Int. J.IndustrialMathematisVol. 3,No. 3(2011)143-156
Iterative methods for solving non-linear
Fokker-Plank equation
Sh. SadighBehzadi
Young Researhers Club,CentralTehranBranh, IslamiAzadUniversity,P.O.Box: 15655/461,
Tehran,Iran.
Reeived17April2011;revised10July2011; aepted 18July2011.
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In this paper, a nonlinear Fokker-Plank equation is solved by using the modied
Ado-miandeompositionmethod(MADM),variationaliterationmethod(VIM)andhomotopy
analysis method (HAM). For eah method, the approximate solution of this equation is
alulatedintheformoftheserieswhihitsomponentsareomputedbyareursive
rela-tion. Insome theorems,theuniquenessofthesolution(ifitexists)andtheonvergeneof
theproposedmethodsare proved. Finally, anumerialexample issolved to demonstrate
theauray of thementionedmethods.
Keywords : Fokker-Plankequation; ModiedAdomiandeompositionmethod; Variational
itera-tionmethod;Homotopyanalysismethod.
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1 Introdution
Fokker-Plank equation arises in a number of dierent elds in natural siene,
inlud-ing solid-state physis, quantum optis, hemial physis, theoretial biology and iruit
theory. The Fokker-Plankequation wasrst used by Fokkerand Plankto desribe the
Brownianmotionofpartiles[21 ℄. Thisequationhasimportantappliationsinthevarious
areas suh asplasmaphysis,surfaephysis,populationdynamis,biophysis,
engineer-ing,polymerphysisandet[11 ,19 ,20 ,24 ,27 ℄. Inreentyears,someworkshavebeendone
inorderto ndthenumerialsolutionofthisequation byapplyingthe eÆient and
pow-erful iterative methods suh as Adomian deomposition method (ADM) [23 ℄,Variational
iterationmethod(VIM)[7,22 ℄andHomotopyperturbationmethod(HPM)[1,6 ℄. Inthis
work, we apply and ompare theiterative methods MADM, VIM and HAM to solve the
non-linearFokker-Plank equationas follows:
u
t =[
x k
1
(x;t;u)+
2
x 2
k
2
(x;t;u)℄:u; (1.1)
withthe initialonditiongiven by:
u(x;0)=f(x); x2R;
where u(x;t) is unknown. In Eq. (1.1), k
2
(x;t;u) is alledthe diusionand k
1
(x;t;u) is
thedriftoeÆient.
We an writeEq.(1.1) asfollows:
L
t u=L
FP
(u); (1.2)
where L
t =
t
and L
FP =
x k
1
(x;t;u)+
2
x 2
k
2
(x;t;u) is the Fokker-Plank operator
withone variable.
The oeÆients k
1
and k
2
an be independent of time, assuming that the inverse
operatorL 1
t
existsand itan betaken onvenientlyasthedeniteintegralwithrespet
to from 0 to tasfollows:
L 1
t (:)=
Z
t
0 (:)d:
Thus,applyingtheinverse operatorL 1
t
to bothsides ofthe Eq. (1.2) yields
L 1
t L
t u=L
1
t L
FP
(u))u(x;t) u(x;0) =L 1
t L
FP (u):
Therefore, usingtheinitialonditionwe have
u(x;t)=f(x)+L 1
t L
FP
(u): (1.3)
Now, we deomposethe unknown funtion u(x;t) by sum of omponents dened by the
followingdeompositionserieswithu
0
identiedasu(x;0).
u(x;t)= 1
X
n=0 u
n
(x;t): (1.4)
The paperis organized asfollows. InSetion 2,the iterative methods MADM, VIMand
HAM are introduedfor solvingEq. (1.1). The existene and uniqueness of thesolution
and onvergene of the proposed methods are proved in this setion, too. Finally, the
algorithm of iterative methods and the numerial example is presented in Setion 3 to
ompare andto illustratethe aurayof these methods.
2 Desription of methods
In Eq. (1.3), we assumef(x) isboundedforall xin J =R. We set,
x k
1
(x;t;u) =k 0
1
(x;t;u);
2
2 k
2
(x;t;u) =k 0
2
(x;t;u):
Inthefollowingtheorem,we supposethenon-lineartermsk 0
1
(x;t;u)uand k 0
2
(x;t;u)uare
Lipshitz ontinuous with
jk 0
1
(x;t;u)u k 0
1
(x;t;z)z jL 0
ju zj; L 0
>0
jk 0
2
(x;t;u)u k 0
2
(x;t;z)z jL 00
ju zj; L 00
>0
and
=b(L 0
+L 00
); 0x;tb; b2 R:
Theorem2.1. Let0<<1,thennon-linearFokker-Plankequation(1.1),hasaunique
solution.
Proof: Letu andu
be two dierentsolutions of(1.3) then
ju u
j =
R
t
0 h
(k 0
1
(x;;u)u k 0
1 (x;;u
)u
)+(k 0
2
(x;;u)u k 0
2 (x;;u
)u
) i
d
R
t
0 h
jk 0
1
(x;;u)u k 0
1 (x;;u
)u
j+jk 0
2
(x;;u)u k 0
2 (x;;u
)u
j i
d
b (L 0
+L 00
)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.
2.1 Desription of the MADM
The ADMisapplied to thefollowinggeneralnon-linear equation
Lu+R u+Nu=g(x); (2.6)
where u is the unknown funtion, L is the highest order derivative operators whih is
assumedto beeasilyinvertible,R isalineardierentialoperatoroforder lessthanL;Nu
representsthe non-linear terms,and g is the soureterm. Applyingthe inverse operator
L 1
to bothsidesof Eq. (2.6),and usingthegiven onditions,we obtain
u=f(x) L 1
(R u) L 1
(Nu); (2.7)
where the funtion f(x) represents the terms arising from integrating the soure term
g(x). The non-linearoperatorNu=G(u) isdeomposed as
G(u)= 1
X
n=0 A
n
; (2.8)
whereA
n
; n0 arethe Adomianpolynomialsdetermined formallyasfollows :
A
n =
1
n! [
d n
d n
[N( 1
X
i=0
i
u
i )℄℄
=0
: (2.9)
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
); (2.10)
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 );:::
The standard deomposition tehnique represents the solution of u in Eq. (1.3) as the
followingseries,
u= 1
X
n=0 u
n
; (2.11)
where, theomponentsu
0 ;u
1
;::: areusuallydetermined reursivelyby
u
0
=f(x)
u
n+1
= L
1
(R u
n ) L
1
(A
n
); n0: (2.12)
SubstitutingEq. (2.10) intotheEq. (2.12) leadsto thedetermination oftheomponents
of u. Having determined the omponents u
0 ;u
1
;::: the solution u in the series form
denedbyEq. (2.11) follows immediately.
The modied deomposition method was introdued by Wazwaz [26 ℄. The modied
form wasestablishedbasedon theassumptionthatthefuntionf(x)an bedividedinto
two parts,namely f
1
(x) and f
2
(x). Underthisassumption we set
f(x)=f
1
(x)+f
2
(x): (2.13)
Aordingly, a slight variation was proposed only on the omponents u
0
and u
1 . The
suggestion was that only the part f
1
be assigned to the zeroth omponent u
0
, whereas
theremainingpart f
2
be ombined withtheother termsgiven inEq. (2.12) to deneu
1 .
Consequently, themodied reursive relation
u
0 =f
1 (x);
u
1 =f
2
(x) L 1
(R u
0 ) L
1
(A
0
); (2.14)
.
.
.
u
n+1
= L
1
(R u
n ) L
1
(A
n
); n1;
was developed. Some of the appliations of the deomposition method an be found in
[2 , 3 ,8 ,16 , 25 ℄.
To obtain theapproximation solution of Eq. (1.1), aording to the MADM, we an
writetheiterative formula (2.14) asfollows:
u
0
(x;t)=f
1 (x);
u
1
(x;t)=f
2
(x)+L 1
t ( L
x [N
0 ℄+L
xx [M
0 ℄);
.
.
.
u
n+1
(x;t)=L 1
( L
x [N
n ℄+L
xx [M
n
℄); n1:
The non-linear expressionN(u)=k
1
(x;t;u)u and M(u)=k
2
(x;t;u)u bythe innite
seriesofthe Adomianpolynomialaregiven by:
N(u)= P 1 n=0 N n ; M(u)= P 1 n=0 M n ; (2.16)
Also,we onsider
k 0
1
(x;t;u)u= P 1 n=0 A n ; k 0 2
(x;t;u)u= P 1 n=0 B n ; (2.17) whereN n ,M n , A n andB n
areAdomianpolynomials. Also,wean writethefollowing
relationsfor A
n and B n [12℄: n X i=0 A i =k 0 1 (x;t;s
n )s n ;; n X i=0 B i =k 0 2 (x;t;s
n )s
n
: (2.18)
Theorem 2.2. Theseries solution u(x;t)= P
1
i=0 u
i
(x;t) of problem (1.3) using MADM
onvergene when 0< <1 and ju
1
(x;t)j<1.
Proof: Let(C[J℄;k:k)betheBanahspae ofallontinuousfuntionsonJ withthe
norm kf(t)k=maxj f(t)j, for all tin J and s
n and s
m
be arbitrarypartialsums with
nm. We willprove that s
n
is aCauhysequene inthisBanah spae.
From Eqs. (2.15) and (2.17), we have,
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 (A i 1 +B i 1 )d =max 8t2J R t 0 ( P n i=m+1 A i 1 )+( P n i=m+1 B i 1 ) d :
From Eq. (2.18),wehave
P n i=m+1 A i 1 =k 0 1
(x;t;(s
n 1 ))s n 1 k 0 1
(x;t;(s
m 1 ))s m 1 ; P n i=m+1 B i 1 =k 0 2
(x;t;(s
n 1 ))s n 1 k 0 2
(x;t;(s
m 1 ))s m 1 : So, ks n s m
k =max
8t2J R t 0 [(k 0 1 (x;;s
n 1 )s n 1 k 0 1 (x;;s
m 1 )s m 1 ) +(k 0 2 (x;;s
n 1 )s n 1 k 0 2 (x;;s
m 1 )s m 1 )℄d max 8t2J R t 0 h (k 0 1 (x;;s
n 1 )s n 1 k 0 1 (x;;s
m 1 )s m 1 ) + k 0 2 (x;;s
n 1 )s n 1 k 0 2 (x;;s
m 1 )s m 1 i d R t 0 (L 0 +L 00 )js n 1 s m 1 jd
Letn=m+1, then
ks
m+1 s
m
k ks
m s
m 1 k
2
ks
m 1 s
m 2 k
.
.
.
m
ks
1 s
0 k:
From thetriangular inequalitywe have
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 ju
1
(x;t)j:
But j u
1
(x;t) j< 1 ( sinef(x) is bounded), so, asm ! 1, then ks
n s
m
k! 0. We
onludethats
n
isaCauhysequene inC[J℄,therefore theseriesisonvergeneandthe
proof isomplete.
2.2 Desription of the VIM
In the VIM [12 , 13 , 14 , 15 ℄ and [10 ℄, we onsider the following non-linear dierential
equation:
L(u)+N(u)=g(t); (2.19)
where L is a linear operator, N is a non-linear operator and g is the known analytial
funtion. Therefore; u = f(x) L 1
(N(u)) where, f = L 1
(g). In this ase, a orret
funtionan beonstruted asfollows:
u
n+1 (t)=u
n (t)+
Z
t
0
()fL(u
n
())+N(u
n
()) g()gd; n0; (2.20)
whereisageneralLagrangemultiplierwhih anbeidentiedoptimallyvia variational
theory. Herethefuntionu
n
()isarestritedvariationswhihmeansÆu
n
=0. Therefore,
we rst determinetheLagrangemultiplierthat willbe identiedoptimallyvia
integra-tion by parts. The suessive approximation u
n
(t), n 0 of the solution u(t) will be
readilyobtained upon usingtheobtained Lagrangemultiplierand byusingany seletive
funtionu
0
. Thezeroth approximationu
0
maybeseleted anyfuntionthatjustsatises
at least theinitialand boundaryonditions. Withdetermined , several approximations
u
n
(t), n 0 follow immediately. Consequently, the exat solution may be obtained by
using
u(t)= lim u
n
The VIM has been shown to solve eetively, easily and aurately a large lass of
non-linear problemswithapproximations onvergerapidlyto aurate solutions.
ToobtaintheapproximationsolutionofEq. (1.3),aordingtotheVIM,weanwrite
iterationformula (2.20) asfollows:
u
n+1
(x;t) =u
n
(x;t)+L 1
t
((x)[u
n
(x;t) f(x)
L 1
t ( L
x N[u
n
(x;)℄+L
xx M[u
n
(x;)℄)℄);
(2.22)
whereN and M arenon-linearoperators orresponding to k
1 and k
2
respetively.
To ndtheoptimal(x), we proeed asfollows:
Æu
n+1
(x;t) =Æu
n
(x;t)+ÆL 1
t
((x)[u
(
x;t) f(x)
L 1
t ( L
x N[u
n
(x;)℄+L
xx M[u
n
(x;)℄)℄)
=Æu
n
(x;t)+(x)Æu
n
(x;t) L 1
t [Æu
n (x;)
0
(x)℄:
(2.23)
From Eq. (2.23),the stationaryonditions an be obtainedasfollows:
0
(x)=0
and
1+(x) j
x=t =0:
Therefore,theLagrangemultipliersan be identiedas(x) = 1andbysubstitutingin
Eq. (2.22),the followingiterationformulais obtained.
u
0
(x;t) =f(x);
u
n+1
(x;t) =u
n
(x;t) L 1
t [u
n
(x;t) f(x)
L 1
t (k
0
1 (x;;u
n
(x;t))u
n
(x;t)+k 0
2 (x;;u
n
(x;t))u
n
(x;t))℄ n0:
(2.24)
Relation (2.24) willenableusto determinetheomponentsu
n
(x;t) reursivelyforn0.
From Eq. (2.24),when ntends to innity, we onludethat
u(x;t) =u(x;t) L 1
t
[u(x;t) f(x)
L 1
t (k
0
1
(x;;u(x;t))u(x;t)+k 0
2
(x;;u(x;t))u(x;t))℄:
(2.25)
In thefollowingtheorem,weassume that
=1 t(1 (L 0
+L 00
)t); t2R +
:
Theorem 2.3. The series solution u(x;t) = P
1
i=0 u
i
(x;t) of problem (1.3) using VIM
onverges when 0<<1.
Proof: Bysubtrating bothsides ofEq. (2.25) from Eq. (2.24),
u
n+1
(x;t) u(x;t) =u
n
(x;t) u(x;t)
L 1
t h
u
n
(x;t) u(x;t) L 1
t
k 0
1 (x;;u
n
(x;t))u
n (x;t)
k 0
1
(x;;u(x;t))u(x;t)+k 0
2 (x;;u
n
(x;t))u
n (x;t)
k 0
(x;;u(x;t))u(x;t) i
Ifwe set,
e
n+1
(x;t)=u
n+1
(x;t) u
n (x;t)
e
n
(x;t)=u
n
(x;t) u(x;t)
j e
n (x;t
)j=max
t je
n (x;t)j
thensine e
n
is adereasing funtionwithrespetto t from themean value theorem,we
an write
e
n+1
(x;t) =e
n
(x;t)+L 1
t [ e
n
(x;t)+L 1
t (k
0
1 (x;;u
n
(x;t))u
n (x;t)
k 0
1
(x;;u(x;t))u(x;t)+k 0
2 (x;;u
n
(x;t))u
n
(x;t) k 0
2
(x;;u(x;t))u(x;t))℄
e
n
(x;t)+L 1
t [ e
n +L
1
t je
n
(x;t)j(L 0
+L 00
)℄
e
n
(x;t) te
n
(x;)+(L 0
+L 00
)L 1
t L
1
t je
n
(x;t)j)
(1 t)e
n
(x;t)+(L 0
+L 00
)t 2
je
n (x;t
)j
(1 t(1 (L 0
+L 00
)t))je
n (x;t
)j;
where0 t. Hene,e
n+1
(x;t) je
n (x;t
)j:
Therefore,
ke
n+1
k =max
8x;t je
n+1 (x;t)j
max
8x;t je
n (x;t)j
=ke
n k:
Sine 0 < <1, then ke
n
k ! 0 asn tends to innity. So, the series onverges and the
proof isomplete.
2.3 Desription of the HAM
Considerthefollowingnon-linearoperator
N[u℄=0; (2.26)
where u(x;t) is unknown funtion. Let u
0
(x;t) denotes an initial guess of the exat
solution u, h 6= 0 an auxiliary parameter, H(x;t) 6= 0, an auxiliary funtion, and L, an
auxiliary non-linear operator with the property L[r(x;t)℄ = 0 when r(x;t) = 0. Then,
usingq2[0;1℄asan embeddingparameter, weonstrut a homotopyasfollows:
(1 q)L[(x;t;q) u
0
(x;t)℄ qhH(x;t)N[(x;t;q)℄= ^
H[(x;t;q);u
0
(x;t);H(x;t);h;q℄:
(2.27)
It shouldbe emphasized that we have great freedom to hoose the initial guess u
0 (x;t),
theauxiliarynon-linearoperatorL,thenon-zeroauxiliaryparameterh,andtheauxiliary
funtionH(x;t) [17 , 18 ,4 ,5 ,9 ℄. Enforingthehomotopy (2.27) to bezero, i.e.,
^
H[(x;t;q);u
0
(x;t);H(x;t);h;q℄=0; (2.28)
we have theso-alledzero-order deformationequation
When q=0, thezero-orderdeformation Eq. (2.29) beomes
(x;t;0) =u
0
(x;t); (2.30)
and when q = 1, sine h 6=0 and H(x;t) 6=0, the zero-order deformation Eq. (2.29) is
equivalent to
(x;t;1)=u(x;t): (2.31)
Thus,aording to Eqs. (2.30) and (2.31), asthe embeddingparameterq inreases from
0 to 1, (x;t;q) varies ontinuously from the initial approximation u
0
(x;t) to the exat
solutionu(x;t). Suh a kindof ontinuousvariationisalleddeformation inhomotopy.
Due to Taylor'stheorem,(x;t;q) an beexpandedina powerseriesof q asfollows:
(x;t;q)=u
0
(x;t)+ 1
X
m=1 u
m (x;t)q
m
; (2.32)
where
u
m
(x;t)= 1
m!
m
(x;t;q)
q m
j
q=0 :
Let theinitialguess u
0
(x;t), theauxiliarynon-linear parameterL,thenonzero
auxil-iaryparameterh andtheauxiliaryfuntionH(x;t) beproperlyhosensothat thepower
series (2.32) of (x;t;q) onverges at q = 1, then, we have under these assumptions the
solutionseries
u(x;t)=(x;t;1)=u
0
(x;t)+ 1
X
m=1 u
m
(x;t): (2.33)
From Eq. (2.32),wean writeEq. (2.29) asfollows:
(1 q)L[(x;t;q) u
0
(x;t)℄ =(1 q)L[ P
1
m=1 u
m (x;t) q
m
℄
=q h H(x;t)N[(x;t;q)℄
(2.34)
then,
L[ 1
X
m=1 u
m (x;t) q
m
℄ q L[ 1
X
m=1 u
m (x;t)q
m
℄=q h H(x;t)N[(x;t;q)℄: (2.35)
By dierentiating(2.35) m times withrespetto q, we obtain
fL[ P
1
m=1 u
m (x;t) q
m
℄ q L[ P
1
m=1 u
m (x;t)q
m
℄g (m)
=fq h H(x;t)N[(x;t;q)℄g (m)
=m!L[u
m
(x;t) u
m 1 (x;t)℄
=h H(x;t) m
m 1
N[(x;t;q)℄
q m 1
j
q=0 :
Therefore,
L[u
m
(x;t)
m u
m 1
(x;t)℄=hH(x;t)<
m (u
m 1
(x;t)); (2.36)
where,
<
m (u
m 1
(x;t))=
1
m 1
N[(x;t;q)℄
m 1 j
q=0
and
m =
0; m1;
1; m>1:
Note that the high-order deformationEq. (2.36) is governing the non-linear operator L,
andtheterm<
m (u
m 1
(x;t))anbeexpressedsimplyby(2.37)foranynon-linearoperator
N.
To obtainthe approximation solutionofEq. (1.3),aording to HAM, let
N[u℄=u(x;t) f(x) L 1
t (
x k
1
(x;;u)u+
2
x 2
k
2
(x;;u)u):
So,
<
m (u
m 1
(x;t)) =u
m 1 (x;t)
L 1
t (
x k
1 (x;;u
m 1 )u
m 1 +
2
x 2
k
2 (x;;u
m 1 )u
m 1 )
(1
m )f(x):
(2.38)
SubstitutingEq. (2.38) into theEq. (2.36)
L[u
m
(x;t)
m u
m 1
(x;t)℄ =hH(x;t)
u
m 1
(x;t) L 1
t
x k
1 (x;;u
m 1 )u
m 1
+
2
x 2
k
2 (x;;u
m 1 )u
m 1
(1
m )f(x)
i
:
(2.39)
We take an initial guess u
0
(x;t) = f(x), an auxiliary non-linear operator Lu = u,
a nonzero auxiliary parameter h = 1, and auxiliary funtion H(x;t) = 1. This is
substituted into theEq. (2.39) to give thereurrenerelation:
u
0
(x;t)=f(x);
u
m
(x;t)=L 1
t (
x k
1 (x;;u
m 1 )u
m 1 +
2
x 2
k
2 (x;;u
m 1 )u
m 1
); m1:
(2.40)
Let
u(x;t)= 1
X
m=0 u
m
(x;t); lim
m!1 u
m
(x;t)=0: (2.41)
If
ju
m
(x;t)j<1 (2.42)
then, theseriessolution(2.41) onvergene uniformly.
Theorem 2.4. If the series solution (2.41) of problem (1.3) using HAM is onvergent
then it onverges tothe exat solution of the problem (1.3).
Proof: We an write,
n
X
m=1 [u
m
(x;t)
m u
m 1
(x;t)℄=u
1 +(u
2 u
1
)+:::+(u
n u
n 1 )=u
n
(x;t): (2.43)
Hene,
lim u
n
So,usingEq. (2.44) and thedenitionofthe non-linearoperatorL,we have
1
X
m=1 L[u
m
(x;t)
m u
m 1
(x;t)℄=L[ 1
X
m=1 [u
m
(x;t)
m u
m 1
(x;t)℄℄=0:
Thereforefrom Eq. (2.36),wean obtainthat,
1
X
m=1 L[u
m
(x;t)
m u
m 1
(x;t)℄=hH(x;t) 1
X
m=1 <
m 1 (y
m 1
(x;t))=0:
Sine h6=0 andH(x;t)6=0, we have
1
X
m=1 <
m 1 (y
m 1
(x;t))=0: (2.45)
By substituting<
m 1 (y
m 1
(x;t)) into therelation(2.45) and simplifyingit, we onlude
that
P
1
m=1 <
m 1 (y
m 1
(x;t)) = P
1
m=1 [u
m 1 (x;t)
L 1
t (
x k
1 (x;;u
m 1 )u
m 1 +
2
x 2
k
2 (x;;u
m 1 )u
m 1 )
(1
m )f(x)℄
=u(x;t) f(x) L 1
t [
x P
1
m=1 k
1 (x;;u
m 1 )u
m 1
+
2
x 2
P
1
m=1 k
2 (x;;u
m 1 )u
m 1 ℄:
(2.46)
From Eqs. (2.45) and (2.46), we have
u(x;t)=f(x)+L 1
t (
x k
1
(x;;u)u+
2
x 2
k
2
(x;;u)u);
therefore,u(x;t) must betheexat solutionofEq. (1.3).
3 Numerial example
Inthissetion,weomputeanumerialexamplewhihissolvedbytheMADM,VIMand
HAM. The programshave beenprovidedwith Mathematia 6 aordingto thefollowing
algorithm. Inthisalgorithm," isa given positivevalue.
Algorithm:
Step 1. Set n 0.
Step 2. Calulate the reursive relation(2.24) forVIM, (2.15) for MADMor (2.40)
forHAM,
Step 3. If ju
n+1 u
n
j<"thengo to step 4,elsen n+1 and go to step2,
Step 4. Print u(x;t)= P
n
i=0 u
i
(x;t) astheapproximateofthe exat solution.
Example 3.1. We onsider the following non-linear Fokker-Plank equation:
k
1
k
2
(x;t;u)=e x
f(x)=x+e x
;
withexat solution u(x;t)=xe t
, "=10 2
and =0:025317.
Table 1
Numerial resultsofExample (3.1)
x=0:01
t Error(HAM, n=3) Error(VIM,n=4) Error(MADM,n=6)
0.05 3:2886810 2
1:8990110 3
3:5304310 2
0.06 3:4960510 2
1:877110 3
3:8410910 2
0.07 3:5226310 2
1:8583710 3
3:9880410 2
0.08 3:355410 2
1:8379810 3
3:9577810 2
0.09 2:9813610 2
1:8175310 3
3:7367810 2
0.1 2:3875410 2
1:7970110 3
3:3115410 2
Table 1 shows that approximate solutionof the non-linear Fokker-Plankequation is
onvergent with 3iterations byusingtheHAM. By omparingthe resultsof Table 1,we
an observethat theHAM ismore rapidonvergene thantheMADMand VIM.
4 Conlusion
Inthispaper,theiterativemethodshavebeensuessfullyemployedtoobtainthe
approx-imatesolutionof thenon-linearFokker-Plankequation. Forthispurpose,weappliedthe
MADM, VIM and HAM and we proved theonvereny of these methods. Also, we
pre-sented that theHAM wasmore rapidonvergene thanthe MADMand VIM bysolving
a numerial example. These methods may be used to solve the nonlinearFokker-Plank
equation intheform of[23 ℄
u
t =[
N
X
i=1
x
i k
i
(x;t;u)+ N
X
i;j=1
2
x
i x
j k
i;j
(x;t;u)℄u;
wherex=(x
1 ;x
2 ;:::;x
N
). Forfurtherresearh,oneanapplythehomotopyperturbation
methodormodiedformofthismethodtosolvetheFokker-Plankequationandompare
theresultswith thementionediterative methods.
Referenes
[1℄ S. Abbasbandy, Modied homotopy perturbation method for nonlinear equations
andomparsionwithAdomiandeompositionmethod,Appl.Math.Comput172(2006)
431-438.
[2℄ E. Babolian, A.Q. Davari, Numerial implemenatation of Adomian deomposition
methodforlinearVolterra integralequationsoftheseondkind,Appl.Math.Comput
165 (2005)223-227.
[3℄ S.H. Behriy, H. Hashish,I.L. E-Kalla, A.Elsaid,A new algorithm for the
[4℄ Sh.S. Behzadi, The onvergene of homotopy methods for nonlinear Klein-Gordon
equation,J.Appl.Math.Informatis, 28(2010)1227-123 7.
[5℄ Sh.S.Behzadi,M.A.FariborziAraghi,Theuseofiterativemethodsforsolving
Naveir-Stokes equation,J.Appl.Math.Informatis, 29(2011) 1-15.
[6℄ J. Biazar, K. Hosseini, P. Gholamin, Homotopy Perturbation Method for F
okker-Plankequations. International Mathematial Forum 3(2008) 945-954.
[7℄ M. Dehgan, M. Tatari, The use of He's variational iteration method forsolving the
Fokker-Plankequation,Phys.Sripta74 (2006) 310-316.
[8℄ M.A.FariborziAraghi,Sh.SadighBehzadi,SolvingnonlinearVolterra-Fredholm
inte-graldierentialequationsusingthemodiedAdomiandeompositionmethod,
Com-put.Methods inAppl.Math.9 (4)(2009) 1-11.
[9℄ M.A. Fariborzi Araghi, Sh.S. Behzadi, Numerial solution of nonlinear
Volterra-Fredholm integro-dierentialequationsusingHomotopyanalysis method.Journal of
AppliedMathematis and Computing,DOI:10.1080/002071 61 003 77 03 94 , 2010.
[10℄ M.A. Fariborzi Araghi, Sh.S. Behzadi,Solvingnonlinear Volterra-Fredholm
integro-dierentialequations using He's variational iteration method. International Journal
ofComputer Mathematis,DOI:10.1007/s12190-01 0- 04 17 -4, 2010.
[11℄ G. Harrison, Numerial solution of the Fokker-Plank equation using moving nite
elements, Numer. MethodsPartialDierentialEquations4 (1998)219-232.
[12℄ J.H. He, Variational iteration method for autonomous ordinary dierential system,
Appl.Math.Comput.114 (2000)115-123.
[13℄ J.H.He,Approximateanalytialsolutionforseepage folwwithfrationalderivatives
inporous media, Comput.Methods. Appl.Meh.Eng., 167 (1998) 57-68.
[14℄ J.H.He,WangShu-Qiang,Variationaliterationmethodforsolvingintegro-dierential
equations,Physis Letters A.367 (2007) 188-191.
[15℄ J.H. He, Variational priniple for some nonlinear partial dierential equations with
variable oÆients, Chaos,Solitonsand Fratals 19 (4)(2004) 847-851.
[16℄ I.L.El. Kalla, Convergene of the Adomian method applied to a lass of nonlinear
integralequations 21(2008) 327-76.
[17℄ S.J. Liao, Beyond Perturbation: Introdution to the Homotopy Analysis
Method.Chapmanand Hall/CRCPress,Boa Raton,2003.
[18℄ S.J. Liao, Notes on the homotopy analysis method:some denitions and theorems,
CommuniationinNonlinearSiene and NumerialSimulation14 (2009) 983-997.
[19℄ V. Palleshi, M.de. Rosa, Numerial solution of the Fokker-Plank equation,
II.Multidimensionalase,Phys.Lett. A 163 (1992) 381-391.
[20℄ V.Palleshi,F.Sarri,G.Marozzi, M.R.Torquati,NumerialsolutionoftheF
[21℄ H.Risken,TheFokker-Plankequation:Methodsofsolutionandappliation,Springer
Verlag, Berlin, Heidelberg,1989.
[22℄ A. Sadighi, D.D. Ganji, Y. Sabzehmeidani, A study on Fokker-Plank equation by
variationaliterationmethodandHomotopy-perturbationmethod,International
Jour-nalof NonlinearSiene4 (2007)92-102.
[23℄ M. Tatari, M. Dehghan, M. Razzaghi, Appliation of the Adomian deomposition
methodfortheFokker-Plankequation,Math.Comput.Modelling,54(2007)639-650.
[24℄ V.Vanaja,NumerialsolutionofsimpleFokker-Plankequation,Appl.Numer.Math.
9(1992) 533-540.
[25℄ A.M. Wazwaz, Constrution of solitary wave solution and rational solutions for the
KdVequation byADM, Chaos,Solutionand fratals 12 (2001) 2283-2293.
[26℄ A.M.Wazwaz, A rst ourseinintegralequations, WSPC, New Jersey; 1997.
[27℄ M.P. Zorzano, H. Mais, L. Vazquez, Numerial solution of two-dimensionalF
