Int. J.IndustrialMathematisVol. 3,No. 1(2011)17-23
On the Laplae Transform Deomposition
Algorithm for Solving Nonlinear Dierential
Equations
A.R.Vahidi
,Gh. AsadiCordshooli
Department ofMathematis,Shahr-e-ReyBranh,IslamiAzadUniversity,Tehran,Iran.
Reeived20Deember2010; revised1Marh 2011;aepted21Marh2011.
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Inthispaper,wepresentaomparativestudybetweentheAdomiandeompositionmethod
(ADM)andtheLaplaetransformdeompositionalgorithm(LTDA)forsolvingnonlinear
dierentialequations. FortheBratu'sboundaryvalueproblemandtheDuÆng'sequation,
we show thattheLTDA isequivalent to theADM.
Keywords: Adomiandeompositionmethod; Laplaetransformdeomposition algorithm;Bratu's
problem;DuÆng'sequation
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1 Introdution
Sometimesusingseveralnumerialmethodstosolveanonlinearproblemmaygive similar
results. Itisnotieablethatapplyingdierentnumerialmethodstosolve aproblemmay
providejustthesameresults. ForexampleitisshownthatusingtheADMand suessive
approximationmethodforlinearintegralequations[8 ℄,givejustthesame resultsandalso
itwillbeholdfortheADMandthepowerseriesmethodfordierentialequations[9℄,and
theADMand theJaobi iterative method forsystemof linearequation [6 ℄.
From the beginning of 1980's that George Adomian introdued his deomposition
method,thismethod hasbeenappliedtoa widelass offuntionalequations[1 , 2℄and it
is demonstrated that the method provides aurate and omputable solutions fora wide
lassof linearor nonlinearfuntionalequations.
The LTDA is an approah based on the ADM, whih is onsidered as an eetive
method in solving many problems beause it provides, in general, a rapidly onvergent
seriessolution. SinetheLaplaetransformonvertsthedierentiationtosimplealgebrai
operationsand as thealgebrai equationsare solvable bythe ADM, we an ombine the
LaplaetransformandtheADMtosolvedierentialequations. TheLTDAapproximates
the exat solution with a high degree of auray using only few terms of the iterative
sheme [5 ℄. Many authors have used this method to solve the Bratu's equation [7 ℄, the
DuÆngequation [11 ℄,and integro-dierentialequation [10 ℄.
In this paper, we use both introdued methods to solve two famous and nonlinear
problems, namely Bratu and DuÆng equations. We will show that these methods are
equivalent.
2 Bratu's boundary value problem
Inthissetion,weapplytheADMandtheLTDAfortheBratu'sboundaryvalueproblem
and showthat theresultsareexatly thesame.
2.1 ADM for solving the Bratu's boundary value problem
ConsidertheBratu's boundaryvalue problemasfollow
u 00
(x)=e u(x)
; u(0)=0; u(1)=0; >0: (2.1)
Denoting d
2
dx 2
byG, we have G 1
astwo-fold integration from0 to x, throughwhih the
dierentialequation in(2.1),an be writtenas
Gu(x)= e u(x)
: (2.2)
Afteroperating withtheinverse operatorG 1
, substitutingtheinitialonditionu(0)=0
and onsideringu 0
(0)=k, one gets
u(x)=kx+G 1
( e u(x)
): (2.3)
In theADM, thesolutionu(x) isonsideredas aninnite series
u(x)= 1
X
n=0 u
n
(x); (2.4)
and thenonlinearpartof theequation (2.3) is replaedby
N(u(x))= 1
X
n=0 A
n
(x); (2.5)
whereA 0
n
saretheAdomian'spolynomialsthatanbealulatedbythefollowingformula
A
n =
1
n! [
d n
d n
N( 1
X
i=0
i
u
i (x))℄
=0
; n=0;1;2;:::: (2.6)
Substituting(2.4) and (2.5) into (2.3), we obtain
1
X
u
n
(x)=u(0)+u 0
(0)x L 1
( 1
X
A
n
Consideringtheinitialonditionsgiven intheproblem (2.1),we anhoosetherstterm
of theseries(2.4) asfollows:
u
0
(x)=kx; (2.8)
and its otherterms willbeobtained bythereursionrelation
u
i+1
(x)= Z
x
0 Z
x
0 A
i
dxdx: (2.9)
The omponents ofu
i
(x)forn=1;2;3an be obtainedusing(2.12) as
u
1
(x)=( 1
k 2
+ x
k e
kx
k 2
); (2.10)
u
2
(x)= 2
( 5
4k 2
+ x
2k 3
e kx
k 4
+ xe
kx
k 3
e 2kx
4k 4
); (2.11)
u
3
(x)=2 3
( e
3kx
24k 6
e 2kx
4k 6
+ xe
2kx
4k 5
5e kx
8k 6
+ 3xe
kx
4k 5
x 2
e kx
4k 4
+ 11
12k 6
+ x
4k 5
): (2.12)
Similarly,theomponentsu
n
(x)arealulatedforn=4;5;::: buttheyarenotlistedhere
forbrevity.
2.2 LTDA for solving Bratu's problem
S.A.Khuriin[7 ℄appliedtheLTDAtosolve theequation(1). Wedesribethisproedure
briey asfollow.
Operating bothsides of the dierential equation given in (2.1) by Laplae transform
integral operator (denotedthroughoutthispaperbyL),gives
L[u 00
(x)℄ = L[e u(x)
℄: (2.13)
Applyingtheformulas oftheLaplae transform,tends to
s 2
L[u(x)℄ u(0)s u 0
(0)= L[e u(x)
℄: (2.14)
Usingtheinitial onditionu(0)=0 and onsideringu 0
(0)=k, yields
s 2
L[u(x)℄=k L[e u(x)
℄; (2.15)
thatan be solved forL[u℄as
L[u(x)℄ = k
s 2
s 2
L[e u(x)
℄: (2.16)
Now, the Bratu's dierential equation is onverted to the algebrai equation (2.16) that
willbesolved usingADM. By substituting(2.4) and (2.5) into (2.16), we obtain
L[ 1
X
u
n (x)℄=
k
s 2
s 2
L[ 1
X
A
n
We an replae the Laplae transformby thesummation and sineLaplae transform is
a linearoperator, thefollowingresult willbe hold
1 X n=0 L[u n (x)℄ = k s 2 s 2 1 X n=0 L[A n (x)℄: (2.18) Choosing L[u 0 ℄ = k s 2 and L[u i+1 (x)℄ = s 2 L[A i
(x)℄ and alulating the inverse of the
Laplaetransform, we obtain
u
0
(x)=kx; (2.19)
u
1
(x)=( 1 k 2 + x k e kx k 2 ); (2.20) u 2 (x)= 2 ( 5 4k 2 + x 2k 3 e kx k 4 + xe kx k 3 e 2kx 4k 4 ); (2.21) u 3
(x)=2 3 ( e 3kx 24k 6 e 2kx 4k 6 + xe 2kx 4k 5 5e kx 8k 6 + 3xe kx 4k 5 x 2 e kx 4k 4 + 11 12k 6 + x 4k 5 ); (2.22)
The omponents u
n
(x) are alulated for n = 4;5;::: but for brevity they will not
be listed. Comparing (2.8)- (2.12) by (2.19)-(2.22), one by one, shows that u
i
(x) for
i=0;1;:::;3 obtainedbytheLTDA arejustasthe same termsobtainedbytheADM.
3 DuÆng problem
Inthissetion,weapplythemethodsADMandLTDAforaspiialversionof theDuÆng
equation. The DuÆngequation
y 00
+3y 2y 3
=osxsin2x; (3.23)
with the initial onditions y(0) = y 0
(0) = 1 [4 ℄ is solved using ADM and LTDA in this
setion.
3.1 ADM for solving the DuÆng equation
Denoting d
2
dt 2
byG,wehaveG 1
asatwo-foldintegrationgivenbyG 1 (:)= R x 0 R x 0 (:)dxdx.
UsingtheoperatorG, theequation (3.23) beomes
Gy=f(x) 3y+2y 3
; (3.24)
where f(x) has seven terms of the Taylor expansion of exitation term about x = 0 as
follow
f(x)=osxsin2x'1 x 3x 2 2 + x 3 6 + 7x 4 8 x 5 120 61x 6 240 : (3.25)
Applying the inverse operator G 1
on both sides of (3.24) and onsidering the initial
ondition,yield
y=t+G 1
[f(x)℄ 3G 1
[y℄+2G 1
[y 3
In order to applytheADMlet y= 1 X n=0 y n ; (3.27) and
N(y)=y 3 = 1 X n=0 A n ; (3.28) where A n
's are the Adomian polynomials depending on y
0 ;y
1 ;:::;y
n
. Replaing (3.27)
and (3.28) into (3.26), weobtain
y
0
=t+G 1
f(x)=x+ 1 3 7 60 x 5 + 61 2520 x 7 547 181440 x 9 (3.29)
and thereurrenerelation
y n+1 = 3G 1 [y n ℄+2G
1
[A
n
℄; n=0;1;2;::: (3.30)
using(3.30), we an obtaintheomponentsof y
i asfollows: y 1 = 1 2 x 3 + 1 20 x 5 + 47 840 x 7 89 60480 x 9
+::: (3.31)
y 2 = 3 40 x 5 3 40 x 7 523 20160 x 9
+::: (3.32)
y 3 = 3 560 x 7 + 29 960 x 9
+::: (3.33)
Similarly, the omponents y
n
are alulated for n = 3;4;::: that was skipped to be
listedhere.
3.2 LTDA for solving the DuÆng problem
E.Yusufogluin[11 ℄ usedtheLTDAto solvetheDuÆngequationthatisexplainedbriey
in this subsetion. Operating both sides of dierential equation given in (3.23) by the
Laplaetransformintegraloperatorgives
L[y 00
℄+L[3y℄ L[2y 3
℄=L[f(x)℄: (3.34)
ApplyingthetheLaplae transformformulas and usingtheinitialonditions,one gets
s 2
L[y℄ 1+3L[y℄ 2L[y 3
℄ =L[f(x)℄; (3.35)
thatan be solved forL[y℄ as
L[y℄= 1 s 2 3 s 2 L[y℄+ 2 s 2 L[y 3 ℄+ 1 s 2 L[f(x)℄: (3.36)
Substituting(3.27) and (3.28) into (3.36), we obtain
L[ 1 X y n ℄= 1 s 2 3 s 2 L[ 1 X y n ℄+ 2 s 2 L[ 1 X A n ℄+ 1 s 2
Mathing bothsides of(3.37), thefollowingiterative algorithm willbe obtained
L[y
0 ℄=
1
s 2
+ 1
s 2
L[f(x)℄; (3.38)
L[y
i+1 ℄=
3
s 2
L[y
i ℄+
2
s 2
L[A
i
℄; i=0;1;2;::: (3.39)
Calulatingtheinverse oftheLaplaetransformsin(3.38) andthreerst termsof(3.39),
we obtain
y
0 =x+
1
3 7
60 x
5
+ 61
2520 x
7
547
181440 x
9
(3.40)
y
1 =
1
2 x
3
+ 1
20 x
5
+ 47
840 x
7
89
60480 x
9
+::: (3.41)
y
2 =
3
40 x
5 3
40 x
7
523
20160 x
9
+::: (3.42)
y
3 =
3
560 x
7
+ 29
960 x
9
+:::: (3.43)
In a similarmanner, the omponents y
i
are alulated fori=4;5;::: butforbrevity
they will not be listed. Note that y
0 ;y
1 ;y
2 ;y
3
, and y
4
obtained by the ADM as given
in equations(3.29), (3.31), (3.32), and (3.33) are equal to similarterms obtained by the
LTDAas an be seenin(3.40)-(3.43) . Thisequalityforother terms takes plae,too.
4 Conlusion
In thispaper, theADM and LTDA were appliedfor solving theBratu's boundaryvalue
problemandtheDuÆng'sequation. We showedthattheADMisequivalenttotheLTDA
from the point of view of estimate the solution for the Bratu's boundaryvalue problem
and theDuÆng'sproblem.
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