A novel iterative scheme and its application to differential equations

Resea ch A icle A No el I e a i e Scheme and I s Applica ion o Di e en ial Equa ions Yasi Khan,1F. Naeem,2and Zdenjk Šma da3 1Depa men o Ma hema ics, Zhejiang Uni e si y, Hangzhou 310027, China 2Mode n Tex ile Ins i u e, Donghua Uni e si y, Shanghai 200051, China 3Depa men o Ma hema ics, Facul y o Elec ical Enginee ing and Communica ion, B no Uni e si y o Technology, Technicka8,61600B no,CzechRepublic Co espondence should be add essed o Yasi Khan; [email p o ec ed]m Recei ed 18 Decembe 2013; Accep ed 18 Feb ua y 2014; Published 16 Ma ch 2014 AcademicEdi o s:H.Ja a iandC.M.Khalique Copy igh © 2014 Yasi Khan e al. This is an open access a icle dis ibu ed unde he C ea i e Commons A ibu ion License, which pe mi s un es ic ed use, dis ibu ion, and ep oduc ion in any medium, p o ided he o iginal wo k is p ope ly ci ed. The pu pose o his pape is o employ an al e na i e app oach o econs uc he s anda d a ia ional i e a ion algo i hm II p oposed by He, including Lag ange mul iplie , and o gi e a simple o mula ion o Adomian decomposi ion and modi ied Adomian decomposi ion me hod in e ms o newly p oposed a ia ional i e a ion me hod-II (VIM). Th ough ca e ul in es iga ion o he ea lie a ia ional i e a ion algo i hm and Adomian decomposi ion me hod, we ind unnecessa y calcula ions o Lag ange mul iplie and also epea ed calcula ions in ol ed in each i e a ion, espec i ely. Se e al examples a e gi en o e i y he eliabili y and e iciency o he me hod. 1. In oduc ion O e he las ew decades se e al analy ical/app oxima e me hods ha e been de eloped o sol e nonlinea o dina y and pa ial di e en ial equa ions. Fo ini ial and bounda y- alue p oblems in o dina y and pa ial di e en ial equa ions, some o hese echniques include he pe u ba ion me hod [1], he a ia ional i e a ion me hod [2–4], he decomposi ion me hod [5–8], and he homo opy me hods [9–11]. The Adomian decomposi ion me hod [12–16] o sol ing di e en ial and in eg al equa ions, linea o nonlinea , has been he subjec o ex ensi e analy ical and nume ical s udies. The me hod, well add essed in [12–16], has a signi ican ad an age in which i p o ides he solu ion in a apid con e gen se ies wi h elegan ly compu able componen s. In ecen yea s, a la ge amoun o li e a u e has been de el- oped conce ning he applica ion o Adomian decomposi ion me hod in applied sciences. In addi ion, he me hod e eals he analy ical s uc u e o he solu ion which is absen in nume ical solu ions. He’s a ia ional i e a ion me hod [2–4]isbasedona Lag ange mul iplie echnique de eloped by Inoku i e al. [17]. This me hod is, in ac , a modi ica ion o he gene al Lag ange mul iplie me hod in o an i e a ion me hod, which is called co ec ion unc ional. The me hod has been shown o sol e e ec i ely, easily, and accu a ely a la ge class o nonlinea p oblems [18–23]. Gene ally, one o wo i e a ions lead o high accu a e solu ions. In he p esen s udy, we ha e linked up a ia ional i e - a ion me hod and Adomian decomposi ion me hod h ough Lag angemul iplie ,whichshows ha VIMisano he o mo exp essing ADM and ice e sa. This s udy e eals ha he e is no need o in eg a e he di e en ial equa ion again and again as we do in Adomian decomposi ion me hod. Ad an- age o new i e a i e scheme o e he a ia ional i e a ion me hod is ha i a oids he unnecessa y calcula ions and we can cons uc Lag ange mul iplie e y easily wi hou cons uc ion o he co ec ional unc ional. 2. New Fo mula ion o Adomian Decomposi ion Me hod and Va ia ional I e a ion Algo i hm II In o de o elucida e he solu ion p ocedu e, we conside he ollowing 𝑛 h o de pa ial di e en ial equa ion: 𝐿𝑛𝑓(𝑥,𝑡)=𝑅𝑓(𝑥,𝑡)+𝑁𝑓(𝑥,𝑡)+𝑔(𝑥,𝑡),𝑡>0,𝑥∈𝐿, (1) Hindawi Publishing Co po a ion e Scien ific Wo ld Jou nal Volume 2014, A icle ID 605376, 4 pages h p://dx.doi.o g/10.1155/2014/605376 2The Scien i ic Wo ld Jou nal whe e 𝐿𝑛=𝜕𝑛/𝜕𝑡𝑛,𝑛≥1,𝑅is a linea di e en ial ope a o , 𝑁 is a nonlinea di e en ial ope a o , 𝑅and 𝑁a e ee o pa ial de i a i e wi h espec o a iable 𝑡,and𝑔is he sou ce e m. As we a e amilia wi h he ac ha in all kinds o i e a ion echniques, excep he ope a o es o he e ms, a e ea ed as a known unc ion on he behal o ini ial guess. In his p esen newlyp oposedidea,weha eused hesameconcep . We ha e bound all e ms in one unc ion excep ope a o . Conside 𝑔+𝑁𝑓+𝑅𝑓=𝐹(𝑡,𝑥,𝑔,𝑓,𝜕𝑓 𝜕𝑥,𝜕2𝑓 𝜕𝑥2,...).(2) By inco po a ing (2)in(1), we ge 𝐿𝑛𝑓=𝐹(𝑡,𝑥,𝑔,𝑓,𝜕𝑓 𝜕𝑥,𝜕2𝑓 𝜕𝑥2,...).(3) On in eg a ing (3), we ob ain 𝐿(𝑛−1)𝑓=∫𝑡 0𝐹(𝜉,𝑥,𝑔,𝑓,𝜕𝑓 𝜕𝑥,𝜕2𝑓 𝜕𝑥2,...)𝑑𝜉+𝑐1(𝑥).(4) Again, by in eg a ing (4), we ha e 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 ha mul iple in eg al can be educe o a single in eg al by using in eg al p ope y. Hence, we can w i e (5)in he ollowing o m: 𝐿(𝑛−2)𝑓=∫𝑡 0(𝑡−𝜉)𝐹(𝜉,𝑥,𝑔,𝑓,𝜕𝑓 𝜕𝑥,𝜕2𝑓 𝜕𝑥2,...)𝑑𝜉+𝑐1(𝑥)𝑡 +𝑐2(𝑥).(6) I we con inue his p ocess o in eg a ion, we can ge inal o m as ollows: 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 w i ing he cons an o in eg a ion in he o m 𝑐𝑘(𝑥) = (𝜕𝑓𝑛−𝑘(𝑥,0+))/𝜕𝑡𝑛−𝑘,𝑘=1,...,𝑛and subs i u ing (2)in(7) hen (7), we ha e 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 i e a ion o m (8),i canbew i enas ollows: 𝑓𝑗+1 (𝑥,𝑡)=𝑓0(𝑥,𝑡)+∫𝑡 0(𝑡−𝜉)𝑛−1 (𝑛−1)!(𝑅𝑓𝑗+𝑁𝑓𝑗+𝑔)𝑑𝜉, 𝑗=0,1,2,...,(9) whe e 𝑓0(𝑥,𝑡)=∑𝑛−1 𝑘=0((𝜕𝑘𝑓(𝑥,0+))/𝜕𝑡𝑘)(𝑡𝑘/𝑘!). In (9), (𝑡−𝜉)𝑛−1/(𝑛−1)!is Lag ange mul iplie o He’s a ia ional i e a ion me hod, deno ed by 𝜆,i 𝑛is an odd in e- ge , and (9) can be w i en in s anda d a ia ional i e a ion algo i hm 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) Equa ion (10)isexac ly hesameas hes anda dHe’s a ia- ional i e a ion algo i hm II [3]. He e is a poin o be no ed, i we change ou ini ial guess by adding sou ce e m in i , he esul ing o mula ion will gi e he esul s ob ained by well- known Adomian decomposi ion me hod by decomposing he nonlinea e m in (10). Conside 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) Equa ion (11) is an al e na i e app oach o Adomian decom- posi ion me hod, whe e 𝐻(𝑥,𝑡)is a e m which a ises om p esc ibed ini ial condi ion and sou ce e m. Fu he mo e, i we decompose he e m 𝐻(𝑥,𝑡)in (11)andw i e he esul ing equa ion in he o m 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) equa ion (12) is an al e na i e o m o modi ied Adomian decomposi ion me hod. The Scien i ic Wo ld Jou nal 3 3. Illus a i e Examples In o de o illus a e he solu ion p ocedu e, we conside he ollowing examples o o dina y and pa ial di e en ial equa- ions. Example 1. Conside he Blasius equa ion 𝑢󸀠󸀠󸀠 (𝑥)+1 2𝑢(𝑥)𝑢󸀠󸀠 (𝑥)=0, (14) subjec o he bounda y condi ions 𝑢(0)=0, 𝑢󸀠(0)=1, 𝑢󸀠󳨀→ 0, 𝑥 󳨀→ ∞. (15) To sol e he abo e gi en p oblem, we conside an ex a ini ial condi ion; ha is, 𝑢󸀠󸀠(0)=𝛼.Ino de osol e(14)wi h his ex a ini ial condi ion, we ollow he o mula ion gi en in (10). Conside 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 ob ain he ollowing successi e app oxima- ions: 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) Equa ion (18)is heexac ly hesameasob ainedbyusing classical VIM in [20]andonecan ind he alueo 𝛼by using Pad´ eapp oximan [21]. Example 2. Conside he nonhomogeneous wa e equa ion 𝜕2𝑢(𝑥,𝑡) 𝜕𝑡2=𝜕2𝑢(𝑥,𝑡) 𝜕𝑥2+𝜂(𝑥,𝑡),(19) whe e 𝜂(𝑥,𝑡)=2𝑒−𝜋𝑡 sin 𝜋𝑥,subjec o heini ialcondi ions 𝑢(𝑥,0)=sin 𝜋𝑥, 𝑢𝑡(𝑥,0)=−𝜋sin 𝜋𝑥, (20) whose exac solu ion is 𝑢(𝑥,𝑡)=𝑒−𝜋𝑡 sin 𝜋𝑥. (21) To sol e (19), we ollow he o mula ion, gi en in (11). Conside 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 hese i e a ions, we obse e ha 𝑢(𝑥,𝑡)=(1−𝜋𝑡+𝜋2𝑡2 2! −𝜋3𝑡3 3! +𝜋4𝑡4 4! −𝜋5𝑡5 5! +𝜋6𝑡6 6! −𝜋7𝑡7 7! +⋅⋅⋅)sin 𝜋𝑥≈𝑒−𝜋𝑡 sin 𝜋𝑥. (23) Solu ion (23) is exac ly he same as ob ained by using ADM in [22]. 4. Conclusion Thispape helpsus ogaininsigh in o heideao Adomian decomposi ion me hod and a ia ional i e a ion me hod. By keeping in iew bo h me hods, we p opose mo e simpli ied o ms o calcula e Lag ange mul iplie s. By in oducing his Lag ange mul iplie in ADM and VIM ollowing he obse a ions ha ha e been made, 4The Scien i ic Wo ld Jou nal (i) he e is no need o do in eg a ion p ocess again and againlikewedoinAdomiandecomposi ionme hod andonecange hesame esul so Adomianme hod. (ii) I is easy o calcula e he Lag ange mul iplie o He’s a ia ional i e a ion me hod. (iii) This new app oach a oids he unnecessa y calcula- ions like we did in He’s a ia ional i e a ion me hod and Adomian decomposi ion me hod. (i ) This s udy shows ha VIM is ano he o m o exp ess- ing ADM and ice e sa. So we can say ha he p esen me hod is pa allel o m o ADM and can gi e good esul s o VIM wi h less e o . Con lic o In e es s The au ho s decla e ha he e is no con lic o in e es s ega ding he publica ion o his pape . Au ho s’ Con ibu ion The au ho s ha e made he same con ibu ion. All au ho s ead and app o ed he inal pape . Acknowledgmen s The au ho s a e g a e ul o he e iewe s o hei commen s and use ul sugges ions and he hi d au ho was suppo ed by P ojec no. FEKT-S-14-2200 o Facul y o Elec ical Engi- nee ing and Communica ion, B no Uni e si y o Technology, Czech Republic. Re e ences [1] J. Ke o kian and J. D. Cole, Mul iple Scale and Singula Pe u - ba ion Me hods, Sp inge , New Yo k, NY, USA, 1996. [2] J.-H. He, “Va ia ional i e a ion me hod—a kind o non-linea analy ical echnique: some examples,” In e na ional Jou nal o Non-Linea Mechanics, ol. 34, no. 4, pp. 699–708, 1999. [3]J.H.He,G.C.Wu,andF.Aus in,“The a ia ionali e a ion me hod which should be ollowed,” Nonlinea Science Le e s A, ol.1,no.1,pp.1–30,2009. [4] N. Fa az, Y. Khan, and F. Aus in, “An al e na i e app oach o di e en ial-di e ence equa ions using he a ia ional i e a ion me hod,” Zei sch i u Na u o schung A, ol. 65, no. 12, pp. 1055–1059, 2010. [5] Y. Khan, “An e ec i e modi ica ion o he laplace decomposi- ion me hod o nonlinea equa ions,” In e na ional Jou nal o Nonlinea Sciences and Nume ical Simula ion, ol.10,no.11-12, pp.1373–1376,2009. [6] Y. Khan and N. Fa az, “Applica ion o modi ied Laplace decom- posi ion me hod o sol ing bounda y laye equa ion,” Jou nal o King Saud Uni e si y, ol.23,no.1,pp.115–119,2011. [7] Y. Khan and F. Aus in, “Applica ion o he laplace decompo- si ion Me hod o Nonlinea Homogeneous and Non-Homog- enous Ad ec ion Equa ions,” Zei sch i u Na u o schung A, ol. 65, no. 10, pp. 849–853, 2010. [8] Y. Khan and H. La i izadeh, “Applica ion o new op imal homo opy pe u ba ion me hod and Adomian decomposi ion me hods o MHD non-New onian luid low o e a s e ching shee ,” In e na ional Jou nal o Nume ical Me hods o Hea and Fluid Flow, ol.24,pp.124–136,2014. [9] C. Chun, H. Ja a i, and Y.-I. Kim, “Nume ical me hod o he wa e and nonlinea di usion equa ions wi h he homo opy pe u ba ion me hod,” Compu e s and Ma hema ics wi h Appli- ca ions, ol.57,no.7,pp.1226–1231,2009. [10] Y. Khan, Q. Wu, N. Fa az, and A. Yildi im, “The e ec s o a iable iscosi y and he mal conduc i i y on a hin ilm low o e a sh inking/s e ching shee ,” Compu e s and Ma hema ics wi h Applica ions, ol. 61, no. 11, pp. 3391–3399, 2011. [11] Y. Khan and Q. Wu, “Homo opy pe u ba ion ans o m me h- od o nonlinea equa ions using He’s polynomials,” Compu e s and Ma hema ics wi h Applica ions, ol.61,no.8,pp.1963–1967, 2011. [12] G. Adomian, Sol ing F on ie P oblems o Physics: The Decom- posi ion Me hod, Kluwe Academic Publishe s, Bos on, Mass, USA, 1994. [13] R. Rach, “On he Adomian (decomposi ion) me hod and compa isons wi h Pica d’s me hod,” Jou nal o Ma hema ical Analysis and Applica ions, ol.128,no.2,pp.480–483,1987. [14] A.-M. Wazwaz, “The decomposi ion me hod applied o sys ems o pa ial di e en ial equa ions and o he eac ion-di usion B ussela o model,” Applied Ma hema ics and Compu a ion, ol. 110, no. 2-3, pp. 251–264, 2000. [15] H. Ja a i and V. Da a -Geijji, “Re ised Adomian decomposi ion me hod o sol ing a sys em o nonlinea equa ions,” Applied Ma hema ics and Compu a ion, ol.189,pp.541–548,2007. [16] R. C. Rach, “A new de ini ion o he Adomian polynomials,” Kybe ne es, ol.37,no.7,pp.910–955,2008. [17] M. Inoku i, H. Sekine, and T. Mu a, “Gene al use o he Lag ange mul iplie in nonlinea ma hema ical physics,” in Va ia ional Me hod in he Mechanics o Solids,S.Nema -Nasee , Ed.,pp.156–162,Pe gamonP ess,NewYo k,NY,USA,1978. [18] L. Xu, J.-H. He, and A.-M. Wazwaz, “Va ia ional i e a ion me hod-Reali y, po en ial, and challenges,” Jou nal o Compu- a ional and Applied Ma hema ics, ol.207,no.1,pp.1–2,2007. [19] J.-H. He, “Va ia ional i e a ion me hod-Some ecen esul s and new in e p e a ions,” Jou nal o Compu a ional and Applied Ma hema ics, ol.207,no.1,pp.3–17,2007. [20] A.-M. Wazwaz, “The a ia ional i e a ion me hod o sol ing linea and nonlinea sys ems o PDEs,” Compu e s and Ma he- ma ics wi h Applica ions, ol.54,no.7-8,pp.895–902,2007. [21] A.-M. Wazwaz, “The a ia ional i e a ion me hod o sol ing wo o ms o Blasius equa ion on a hal -in ini e domain,” Applied Ma hema ics and Compu a ion, ol.188,no.1,pp.485– 491, 2007. [22] M. El-Gamel, “Compa ison o he solu ions ob ained by Adomian decomposi ion and wa ele -Gale kin me hods o bounda y- alue p oblems,” Applied Ma hema ics and Compu- a ion, ol.186,no.1,pp.652–664,2007. [23] H. Ja a i, H. Tajadodi, and D. Baleanu, “A modi ied a ia ional i e a ion me hod o sol ing ac ional Ricca i di e en ial equa- ion by Adomian polynomials,” F ac ional Calculus and Applied Analysis, ol.16,pp.109–122,2013. 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