Int. J.IndustrialMathematisVol. 3,No. 2(2011)79-89
Numerial Solution of Klein-Gordon Equation by
Using the Adomian's Deomposition and
Variational Iterative Methods
Sh. SadighBehzadi
Young Researhers Club,CentralTehranBranh, IslamiAzadUniversity,P.O.Box: 15655/461,
Tehran,Iran.
Reeived7February2011; revised2May2011;aepted5May2011.
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In this paper, a Klein-Gordon equation is solved by usingthe Adomian's deomposition
method,variational iteration method and modied form ofthese methods. The
approxi-mate solutionof thisequation isalulated intheform of seriesinwhihits omponents
are omputed by applying a reursive relation. The existene and uniqueness of the
so-lution andthe onvergene ofthe proposed methodsareproved. A numerial exampleis
studiedto demonstrate theauray of thepresentedmethods.
Keywords: Klein-Gordonequation,Adomian deomposition method(ADM) ,Modied Adomian
deompositionmethod(MADM),Variationaliterationmethod(VIM),Modiedvariational
itera-tionmethod(MVIM).
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1 Introdution
Klein-Gordonequationplaysanimportantroleinmathematialphysis. Theequationhas
attrated muh attention instudyingsolitonsand ondensed matterphysis [6 ℄,in
inves-tigatingtheinterationofsolitonsinaollisionlessplasma,thereurreneofinitialstates,
and inexamining thenonlinearwave equations[8 ℄. Inreentyearssome workshave been
doneinordertondthenumerialsolutionofthisequation,forexample,splinedierene
method for solvingKlein-Gordonequations [16 ℄, invariant-onserving nitedierene
al-gorithms for the nonlinear Klein-Gordon equation [21 ℄, a Legendre spetral method [5 ℄,
Adomian deomposition method [10, 17 , 18 ℄, the variational iteration method [24 ℄,
ap-pliation of homotopy perturbation method to Klein-Gordon equation [4℄. In this work,
we ompare the ADM, VIM, MADM and MVIM to solve the Klein-Gordon equation as
follows:
2
u
t 2
2
u
x 2
= F(u); (1.1)
withthe initialonditionsgiven by:
u(x;0)=f(x);
u(x;t)
t j
t=0
=g(x):
Where, F(u) is a linear or nonlinear funtion and u(x;t) is unknown. The paper is
organized as follows. In setion 2, the mentioned iterative methods are introdued for
solving Eq. (1.1). Also, the existene and uniqueness of the solutionand onvergene of
theproposedmethodareprovedinsetion3. Finally, thenumerialexampleispresented
insetion 4to illustrate theauray ofthese methods.
ToobtaintheapproximatesolutionofEq. (1.1),byintegrating2timesfromEq. (1.1)
withrespetto t andusingthe initialonditionswe obtain,
u(x;t)=G(x;t)+ Z
t
0 Z
t
0
2
u(x;)
x 2
d d Z
t
0 Z
t
0
F(u(x;))d d; (1.2)
where,
G(x;t)=f(x)+tg(x):
The doubleintegralsin Eq. (1.2) an bewritten as[22 ℄:
R
t
0 R
t
0
2
u(x;)
x 2
d d = R
t
0 (t )
2
u(x;)
x 2
d;
R
t
0 R
t
0
F(u(x;))d d = R
t
0
(t ) F(u(x;))d:
So,wean writeEq. (1.2) asfollows:
u(x;t)=G(x;t)+ Z
t
0
(t )
2
u(x;)
x 2
d Z
t
0
(t ) F(u(x;))d: (1.3)
In Eq. (1.3), we assumeG(x;t) isboundedforall ;t inJ =[0;T℄(T 2R) and
jt jM 0
; 8 0t; T;M 0
2R:
We assume theterms D 2
(u(x;))= d
2
dx 2
u(x;t) andF(u) areLipshitz ontinuous with
j D 2
(u) D 2
(u
)jL
1
ju u
jjF(u) F(u
)jL
2
ju u
j
and
:=T (M 0
L
1 +M
0
L
2 );
:=1 T 2
(1 );
2 The iterative methods
2.1 Desription of the MADM and ADM
TheAdomiandeompositionmethodisappliedtothefollowinggeneralnonlinearequation
Lu+R u+Nu=g
1
(x); (2.4)
where u is the unknown funtion, L is the highest order derivative operator whih is
assumedto beeasilyinvertible,R isalineardierentialoperatoroforder lessthanL;Nu
represents the nonlinear terms, and g is the soure term [4, 11 ℄. Applying the inverse
operatorL 1
to bothsidesof Eq. (2.4),and usingthegiven onditionsweobtain
u=f(x) L 1
(R u) L 1
(Nu); (2.5)
where the funtion f(x) represents the terms arising from integrating the soure term
g
1
(x). ThenonlinearoperatorNu=G(u) isdeomposed as
G(u)= 1
X
n=0 A
n
; (2.6)
whereA
n
; n0 arethe Adomianpolynomialsdetermined formallyasfollows [9 ℄:
A
n =
1
n! [
d n
d n
[N( 1
X
i=0
i
u
i )℄℄
=0
: (2.7)
Thesepolynomialsan be obtainedas[19, 20 ,23 ℄:
A
0
=G(u
0 );
A
1 =u
1 G
0
(u
0 );
A
2 =u
2 G
0
(u
0 )+
1
2! u
2
1 G
00
(u
0
); (2.8)
A
3 =u
3 G
0
(u
0 )+u
1 u
2 G
00
(u
0 )+
1
3! u
3
1 G
000
(u
0 );:::
2.1.1 Adomian deomposition method
The standard deomposition tehnique represents the solution of u in Eq. (2.4) as the
followingseries,
u(x;t)= 1
X
i=0 u
i
(x;t); (2.9)
where, theomponentsu
0 ;u
1
;::: areusuallydetermined reursivelyby
u
0
=G(x;t)
u
1 =
Z
t
0
(t )L
0
(x;) d Z
t
0
(t ) A
0
(x;) d;
.
.
.
u
n+1 =
Z
t
(t )L
n
(x;) dtau Z
t
(t ) A
n
SubstitutingEq. (2.8) into Eq. (2.10) leadsto thedeterminationof theomponents ofu.
Having determined the omponents u
0 ;u
1
;::: the solution u in a series form dened by
Eq. (2.9) followsimmediately .
2.1.2 The modied Adomian deomposition method
ThemodieddeompositionmethodwasintroduedbyWazwaz [22 ℄. Themodiedforms
were established based on the assumption that the funtion G(x;t) an be divided into
two parts,namely G
1
(x;t) and G
2
(x;t). Underthisassumption we set
G(x;t)=G
1
(x;t)+G
2
(x;t): (2.11)
Aordingly, a slight variation was proposed only on the omponents u
0
and u
1 . The
suggestion was that only the part G
1
is assigned to the zeroth omponent u
0
, whereas
theremaining partG
2
is ombined withthe otherterms given inEq. (2.10) to deneu
1 .
Consequently, themodied reursive relation
u
0 =G
1 (x;t);
u
1 =G
2
(x;t) L 1
(R u
0 ) L
1
(A
0
); (2.12)
.
.
.
u
n+1
= L
1
(R u
n ) L
1
(A
n
); n1;
was developed.
To obtain theapproximation solution of Eq. (1.1), aording to the MADM, we an
writetheiterative formula (2.12) asfollows:
u
0
(x;t)=G
1 (x;t);
u
1
(x;t)=G
2
(x;t)+ R
t
0
(t ) L
0
(x;) d R
t
0
(t ) A
0
(x;) d;
.
.
.
u
n+1
(x;t)= R
t
0
(t ) L
n
(x;) d R
t
0
(t )A
n
(x;) d:
(2.13)
The operators D 2
(u(x;))= d
2
dx 2
u(x;t) and F(u(x;)) are usuallyrepresentedbythe
inniteseriesof theAdomian polynomials asfollows:
F(u)= 1
X
i=0 A
i ; D
2
(u)= 1
X
i=0 L
i ;
whereA
i and L
i
(i0) arethe Adomianpolynomials.
Also,we an usethefollowingformulafortheAdomian polynomials[9℄:
L
n =D
2
(s
n )
P
n 1
i=0 L
i ;
A
n =F(s
n )
P
n 1
i=0 A
i :
(2.14)
wherethepartialsum iss
n =
P
n
u
2.2 Desription of the VIM and MVIM
In theVIM[12 ℄-[15℄, we onsider thefollowingnonlineardierentialequation:
Lu+Nu=g; (2.15)
whereLisalinearoperator,N isanonlinearoperatorandgisaknownanalytialfuntion.
In thisase, aorretion funtionalan be onstrutedasfollows:
u
n+1 (t)=u
n (t)+
Z
t
0
()fL(u
n
())+N(u
n
()) g()gd; n0; (2.16)
whereisageneralLagrangemultiplierwhih anbeidentiedoptimallyvia variational
theory. Here thefuntionu
n
() isarestrited variationwhihmeansÆu
n
=0. Therefore,
we rst determine the Lagrange multiplier that will be identied optimally via
inte-gration byparts. The suessive approximation u
n
(t), n 0 of the solutionu(t) willbe
readilyobtained upon usingtheobtained Lagrangemultiplierand byusingany seletive
funtionu
0
. The zeroth approximation u
0
may selete anyfuntion that just satisesat
leasttheinitialandboundaryonditions. Withdetermined,thenseveralapproximation
u
n
(t), n 0 follow immediately. Consequently, the exat solution may be obtained by
using
u(x;t)= lim
n!1 u
n
(x;t): (2.17)
The VIM has been shown to solve eetively, easily and aurately a large lass of
nonlinearproblemswithapproximationsonverge rapidlyto auratesolutions.
ToobtaintheapproximationsolutionofEq. (1.1),aordingtotheVIM,weanwrite
iterationformula (2.16) asfollows:
u
n+1
(x;t) =u
n
(x;t)+L 1
t
h
u
n
(x;t) G(x;t) R
t
0
(t ) D 2
(u
n
(x;))d
+ R
t
0
(t )F(u
n
(x;))d i
;
(2.18)
where,
L 1
t (:)=
Z
t
0 Z
t
0
(:) d d:
To ndtheoptimal, we proeedas
Æu
n+1
(x;t) =Æu
n
(x;t)+ÆL 1
t
h
u
n
(x;t) G(x;t) R
t
0
(t ) D 2
(u
n
(x;)) d
+ R
t
0
(t ) F(u
n
(x;))d i
=Æu
n
(x;t)+Æu
n
(x;t) L 1
t [Æu
n (x;t)
0
℄:
(2.19)
From Eq. (2.19),the stationaryonditions an be obtainedasfollows:
Therefore, the Lagrange multipliersan be identied as = 1 and by substituting in
Eq. (2.18),the followingiterationformulais obtained:
u
0
(x;t) =G(x;t);
u
n+1
(x;t) =u
n
(x;t) L 1
t h
u
n
(x;t) G(x;t) R
t
0
(t )D 2
(u
n
(x;))d
+ R
t
0
(t )F(u
n
(x;))d i
;n0:
(2.20)
To obtain theapproximation solutionof Eq. (1.1), based on the MVIM [1 , 2, 3℄, we an
writethefollowingiteration formula:
u
0
(x;t) =G(x;t);
u
n+1
(x;t) =u
n
(x;t) L 1
t h
R
t
0
(t ) D 2
(u
n
(x;) u
n 1
(x;)) d
+ R
t
0
(t ) F(u
n
(x;) u
n 1
(x;))d i
;n0:
(2.21)
Relations(2.20)and(2.21)willenableustodeterminetheomponentsu
n
(x;t)reursively
forn0.
3 Existene and onvergeny of iterative methods
Theorem 3.1. Let 0<<1, then Klein-Gordon equation(1), has a unique solution.
Proof: Letu andu
be two dierentsolutions of(1.3) then
ju u
j =
R
t
0 (t )
D 2
(u(x;)) D 2
(u
(x;))
d
R
t
0
(t )[F(u(x;)) F(u
(x;))℄ d
R
t
0
j(t )jjD 2
(u(x;)) D 2
(u
(x;))j d
+ R
t
0
j(t )jjF(u(x;)) F(u
(x;))j d
T (M 0
L
1 +M
0
L
2
)ju u
j
=ju u
j
From whih we get (1 ) j u u
j 0. Sine 0 < <1. then j u u
j= 0. Implies
u=u
and ompletesthe proof.
Theorem 3.2. Theseries solution u(x;t)= P
1
i=0 u
i
(x;t) of problem (1.1) using MADM
and ADM onvergene when
0<<1; ju
1
(x;t)j<1
Proof: We denote (C[J℄;k : k) as the Banah spae of all ontinuous funtions on
arbitrary partialsums withnm. We are going to prove that s
n
isa Cauhysequene
inthisBanahspae:
ks
n s
m
k =max
8t2J js n s m j =max 8t2J P n i=m+1 u i (x;t) =max 8t2J P n i=m+1 R t 0
(t )L
i d P n i=m+1 R t 0
(t )A
i d =max 8t2J R t 0
(t ) ( P
n 1
i=m L
i ) d+
R
t
0
(t ) ( P n 1 i=m A i ) d :
FromEq. (2.14), we have
P n 1 i=m L i =D 2 (s n 1 s m 1 ); P n 1 i=m A i =F(s
n 1 s m 1 ): So, ks n s m
k =max
8t2J R t 0
(t ) [D 2 (s n 1 s m 1 )℄d R t 0
(t )[F(s
n 1 s
m 1 )℄d)
R t 0
j(t )jjD 2 (s n 1 s m 1 )jd + R t 0
j(t )jjF(s
n 1 s m 1 )jd ks n 1 s m 1 k:
Letn=m+1, then
ks
n s
m
k ks
m s m 1 k 2 ks m 1 s m 2 k . . . m ks 1 s 0 k
From thetriangleinequalitywehave
ks
n s
m
k ks
m+1 s
m k+ks
m+2 s
m+1
k++ks
n s n 1 k [ m + m+1 ++ n 1 ℄ks 1 s 0 k m
[1++ 2
++ n m 1
℄ks 1 s 0 k m [ 1 n m 1 ℄ku 1 (x;t)k
Sine 0<<1,wehave (1 n m
)<1,then
ks n s m k m 1 max 8t2J j u 1
(x;t)j:
But ju
1
(x;t)j<1 ( sine G(x;t) is bounded),so,as m!1,then ks
n s
m
k! 0. We
onludethats
n
isaCauhysequene inC[J℄,therefore theseriesisonvergeneandthe
proof isomplete.
Theorem 3.3. The solution u
n
on-Proof:
u
n+1
(x;t) =u
n
(x;t) L 1
t h
u
n
(x;t) G(x;t) R
t
0
(t )D 2
(u
n
(x;))d
+ R
t
0
(t )F(u
n
(x;))d i
(3.22)
and
u(x;t) =u(x;t) L 1
t h
u(x;t) G(x;t) R
t
0
(t )D 2
(u(x;))d
+ R
t
0
(t ) F(u(x;)) d i
(3.23)
By subtratingEq. (3.22) from Eq. (3.23),
u
n+1
(x;t) u(x;t) =u
n
(x;t) u(x;t)
L 1
t
u
n
(x;t) u(x;t) R
t
0
(t )[D 2
(u
n
(x;)) D 2
(u(x;))℄d
+ R
t
0
(t )[F(u
n
(x;)) F(u(x;))℄ d
;
ifwe set, e
n+1 =u
n+1
(x;t) u
n
(x;t), e
n =u
n
(x;t) u(x;t),then
e
n+1 =e
n L
1
t
e
n R
t
0
(t )[D 2
(u
n
(x;)) D 2
(u(x;))℄ d
+ R
t
0
(t )[F(u
n
(x;)) F(u(x;))℄ d
e
n T
2
(e
n je
n j(M
0
L
1 +M
0
L
2 ))
Ife
n
>0 thenje
n j=e
n
sowe have
e
n+1 =e
n (1 T
2
(1 ))=e
n
Therefore,
ke
n+1
k =max
8t2J je
n+1 j
max
8t2J je
n j
=ke
n k
Sine 0<<1,thenke
n
k!0. So,theseriesonverges andthe proofis omplete.
Theorem 3.4. The solution u
n
(x;t) obtained from the relation (2.21) using MVIM for
the problem (1.1) onvergeswhen 0<<1 ,0< <1.
4 Numerial example
In this setion, we ompute a numerial example whih is solved by the MADM, VIM,
ADMand MVIM. Theprogram hasbeenprovidedwith Mathematia6 aording to the
followingalgorithm where"is agiven positive value.
Algorithm (ADM and MADM)
Step 1. Set n 0.
Step 2. Calulate thereursive relation(10)forADM and(13) forMADM.
Step 3. If ju
n+1 u
n
j<"thengo to step 4,
elsen n+1and go to step2.
Step 4. Print u(x;t)= P
n
i=0 u
i
(x;t) astheapproximateofthe exat solution.
Algorithm (VIM and MVIM)
Step 1. Set n 0.
Step 2. Calulate thereursive relation(20)forVIM and (21)forMVIM.
Step 3. If ju
n+1 u
n
j<"thengo to step 4,
elsen n+1and go to step2.
Step 4. Print u
n
(x;t) astheapproximateof theexat solution.
Example 4.1. [7, 24℄,Consider the nonlinear Klein-Gordon equation
u
tt u
xx = u
2
;
withthe initial onditions
u(x;0)=1+sinx; u
t
(x;0)=0:
Table 1
Numerial resultsfor Example(4.1)
x t=0.3 t=0.4
MADM(n=11) VIM(n=7) ADM(n=13) MVIM(n=3) MADM(n=12) VIM(n=7) ADM(n=14) MVIM(n=4)
0.0 0:9849999861 0:992000024 0:98133425 0:99833421 0:986699116 0:99234421 0:981234262 0:9979833
0.1 1:092291132 1:0931672174 1:09179866 1:09383508 1:072423730 1:073226319 1:07189354 1:0738826
0.2 1:189702983 1:190103087 1:18885990 1:19085859 1:169634875 1:170138050 1:16869202 1:17078255
0.3 1:277668610 1:282668848 1:26899334 1:28858484 1:247326130 1:252361032 1:23779468 1:25878909
0.4 1:367844211 1:371844710 1:36019967 1:37860068 1:337423788 1:34042104 1:32739734 1:34798688
Table 1 shows that, approximate solution of the nonlinear Klein-Gordon equation is
onvergene with 3 iterations in t = 0:3 and with 4 iterations in t = 0:4. by using the
MVIM. Comparing theresults of table 1 , we an observe that theMVIM is more rapid
onvergene thantheMADM, VIMand ADM.
5 Conlusion
The MVIM has been shown to solve eetively, easily and aurately a large lass of
nonlinearproblems with the approximations whih are rapidlyonvergent to exat
solu-tions. In thiswork, theMVIM hasbeensuessfullyemployedto obtaintheapproximate
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