideaofAdomian 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, 4The Scientific World Journal (i) there is no need to do integration process again and againlikewedoinAdomiandecompositionmethod andonecangetthesameresultsofAdomianmethod. (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. 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