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.
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