research article solution of the differential-difference
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Research ArticleSolution of the Differential-Difference Equations byOptimal Homotopy Asymptotic Method
H Ullah S Islam M Idrees and M Fiza
Department of Mathematics Abdul Wali Khan University Mardan 23200 Pakistan
Correspondence should be addressed to H Ullah hakeemullah1gmailcom
Received 20 January 2014 Revised 1 May 2014 Accepted 1 May 2014 Published 29 May 2014
Academic Editor Fuding Xie
Copyright copy 2014 H Ullah et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
We applied a new analytic approximate technique optimal homotopy asymptotic method (OHAM) for treatment of differential-difference equations (DDEs) To see the efficiency and reliability of the method we consider Volterra equation in different form Itprovides us with a convenient way to control the convergence of approximate solutions when it is compared with other methods ofsolution found in the literature The obtained solutions show that OHAM is effective simpler easier and explicit
1 Introduction
In physical science the DDEs play a vital role in modelingof the complex physical phenomena The DDEs modelsare used in vibration of particles in lattices the flow ofcurrent in a network and the pulses in biological chainsThe models containing DDEs have been investigated bynumerical techniques such as discretizations in solid statephysics and quantum mechanics In the last decades mostof the research work has been done on DDEs Levi andYamilov [1 2] attribute their work to classification of DDEsand connection of integrable partial differential equation(PDEs) and DDEs [3 4] In [5ndash8] the exact solutionsof the DDEs have been studied For the solution of theDDEs Zou et al extended the homotopy analysis method(HAM) [9 10]Wang et al extendedAdomian decompositionmethod (ADM) for solving nonlinear difference-differentialequations (NDDEs) and got a good accuracy with theanalytic solution [11] Recently Marinca et al introducedoptimal homotopy asymptotic method (OHAM) [12ndash16] forthe solution of nonlinear problems The validity of OHAMis independent of whether or not the nonlinear problemscontain small parameters
The motivation of this paper is to extend OHAM for thesolution of NDDEs In [17ndash20] OHAM has been proved tobe useful for obtaining an approximate solution of nonlinearboundary value problems of differential equations In thiswork we have proved that OHAM is also useful and reliable
for the solution of NDDEs hence showing its validity andgreat potential for the solution of NDDEs phenomenon inscience and engineering
In the succeeding section the basic idea of OHAM [12ndash16] is formulated for the solution of boundary value problemsIn Section 3 the effectiveness of the modified formulation ofOHAM for NDDEs has been studied
2 Basic Mathematical Theory ofOHAM for NDDE
Let us take OHAM to the following differential-differenceequation
A (V119899(119905)) + 119891 (119905) = 0 119899 isin Ω (1)
with boundary conditions
B(V119899(119905)
120597V119899(119905)
120597119905) = 0 119899 isin Γ (2)
where A is a differential operator V119899(119905) is an unknown
function 119899 and 119905 denote spatial and temporal independentvariables respectively Γ is the boundary of Ω and 119891
119899(119905) is a
known analytic functionA can be divided into two partsLandN such that
A = L +N (3)
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 520467 7 pageshttpdxdoiorg1011552014520467
2 Abstract and Applied Analysis
where L is simpler part of the partial differential equationwhich is easier to solve andN contains the remaining part ofA So (1) can be written as
L (V119899(119905)) + 119891 (119905) +N (V
119899(119905) V119899minus119896
(119905) V119899+119896
(119905)) = 0 (4)
According to OHAM one can construct an optimal homo-topy 120601
119899(119905 119902) Ω times [0 1] rarr R satisfying
(1 minus 119902) [L (120601119899(119905 119902)) + 119891 (119905)]
= 119867 (119902) [L (120601119899(119905 119902)) + 119891 (119905)
+N (120601119899(119905 119902) 120601
119899minus119896(119905 119902) 120601
119899+119896(119905 119902))]
(5)
B(120601119899(119905 119902)
120597120601119899(119905 119902)
120597119905) = 0 (6)
where 119902 isin [0 1] is an embedding parameter 120601119899(119905 119902) is an
unknown function and119867(119902) is a nonzero auxiliary functionThe auxiliary function119867(119902) is nonzero for 119902 = 0 and119867(0) =0 Equation (5) is the structure of OHAM homotopy Clearlywe have
119902 = 0 997904rArr 119867(120601119899(119905 0) 0)
= L (120601119899(119905 0)) + 119891 (119905) = 0
119902 = 1 997904rArr 119867(120601119899(119905 1) 1)
= 119867 (1) A (120601119899(119905 1)) + 119891 (119905) = 0
(7)
Obviously when 119902 = 0 and 119902 = 1 we obtain
119902 = 0 997904rArr 120601119899(119905 0) = V
1198990(119905)
119902 = 1 997904rArr 120601119899(119905 1) = V
119899(119905)
(8)
respectivelyThus as 119902 varies from 0 to 1 the solution 120601119899(119905 119902)
varies from V1198990(119905) to V
119899(119905) where V
1198990(119905) is the zeroth order
solution and can be obtained from (10)Expanding 120601
119899(119905 119902 119862
119894) 120601119899minus119896
(119905 119902 119862119894) 120601119899+119896
(119905 119902 119862119894) by
Taylors series and choosing119867(119902) as given
119867(119902) = 1199021198621+ 1199022119862
2+ 1199023119862
3+ sdot sdot sdot + 119902119898119862
119898
120601119899minus119896
(119905 119902 119862119894) = V(119899minus119896)0
(119905)
+infin
sum119898=1
V(119899minus119896)119898
(119905 119862119894) 119902119898
120601119899(119905 119902 119862
119894) = V(119899)0
(119905)
+infin
sum119898=1
V(119899)119898
(119905 119862119894) 119902119898
120601119899+119896
(119905 119902 119862119894) = V(119899+119896)0
(119905)
+infin
sum119898=1
V(119899+119896)119898
(119905 119862119894) 119902119898
(9)
where 119862119894 119894 = 1 2 3 are constants and to be determined
and119898 isin 119873
Now substituting (9) into (5) and equating the coefficientof like powers of 119902 we obtain
L (V1198990(119905)) + 119891 (119905) = 0 B(V
1198990(119905)
120597V1198990(119905)
120597119905) = 0
(10)L (V1198991(119905)) minusL (V
1198990(119905))
= 1198621[L (V
1198990(119905)) +N (V
(119899minus119896)0(119905) V1198990(119905) V(119899+119896)0
(119905))]
B(V1198991(119905)
120597V1198991(119905)
120597119905) = 0
(11)L (V1198992(119905)) minusL (V
1198991(119905))
= 1198621[L (V
1198991(119905)) +N (V
(119899minus119896)1(119905) V1198991(119905) V(119899+119896)1
(119905))]
+ 1198622[L (V
1198990(119905)) +N (V
(119899minus119896)0(119905) V1198990(119905) V(119899+119896)0
(119905))]
B(1199061198992(119899 119905)
1205971199061198992(119899 119905)
120597119905) = 0
(12)L (V1198993(119905)) minusL (V
1198992(119905))
= 1198621[L (V
1198992(119905)) +N (V
(119899minus119896)2(119905) V1198992(119905) V(119899+119896)2
(119905))]
+ 1198622[L (V
1198991(119905)) +N (V
(119899minus119896)1(119905) V1198991(119905) V(119899+119896)1
(119905))]
+ 1198623[L (V
1198990(119905)) +N (V
(119899minus119896)0(119905) V1198990(119905) V(119899+119896)0
(119905))]
B(V1198993(119905)
120597V1198993(119905)
120597119905) = 0
(13)
L (V119899119895(119905)) minusL (V
119899(119895minus1)(119905))
=119898
sum119894=1
0
sum119895=119898minus1
119862119894[L (V
119899119895(119905))
+N (V119899119895(119905) V(119899minus119896)119895
(119905) V(119899+119896)119895
(119905))]
B(V119899119895(119905)
120597V119899119895(119905)
120597119905) = 0
(14)
We obtained the zeroth order solution and first second thirdand the general order solutions by solving (10)ndash(14) In thesame way the remaining solutions can be determined It hasbeen observed that the convergence of the series (9) dependson the auxiliary constants 119862
119894 If it is convergent at 119902 = 1 one
has
V119899(119905 119862119894) = V
1198990(119909 119905) + sum
119896ge1
V119899119896(119905 119862119894) (15)
Substituting (15) into (1) it results in the following expressionfor residual
119877 (119905 119862119894) = L (V
119899(119905 119862119894)) + 119891 (119905) +N (V
119899(119905 119862119894)) (16)
Abstract and Applied Analysis 3
If 119877(119905 119862119894) = 0 then V
119899(119905 119862119894) is the exact solution of
the problem Generally it does not happen especially innonlinear problems
For the determinations of auxiliary constants 119862119894 119894 =
1 2 3 there are different methods like Galerkinrsquos methodRitz method least squares method and collocation methodOne can apply the method of least squares as follows
119869 (119862119894) = int119887
119886
intΩ
1198772 (119905 119862119894) 119889119899 119889119905 (17)
where 119886 and 119887 are two values depending on the nature of thegiven problem
The auxiliary constants 119862119894can be optimally calculated as
120597119869
1205971198621
=120597119869
1205971198622
=120597119869
1205971198623
sdot sdot sdot120597119869
120597119862119898
= 0 (18)
The119898th order approximate solution can be obtained by theseobtained constants
The convergence of OHAM is directly proportional to thenumber of optimal constants 119862
119894
3 Application of OHAM toDifference-Differential Equations
To show the validity and effectiveness of OHAM formulationto difference-differential equation we apply it to Volteraequations in different form
Model 1 We consider Voltera equation of the form
120597V119899(119905)
120597119905= V119899(119905) (V119899minus1
(119905) minus V119899+1
(119905)) 119899 ≻ 0 119905 isin [0 1]
(19)
with initial conditions
V119899(0) = 119899 (20)
The exact solution of (31) is given by
V119899(119905) =
119899
1 + 119899119905 (21)
and we take
V119899(119905) = V
1198990(119905) + 119901V
1198991(119905) + 1199012V
1198992(119905) + 1199013V
1198993(119905)
V119899+1
(119905) = V(119899+1)0
(119905) + 119901V(119899+1)1
(119905)
+ 1199012V(119899+1)2
(119905) + 1199013V(119899+1)3
(119905)
V119899minus1
(119905) = V(119899minus1)0
(119905) + 119901V(119899minus1)1
(119905)
+ 1199012V(119899minus1)2
(119905) + 1199013V(119899minus1)3
(119905)
(22)
According to (1) we have
L =120597V119899(119905)
120597119905 119891 (119905) = 0
N = minusV119899(119905) (V119899minus1
(119905) minus V119899+1
(119905))
(23)
Using (19) (20) and (22) into (5) and using the methoddiscussed in Section 2 lead to the following
Zeroth Order Problem Consider120597V1198990(119905)
120597119905= 0 (24)
with
V1198990(0) =119899 (25)
from which we obtain
V1198990(119905) =119899 (26)
First Order Problem One has120597V1198991(119905)
120597119905= (1 + 119862
1)120597V1198990(119905)
120597119905
+ 1198621(V(119899+1)0
(119905) minus V(119899minus1)0
(119905)) V1198990(119905)
(27)
with
V1198991(0) = 0 (28)
Its solution is
V1198991(119905) = 2119862
1119899119905 (29)
Second Order Problem Consider120597V1198992(119905)
120597119905= (1 + 119862
1)120597V1198991(119905)
120597119905+ 1198622
120597V1198990(119905)
120597119905
+ 1198621(V(119899+1)0
(119905) minus V(119899minus1)0
(119905)) V1198991(119905)
+ 1198621(V(119899+1)1
(119905) minus V(119899minus1)1
(119905)) V1198990(119905)
+ 1198622(V(119899+1)0
(119905) minus V(119899minus1)0
(119905)) V1198990(119905)
(30)
with
V1198992(0) = 0 (31)
V1198992(119905) = 2 (119862
1119899119905 + 1198622
1119899119905 + 119862
2119899119905 + 21198622
11198991199052) (32)
Third Order Problem One has120597V1198993(119905)
120597119905= (1 + 119862
1)120597V1198992(119905)
120597119905+ 1198622
120597V1198991(119905)
120597119905+ 1198623
120597V1198990(119905)
120597119905
+ 1198621(V(119899+1)0
(119905) minus V(119899minus1)0
(119905)) V1198992(119905)
+ 1198621(V(119899+1)1
(119905) minus V(119899minus1)1
(119905)) V1198991(119905)
+ 1198622(V(119899+1)0
(119905) minus V(119899minus1)0
(119905)) V1198991(119905)
+ 1198621(V(119899+1)2
(119905) minus V(119899minus1)2
(119905)) V1198990
+ 1198622(V(119899+1)1
(119905) minus V(119899minus1)1
(119905)) V1198990(119905)
+ 1198623(V(119899+1)0
(119905) minus V(119899minus1)0
(119905)) V1198990(119905)
(33)
4 Abstract and Applied Analysis
Table 1 Comparisons of OHAM exact and ADM results atdifferent values of 119899 at 119905 = 1
119899 OHAM Exact ADM Abs error1 0333333 0333333 0333333 717103 times 10minus7
2 0666665 0666665 0666665 143421 times 10minus6
3 0999998 0999998 0999998 215131 times 10minus6
4 133333 133333 133333 286841 times 10minus6
5 166666 166666 166666 358551 times 10minus6
6 2 2 2 430262 times 10minus6
7 233333 233333 233333 501972 times 10minus6
8 266666 266666 266666 573682 times 10minus6
9 299999 299999 299999 645392 times 10minus6
10 333333 333333 333333 717103 times 10minus6
Table 2 Comparisons of first second and third order and exactsolutions at different values of 119899 at 119905 = 1
119899 First order Second order Third order Exact sol1 0527142 0858619 0333333 03333332 105428 171724 0666665 06666653 158143 257586 0999998 09999984 210857 343447 133333 1333335 263571 429309 166666 1666666 316285 515171 2 27 368999 601033 233333 2333338 421713 686895 266666 2666669 474428 772757 299999 29999910 527142 858619 333333 333333
with
V1198993(0) = 0 (34)
V1198993(119905) =
2
3(31198621119899119905 + 61198622
1119899119905 + 31198623
1119899119905 + 3119862
2119899119905 + 3119862
3119899119905
+12119862211198991199052 + 121198623
11198991199052 + 6119862
111986221198991199052 + 41198623
11198991199053)
(35)
Adding (26) (29) (32) and (35) we obtain
V119899(119905) = V
1198990(119905) + V
1198991(119905 1198621) + V1198992(119905 1198621 1198622)
+ V1198993(119905 1198621 1198622 1198623)
(36)
V119899(119905) = 119899 + 2119862
1119899119905 + 2 (119862
1119899119905 + 1198622
1119899119905 + 119862
2119899119905 + 21198622
11198991199052)
+2
3(31198621119899119905 + 61198622
1119899119905 + 31198623
1119899119905 + 3119862
2119899119905 + 3119862
3119899119905
+12119862211198991199052 + 121198623
11198991199052 + 6119862
111986221198991199052 + 41198623
11198991199053) (37)
For the computation of the constants1198621 1198622 and119862
3applying
the method of least square mentioned in (16)ndash(18) we get
1198621= minus023642908394894221
1198622= 02344713680405294
1198623= minus04015040325725273 for 119899 = 1
(38)
2 4 6 8 1000
05
10
15
20
25
30
ADMExactOHAM
n
n
Figure 1 Comparison ofOHAM exact andADMsolutions at 119905 = 1
2 4 6 8 10minus6
minus5
minus4
minus3
minus2
minus1
0Re
sidual
Residual
n
Figure 2 Residual of (19) at 119905 = 1
Putting these values in (37) we obtained the approximatesolution of the form
V119899(119905) = 119899 minus 0472858119899119905 + 2 (005119899119905 + 01117971198991199052)
+2
3(minus0914641119899119905 + 01795761198991199052 minus 005286431198991199053)
(39)
The exact solution of the problem (31) is [11]
V119899(119905) =
119899
1 + 119899119905 (40)
The Adomian solution with Pade approximation of (31) isgiven by [11]
V119899(119905) =
119899
1 + 119899119905 (41)
Abstract and Applied Analysis 5
Model 2 Now consider another form of Volterra equation
120597V119899(119905)
120597119905= V119899(119905) (V119899minus1
(119905) minus V119899+1
(119905))
+ V119899(119905) (V119899minus2
(119905) minus V119899+2
(119905)) 119899 ≻ 0 119905 isin [0 1] (42)
with initial conditions
V119899(0) = 119899 (43)
The exact solution of (42) is given by
V119899(119905) =
119899
1 + 6119905 (44)
We take
V119899(119905) = V
1198990(119905) + 119901V
1198991(119905) + 1199012V
1198992(119905)
+ 1199013V1198993(119905)
V119899+1
(119905) = V(119899+1)0
(119905) + 119901V(119899+1)1
(119905)
+ 1199012V(119899+1)2
(119905) + 1199013V(119899+1)3
(119905)
V119899minus1
(119905) = V(119899minus1)0
(119905) + 119901V(119899minus1)1
(119905)
+ 1199012V(119899minus1)2
(119905) + 1199013V(119899minus1)3
(119905)
V119899+2
(119905) = V(119899+2)0
(119905) + 119901V(119899+2)1
(119905)
+ 1199012V(119899+2)2
(119905) + 1199013V(119899+2)3
(119905)
V119899minus2
(119905) = V(119899minus2)0
(119905) + 119901V(119899minus2)1
(119905)
+ 1199012V(119899minus2)2
(119905) + 1199013V(119899minus2)3
(119905)
(45)
According to (1) we have
L =120597V119899(119905)
120597119905 119891 (119905) = 0
N = minusV119899(119905) (V119899minus1
(119905) minus V119899+1
(119905))
minus V119899(119905) (V119899minus2
(119905) minus V119899+2
(119905))
(46)
Using (42) (43) and (45) into (5) and using the methoddiscussed in Section 2 lead to the following
Zeroth Order Problem Consider
120597V1198990(119905)
120597119905= 0 (47)
with
V1198990(0) = 119899 (48)
from which we obtain
V1198990(119905) = 119899 (49)
First Order Problem One has the following
120597V1198991(119905)
120597119905= (1 + 119862
1)120597V1198990(119905)
120597119905
+ 1198621(V(119899+2)0
(119905) minus 1198621V(119899minus2)0
(119905)) V1198990(119905)
+ 1198621(V(119899+1)0
(119905) minus 1198621V(119899minus1)0
(119905)) V1198990(119905)
(50)
with
V1198991(0) = 0 (51)
Its solution is
V1198991(119905) = 6119862
1119899119905 (52)
Second Order Problem Consider
120597V1198992(119905)
120597119905= (1 + 119862
1)120597V1198991(119905)
120597119905+ 1198622
120597V1198990(119905)
120597119905
+ 1198621(V(119899+2)1
(119905) minus V(119899minus2)1
(119905)) V1198990(119905)
+ 1198622(V(119899+2)0
(119905) minus V(119899minus2)0
(119905)) V1198990
+ 1198621(V(119899+2)0
(119905) minus V(119899minus2)0
(119905)) V1198991(119905)
+ 1198621(V(119899+1)0
(119905) minus V(119899minus1)0
(119905)) V1198991(119905)
+ 1198622(V(119899+1)0
(119905) minus V(119899minus1)0
(119905)) V1198990(119905)
+ 1198621(V(119899+1)1
(119905) minus V(119899minus1)1
(119905)) V1198990(119905)
(53)
with
V1198992(0) = 0 (54)
V1198992(119905) = 6119899119905 (119862
1+ 1198622+ 11986221(1 + 6119905)) (55)
Third Order Problem One has
120597V1198993(119905)
120597119905= (1 + 119862
1)120597V1198992(119905)
120597119905+ 1198622
120597V1198991(119905)
120597119905+ 1198623
120597V1198990(119905)
120597119905
+ 1198621(V(119899+2)2
(119905) minus V(119899minus2)2
(119905)) V1198990(119905)
+ 1198622(V(119899+2)1
(119905) minus V(119899minus2)1
(119905)) V1198990(119905)
+ 1198623(V(119899+2)0
(119905) minus V(119899minus2)0
(119905)) V1198990(119905)
+ 1198621(V(119899+2)1
(119905) minus V(119899minus2)1
(119905)) V1198991(119905)
6 Abstract and Applied Analysis
+ 1198622(V(119899+2)0
(119905) minus V(119899minus2)0
(119905)) V1198991(119905)
+ 1198621(V(119899+2)0
(119905) minus V(119899minus2)0
(119905)) V1198992(119905)
+ 1198621(V(119899+1)2
(119905) minus V(119899minus1)2
(119905)) V1198990(119905)
+ 1198621(V(119899+1)0
(119905) minus V(119899minus1)0
(119905)) V1198992(119905)
+ 1198621(V(119899+1)1
(119905) minus V(119899minus1)1
(119905)) V1198991(119905)
+ 1198622(V(119899+1)0
(119905) minus V(119899minus1)0
(119905)) V1198991(119905)
+ 1198621(V(119899+1)2
(119905) minus V(119899minus1)2
(119905)) V1198990(119905)
+ 1198622(V(119899+1)1
minus V(119899minus1)1
) V1198990(119899 119905)
+ 1198623(V(119899+1)0
(119899 119905) minus V(119899minus1)0
(119899 119905)) V1198990(119905)
(56)
with
V1198993(0) = 0 (57)
V1198993(119905) = 6119899119905 (119862
2+ 1198623+ 211986221(1 + 6119905)
+11986231(1 + 6119905)2 + 119862
1(1 + 2119862
2(1 + 6119905)))
(58)
Adding (49) (52) (55) and (58) we obtain
V119899(119905) = V
1198990(119905) + V
1198991(119905 1198621) + V1198992(119905 1198621 1198622)
+ V1198993(119905 1198621 1198622 1198623)
V1198993(119905) = 119899 + 6119862
1119899119905 + 6119899119905 (119862
1+ 1198622+ 11986221(1 + 6119905))
+ 6119899119905 (1198622+ 1198623+ 211986221(1 + 6119905) + 1198623
1(1 + 6119905)2
+ 11986231(1 + 6119905)2 + 119862
1(1 + 2119862
2(1 + 6119905)))
(59)
For the computation of the constants11986211198622 and119862
3applying
the method of least square mentioned in (16)ndash(18) we get
1198621= minus023640805995815892
1198622= minus0039496242739004025
1198623= minus0012135699742215487 for 119899 = 1
(60)
The exact solution of problem (42) is [9]
V119899(119905) =
119899
1 + 6119905 (61)
The HAM solution of (42) is given by [9]
V119899(119905) = (119899 + 119905) + ℎ (119899 + 1) 119905 + 3ℎ1199052 + 119905ℎ (1 + ℎ) (1 + 6119899)
+ (36ℎ2119899 + 6ℎ2 + 3ℎ) 1199052 + 18ℎ21199053(62)
2 4 6 8 1000
02
04
06
08
10
12
14
HAMExactOHAM
n
n
Figure 3 Comparisons of OHAM HAM and exact solutions at 119905 =1 ℎ = minus1
0 2 4 6 8 1000
05
10
15
20
25
ExactThird
SecondFirst
n
n
Figure 4 Comparison of first second third of OHAM HAM andexact solutions at 119905 = 1 ℎ = minus1
0 2 4 6 8 10minus7
minus6
minus5
minus4
minus3
minus2
minus1
0
Resid
ual
Residual
n
Figure 5 Residual of (42) at 119905 = 1
Abstract and Applied Analysis 7
4 Results and Discussions
The formulation presented in Section 2 provides highly accu-rate solutions for the problems demonstrated in Section 3We have used Mathematica 7 for most of our computationalwork Table 1 and Figure 1 give the comparisons of OHAMwith ADM and exact solutions for Model 1 Also the absoluteerrors at different values of 119899 at 119905 = 1 are given (Figure 4)Figure 3 shows the comparisons ofOHAMresults withHAMexact at different values of 119899 at 119905 = 1 for Model 2 Theconvergence of approximate orders to exact solutions is givenin Table 2 for Model 1 and Figure 5 for Model 2The residualshave been plotted for Model 1 in Figure 2 and for Model 2in Figure 5 We have concluded that the results obtained byOHAM are identical to the results obtained by ADM HAMand exact OHAM converges rapidly with increasing of theorder of approximation
5 Conclusion
In this work we have seen the effectiveness of OHAM[12ndash16] to differential-difference equations By applying thebasic idea of OHAM to differential-difference equations wefound it simpler in applicability more convenient to con-trol convergence and involved less computational overheadTherefore OHAM shows its validity and great potential forthe differential-difference equations arising in science andengineering
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Theauthors are thankful to the anonymous reviewers for theirconstructive and valuable suggestions
References
[1] D Levi and R Yamilov ldquoConditions for the existence of highersymmetries of evolutionary equations on the latticerdquo Journal ofMathematical Physics vol 38 no 12 pp 6648ndash6674 1997
[2] R I Yamilov ldquoConstruction scheme for discreteMiura transfor-mationsrdquo Journal of Physics A Mathematical and General vol27 no 20 pp 6839ndash6851 1994
[3] I Yu Cherdantsev and R I Yamilov ldquoMaster symmetries fordifferential-difference equations of the Volterra typerdquo PhysicaD Nonlinear Phenomena vol 87 no 1ndash4 pp 140ndash144 1995
[4] A B Shabat and R I Yamilov ldquoSymmetries of nonlinearlatticesrdquo Leningrad Mathematical Journal vol 2 p 377 1991
[5] D-J Zhang ldquoSingular solutions in Casoratian form for twodifferential-difference equationsrdquo Chaos Solitons and Fractalsvol 23 no 4 pp 1333ndash1350 2005
[6] K Narita ldquoSoliton solution for a highly nonlinear difference-differential equationrdquo Chaos Solitons and Fractals vol 3 no 3pp 279ndash283 1993
[7] Z Wang and H-Q Zhang ldquoNew exact solutions to somedifference differential equationsrdquoChinese Physics vol 15 no 10pp 2210ndash2215 2006
[8] W Zhen and H Zhang ldquoA symbolic computational methodfor constructing exact solutions to difference-differential equa-tionsrdquoAppliedMathematics andComputation vol 178 no 2 pp431ndash440 2006
[9] L Zou Z Zong and G H Dong ldquoGeneralizing homotopyanalysis method to solve Lotka-Volterra equationrdquo Computersamp Mathematics with Applications vol 56 no 9 pp 2289ndash22932008
[10] Z Wang L Zou and H Zhang ldquoApplying homotopy analysismethod for solving differential-difference equationrdquo PhysicsLetters A General Atomic and Solid State Physics vol 369 no1-2 pp 77ndash84 2007
[11] Z Wang L Zou and Z Zong ldquoAdomian decomposition andPade approximate for solving differential-difference equationrdquoApplied Mathematics and Computation vol 218 no 4 pp 1371ndash1378 2011
[12] N Herisanu and VMarinca ldquoExplicit analytical approximationto large-amplitude non-linear oscillations of a uniform can-tilever beam carrying an intermediate lumped mass and rotaryinertiardquoMeccanica vol 45 no 6 pp 847ndash855 2010
[13] V Marinca and N Herisanu ldquoApplication of optimal homotopyasymptotic method for solving nonlinear equations arising inheat transferrdquo International Communications in Heat and MassTransfer vol 35 no 6 pp 710ndash715 2008
[14] V Marinca N Herisanu C Bota and B Marinca ldquoAn optimalhomotopy asymptotic method applied to the steady flow of afourth-grade fluid past a porous platerdquo Applied MathematicsLetters vol 22 no 2 pp 245ndash251 2009
[15] V Marinca N Herisanu and I Nemes ldquoA new analyticapproach to nonlinear vibration of an electrical machinerdquoProceedings of the RomanianAcademy vol 9 pp 229ndash236 2008
[16] V Marinca and N Herisanu ldquoDetermination of periodicsolutions for the motion of a particle on a rotating parabola bymeans of the optimal homotopy asymptotic methodrdquo Journal ofSound and Vibration vol 329 no 9 pp 1450ndash1459 2010
[17] M Idrees S Islam and A M Tirmizi ldquoApplication of optimalhomotopy asymptotic method of the Korteqag-de-Varies equa-tionrdquo Computers amp Mathematics with Applications vol 63 pp695ndash707 2012
[18] HUllah S IslamM Idrees andMArif ldquoSolution of boundarylayer problems with heat transfer by optimal homotopy asymp-totic methodrdquo Abstract and Applied Analysis vol 2013 ArticleID 324869 10 pages 2013
[19] R Nawaz H Ullah S Islam and M Idrees ldquoApplication ofoptimal homotopy asymptotic method to Burger equationsrdquoJournal of Applied Mathematics vol 2013 Article ID 387478 8pages 2013
[20] H Ullah S Islam M Idrees and R Nawaz ldquoApplicationof optimal homotopy asymptotic method to doubly wavesolutions of the coupled Drinfelrsquod-Sokolov-Wilson EquationsrdquoMathematical Problems in Engineering vol 2013 Article ID362816 8 pages 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Abstract and Applied Analysis
where L is simpler part of the partial differential equationwhich is easier to solve andN contains the remaining part ofA So (1) can be written as
L (V119899(119905)) + 119891 (119905) +N (V
119899(119905) V119899minus119896
(119905) V119899+119896
(119905)) = 0 (4)
According to OHAM one can construct an optimal homo-topy 120601
119899(119905 119902) Ω times [0 1] rarr R satisfying
(1 minus 119902) [L (120601119899(119905 119902)) + 119891 (119905)]
= 119867 (119902) [L (120601119899(119905 119902)) + 119891 (119905)
+N (120601119899(119905 119902) 120601
119899minus119896(119905 119902) 120601
119899+119896(119905 119902))]
(5)
B(120601119899(119905 119902)
120597120601119899(119905 119902)
120597119905) = 0 (6)
where 119902 isin [0 1] is an embedding parameter 120601119899(119905 119902) is an
unknown function and119867(119902) is a nonzero auxiliary functionThe auxiliary function119867(119902) is nonzero for 119902 = 0 and119867(0) =0 Equation (5) is the structure of OHAM homotopy Clearlywe have
119902 = 0 997904rArr 119867(120601119899(119905 0) 0)
= L (120601119899(119905 0)) + 119891 (119905) = 0
119902 = 1 997904rArr 119867(120601119899(119905 1) 1)
= 119867 (1) A (120601119899(119905 1)) + 119891 (119905) = 0
(7)
Obviously when 119902 = 0 and 119902 = 1 we obtain
119902 = 0 997904rArr 120601119899(119905 0) = V
1198990(119905)
119902 = 1 997904rArr 120601119899(119905 1) = V
119899(119905)
(8)
respectivelyThus as 119902 varies from 0 to 1 the solution 120601119899(119905 119902)
varies from V1198990(119905) to V
119899(119905) where V
1198990(119905) is the zeroth order
solution and can be obtained from (10)Expanding 120601
119899(119905 119902 119862
119894) 120601119899minus119896
(119905 119902 119862119894) 120601119899+119896
(119905 119902 119862119894) by
Taylors series and choosing119867(119902) as given
119867(119902) = 1199021198621+ 1199022119862
2+ 1199023119862
3+ sdot sdot sdot + 119902119898119862
119898
120601119899minus119896
(119905 119902 119862119894) = V(119899minus119896)0
(119905)
+infin
sum119898=1
V(119899minus119896)119898
(119905 119862119894) 119902119898
120601119899(119905 119902 119862
119894) = V(119899)0
(119905)
+infin
sum119898=1
V(119899)119898
(119905 119862119894) 119902119898
120601119899+119896
(119905 119902 119862119894) = V(119899+119896)0
(119905)
+infin
sum119898=1
V(119899+119896)119898
(119905 119862119894) 119902119898
(9)
where 119862119894 119894 = 1 2 3 are constants and to be determined
and119898 isin 119873
Now substituting (9) into (5) and equating the coefficientof like powers of 119902 we obtain
L (V1198990(119905)) + 119891 (119905) = 0 B(V
1198990(119905)
120597V1198990(119905)
120597119905) = 0
(10)L (V1198991(119905)) minusL (V
1198990(119905))
= 1198621[L (V
1198990(119905)) +N (V
(119899minus119896)0(119905) V1198990(119905) V(119899+119896)0
(119905))]
B(V1198991(119905)
120597V1198991(119905)
120597119905) = 0
(11)L (V1198992(119905)) minusL (V
1198991(119905))
= 1198621[L (V
1198991(119905)) +N (V
(119899minus119896)1(119905) V1198991(119905) V(119899+119896)1
(119905))]
+ 1198622[L (V
1198990(119905)) +N (V
(119899minus119896)0(119905) V1198990(119905) V(119899+119896)0
(119905))]
B(1199061198992(119899 119905)
1205971199061198992(119899 119905)
120597119905) = 0
(12)L (V1198993(119905)) minusL (V
1198992(119905))
= 1198621[L (V
1198992(119905)) +N (V
(119899minus119896)2(119905) V1198992(119905) V(119899+119896)2
(119905))]
+ 1198622[L (V
1198991(119905)) +N (V
(119899minus119896)1(119905) V1198991(119905) V(119899+119896)1
(119905))]
+ 1198623[L (V
1198990(119905)) +N (V
(119899minus119896)0(119905) V1198990(119905) V(119899+119896)0
(119905))]
B(V1198993(119905)
120597V1198993(119905)
120597119905) = 0
(13)
L (V119899119895(119905)) minusL (V
119899(119895minus1)(119905))
=119898
sum119894=1
0
sum119895=119898minus1
119862119894[L (V
119899119895(119905))
+N (V119899119895(119905) V(119899minus119896)119895
(119905) V(119899+119896)119895
(119905))]
B(V119899119895(119905)
120597V119899119895(119905)
120597119905) = 0
(14)
We obtained the zeroth order solution and first second thirdand the general order solutions by solving (10)ndash(14) In thesame way the remaining solutions can be determined It hasbeen observed that the convergence of the series (9) dependson the auxiliary constants 119862
119894 If it is convergent at 119902 = 1 one
has
V119899(119905 119862119894) = V
1198990(119909 119905) + sum
119896ge1
V119899119896(119905 119862119894) (15)
Substituting (15) into (1) it results in the following expressionfor residual
119877 (119905 119862119894) = L (V
119899(119905 119862119894)) + 119891 (119905) +N (V
119899(119905 119862119894)) (16)
Abstract and Applied Analysis 3
If 119877(119905 119862119894) = 0 then V
119899(119905 119862119894) is the exact solution of
the problem Generally it does not happen especially innonlinear problems
For the determinations of auxiliary constants 119862119894 119894 =
1 2 3 there are different methods like Galerkinrsquos methodRitz method least squares method and collocation methodOne can apply the method of least squares as follows
119869 (119862119894) = int119887
119886
intΩ
1198772 (119905 119862119894) 119889119899 119889119905 (17)
where 119886 and 119887 are two values depending on the nature of thegiven problem
The auxiliary constants 119862119894can be optimally calculated as
120597119869
1205971198621
=120597119869
1205971198622
=120597119869
1205971198623
sdot sdot sdot120597119869
120597119862119898
= 0 (18)
The119898th order approximate solution can be obtained by theseobtained constants
The convergence of OHAM is directly proportional to thenumber of optimal constants 119862
119894
3 Application of OHAM toDifference-Differential Equations
To show the validity and effectiveness of OHAM formulationto difference-differential equation we apply it to Volteraequations in different form
Model 1 We consider Voltera equation of the form
120597V119899(119905)
120597119905= V119899(119905) (V119899minus1
(119905) minus V119899+1
(119905)) 119899 ≻ 0 119905 isin [0 1]
(19)
with initial conditions
V119899(0) = 119899 (20)
The exact solution of (31) is given by
V119899(119905) =
119899
1 + 119899119905 (21)
and we take
V119899(119905) = V
1198990(119905) + 119901V
1198991(119905) + 1199012V
1198992(119905) + 1199013V
1198993(119905)
V119899+1
(119905) = V(119899+1)0
(119905) + 119901V(119899+1)1
(119905)
+ 1199012V(119899+1)2
(119905) + 1199013V(119899+1)3
(119905)
V119899minus1
(119905) = V(119899minus1)0
(119905) + 119901V(119899minus1)1
(119905)
+ 1199012V(119899minus1)2
(119905) + 1199013V(119899minus1)3
(119905)
(22)
According to (1) we have
L =120597V119899(119905)
120597119905 119891 (119905) = 0
N = minusV119899(119905) (V119899minus1
(119905) minus V119899+1
(119905))
(23)
Using (19) (20) and (22) into (5) and using the methoddiscussed in Section 2 lead to the following
Zeroth Order Problem Consider120597V1198990(119905)
120597119905= 0 (24)
with
V1198990(0) =119899 (25)
from which we obtain
V1198990(119905) =119899 (26)
First Order Problem One has120597V1198991(119905)
120597119905= (1 + 119862
1)120597V1198990(119905)
120597119905
+ 1198621(V(119899+1)0
(119905) minus V(119899minus1)0
(119905)) V1198990(119905)
(27)
with
V1198991(0) = 0 (28)
Its solution is
V1198991(119905) = 2119862
1119899119905 (29)
Second Order Problem Consider120597V1198992(119905)
120597119905= (1 + 119862
1)120597V1198991(119905)
120597119905+ 1198622
120597V1198990(119905)
120597119905
+ 1198621(V(119899+1)0
(119905) minus V(119899minus1)0
(119905)) V1198991(119905)
+ 1198621(V(119899+1)1
(119905) minus V(119899minus1)1
(119905)) V1198990(119905)
+ 1198622(V(119899+1)0
(119905) minus V(119899minus1)0
(119905)) V1198990(119905)
(30)
with
V1198992(0) = 0 (31)
V1198992(119905) = 2 (119862
1119899119905 + 1198622
1119899119905 + 119862
2119899119905 + 21198622
11198991199052) (32)
Third Order Problem One has120597V1198993(119905)
120597119905= (1 + 119862
1)120597V1198992(119905)
120597119905+ 1198622
120597V1198991(119905)
120597119905+ 1198623
120597V1198990(119905)
120597119905
+ 1198621(V(119899+1)0
(119905) minus V(119899minus1)0
(119905)) V1198992(119905)
+ 1198621(V(119899+1)1
(119905) minus V(119899minus1)1
(119905)) V1198991(119905)
+ 1198622(V(119899+1)0
(119905) minus V(119899minus1)0
(119905)) V1198991(119905)
+ 1198621(V(119899+1)2
(119905) minus V(119899minus1)2
(119905)) V1198990
+ 1198622(V(119899+1)1
(119905) minus V(119899minus1)1
(119905)) V1198990(119905)
+ 1198623(V(119899+1)0
(119905) minus V(119899minus1)0
(119905)) V1198990(119905)
(33)
4 Abstract and Applied Analysis
Table 1 Comparisons of OHAM exact and ADM results atdifferent values of 119899 at 119905 = 1
119899 OHAM Exact ADM Abs error1 0333333 0333333 0333333 717103 times 10minus7
2 0666665 0666665 0666665 143421 times 10minus6
3 0999998 0999998 0999998 215131 times 10minus6
4 133333 133333 133333 286841 times 10minus6
5 166666 166666 166666 358551 times 10minus6
6 2 2 2 430262 times 10minus6
7 233333 233333 233333 501972 times 10minus6
8 266666 266666 266666 573682 times 10minus6
9 299999 299999 299999 645392 times 10minus6
10 333333 333333 333333 717103 times 10minus6
Table 2 Comparisons of first second and third order and exactsolutions at different values of 119899 at 119905 = 1
119899 First order Second order Third order Exact sol1 0527142 0858619 0333333 03333332 105428 171724 0666665 06666653 158143 257586 0999998 09999984 210857 343447 133333 1333335 263571 429309 166666 1666666 316285 515171 2 27 368999 601033 233333 2333338 421713 686895 266666 2666669 474428 772757 299999 29999910 527142 858619 333333 333333
with
V1198993(0) = 0 (34)
V1198993(119905) =
2
3(31198621119899119905 + 61198622
1119899119905 + 31198623
1119899119905 + 3119862
2119899119905 + 3119862
3119899119905
+12119862211198991199052 + 121198623
11198991199052 + 6119862
111986221198991199052 + 41198623
11198991199053)
(35)
Adding (26) (29) (32) and (35) we obtain
V119899(119905) = V
1198990(119905) + V
1198991(119905 1198621) + V1198992(119905 1198621 1198622)
+ V1198993(119905 1198621 1198622 1198623)
(36)
V119899(119905) = 119899 + 2119862
1119899119905 + 2 (119862
1119899119905 + 1198622
1119899119905 + 119862
2119899119905 + 21198622
11198991199052)
+2
3(31198621119899119905 + 61198622
1119899119905 + 31198623
1119899119905 + 3119862
2119899119905 + 3119862
3119899119905
+12119862211198991199052 + 121198623
11198991199052 + 6119862
111986221198991199052 + 41198623
11198991199053) (37)
For the computation of the constants1198621 1198622 and119862
3applying
the method of least square mentioned in (16)ndash(18) we get
1198621= minus023642908394894221
1198622= 02344713680405294
1198623= minus04015040325725273 for 119899 = 1
(38)
2 4 6 8 1000
05
10
15
20
25
30
ADMExactOHAM
n
n
Figure 1 Comparison ofOHAM exact andADMsolutions at 119905 = 1
2 4 6 8 10minus6
minus5
minus4
minus3
minus2
minus1
0Re
sidual
Residual
n
Figure 2 Residual of (19) at 119905 = 1
Putting these values in (37) we obtained the approximatesolution of the form
V119899(119905) = 119899 minus 0472858119899119905 + 2 (005119899119905 + 01117971198991199052)
+2
3(minus0914641119899119905 + 01795761198991199052 minus 005286431198991199053)
(39)
The exact solution of the problem (31) is [11]
V119899(119905) =
119899
1 + 119899119905 (40)
The Adomian solution with Pade approximation of (31) isgiven by [11]
V119899(119905) =
119899
1 + 119899119905 (41)
Abstract and Applied Analysis 5
Model 2 Now consider another form of Volterra equation
120597V119899(119905)
120597119905= V119899(119905) (V119899minus1
(119905) minus V119899+1
(119905))
+ V119899(119905) (V119899minus2
(119905) minus V119899+2
(119905)) 119899 ≻ 0 119905 isin [0 1] (42)
with initial conditions
V119899(0) = 119899 (43)
The exact solution of (42) is given by
V119899(119905) =
119899
1 + 6119905 (44)
We take
V119899(119905) = V
1198990(119905) + 119901V
1198991(119905) + 1199012V
1198992(119905)
+ 1199013V1198993(119905)
V119899+1
(119905) = V(119899+1)0
(119905) + 119901V(119899+1)1
(119905)
+ 1199012V(119899+1)2
(119905) + 1199013V(119899+1)3
(119905)
V119899minus1
(119905) = V(119899minus1)0
(119905) + 119901V(119899minus1)1
(119905)
+ 1199012V(119899minus1)2
(119905) + 1199013V(119899minus1)3
(119905)
V119899+2
(119905) = V(119899+2)0
(119905) + 119901V(119899+2)1
(119905)
+ 1199012V(119899+2)2
(119905) + 1199013V(119899+2)3
(119905)
V119899minus2
(119905) = V(119899minus2)0
(119905) + 119901V(119899minus2)1
(119905)
+ 1199012V(119899minus2)2
(119905) + 1199013V(119899minus2)3
(119905)
(45)
According to (1) we have
L =120597V119899(119905)
120597119905 119891 (119905) = 0
N = minusV119899(119905) (V119899minus1
(119905) minus V119899+1
(119905))
minus V119899(119905) (V119899minus2
(119905) minus V119899+2
(119905))
(46)
Using (42) (43) and (45) into (5) and using the methoddiscussed in Section 2 lead to the following
Zeroth Order Problem Consider
120597V1198990(119905)
120597119905= 0 (47)
with
V1198990(0) = 119899 (48)
from which we obtain
V1198990(119905) = 119899 (49)
First Order Problem One has the following
120597V1198991(119905)
120597119905= (1 + 119862
1)120597V1198990(119905)
120597119905
+ 1198621(V(119899+2)0
(119905) minus 1198621V(119899minus2)0
(119905)) V1198990(119905)
+ 1198621(V(119899+1)0
(119905) minus 1198621V(119899minus1)0
(119905)) V1198990(119905)
(50)
with
V1198991(0) = 0 (51)
Its solution is
V1198991(119905) = 6119862
1119899119905 (52)
Second Order Problem Consider
120597V1198992(119905)
120597119905= (1 + 119862
1)120597V1198991(119905)
120597119905+ 1198622
120597V1198990(119905)
120597119905
+ 1198621(V(119899+2)1
(119905) minus V(119899minus2)1
(119905)) V1198990(119905)
+ 1198622(V(119899+2)0
(119905) minus V(119899minus2)0
(119905)) V1198990
+ 1198621(V(119899+2)0
(119905) minus V(119899minus2)0
(119905)) V1198991(119905)
+ 1198621(V(119899+1)0
(119905) minus V(119899minus1)0
(119905)) V1198991(119905)
+ 1198622(V(119899+1)0
(119905) minus V(119899minus1)0
(119905)) V1198990(119905)
+ 1198621(V(119899+1)1
(119905) minus V(119899minus1)1
(119905)) V1198990(119905)
(53)
with
V1198992(0) = 0 (54)
V1198992(119905) = 6119899119905 (119862
1+ 1198622+ 11986221(1 + 6119905)) (55)
Third Order Problem One has
120597V1198993(119905)
120597119905= (1 + 119862
1)120597V1198992(119905)
120597119905+ 1198622
120597V1198991(119905)
120597119905+ 1198623
120597V1198990(119905)
120597119905
+ 1198621(V(119899+2)2
(119905) minus V(119899minus2)2
(119905)) V1198990(119905)
+ 1198622(V(119899+2)1
(119905) minus V(119899minus2)1
(119905)) V1198990(119905)
+ 1198623(V(119899+2)0
(119905) minus V(119899minus2)0
(119905)) V1198990(119905)
+ 1198621(V(119899+2)1
(119905) minus V(119899minus2)1
(119905)) V1198991(119905)
6 Abstract and Applied Analysis
+ 1198622(V(119899+2)0
(119905) minus V(119899minus2)0
(119905)) V1198991(119905)
+ 1198621(V(119899+2)0
(119905) minus V(119899minus2)0
(119905)) V1198992(119905)
+ 1198621(V(119899+1)2
(119905) minus V(119899minus1)2
(119905)) V1198990(119905)
+ 1198621(V(119899+1)0
(119905) minus V(119899minus1)0
(119905)) V1198992(119905)
+ 1198621(V(119899+1)1
(119905) minus V(119899minus1)1
(119905)) V1198991(119905)
+ 1198622(V(119899+1)0
(119905) minus V(119899minus1)0
(119905)) V1198991(119905)
+ 1198621(V(119899+1)2
(119905) minus V(119899minus1)2
(119905)) V1198990(119905)
+ 1198622(V(119899+1)1
minus V(119899minus1)1
) V1198990(119899 119905)
+ 1198623(V(119899+1)0
(119899 119905) minus V(119899minus1)0
(119899 119905)) V1198990(119905)
(56)
with
V1198993(0) = 0 (57)
V1198993(119905) = 6119899119905 (119862
2+ 1198623+ 211986221(1 + 6119905)
+11986231(1 + 6119905)2 + 119862
1(1 + 2119862
2(1 + 6119905)))
(58)
Adding (49) (52) (55) and (58) we obtain
V119899(119905) = V
1198990(119905) + V
1198991(119905 1198621) + V1198992(119905 1198621 1198622)
+ V1198993(119905 1198621 1198622 1198623)
V1198993(119905) = 119899 + 6119862
1119899119905 + 6119899119905 (119862
1+ 1198622+ 11986221(1 + 6119905))
+ 6119899119905 (1198622+ 1198623+ 211986221(1 + 6119905) + 1198623
1(1 + 6119905)2
+ 11986231(1 + 6119905)2 + 119862
1(1 + 2119862
2(1 + 6119905)))
(59)
For the computation of the constants11986211198622 and119862
3applying
the method of least square mentioned in (16)ndash(18) we get
1198621= minus023640805995815892
1198622= minus0039496242739004025
1198623= minus0012135699742215487 for 119899 = 1
(60)
The exact solution of problem (42) is [9]
V119899(119905) =
119899
1 + 6119905 (61)
The HAM solution of (42) is given by [9]
V119899(119905) = (119899 + 119905) + ℎ (119899 + 1) 119905 + 3ℎ1199052 + 119905ℎ (1 + ℎ) (1 + 6119899)
+ (36ℎ2119899 + 6ℎ2 + 3ℎ) 1199052 + 18ℎ21199053(62)
2 4 6 8 1000
02
04
06
08
10
12
14
HAMExactOHAM
n
n
Figure 3 Comparisons of OHAM HAM and exact solutions at 119905 =1 ℎ = minus1
0 2 4 6 8 1000
05
10
15
20
25
ExactThird
SecondFirst
n
n
Figure 4 Comparison of first second third of OHAM HAM andexact solutions at 119905 = 1 ℎ = minus1
0 2 4 6 8 10minus7
minus6
minus5
minus4
minus3
minus2
minus1
0
Resid
ual
Residual
n
Figure 5 Residual of (42) at 119905 = 1
Abstract and Applied Analysis 7
4 Results and Discussions
The formulation presented in Section 2 provides highly accu-rate solutions for the problems demonstrated in Section 3We have used Mathematica 7 for most of our computationalwork Table 1 and Figure 1 give the comparisons of OHAMwith ADM and exact solutions for Model 1 Also the absoluteerrors at different values of 119899 at 119905 = 1 are given (Figure 4)Figure 3 shows the comparisons ofOHAMresults withHAMexact at different values of 119899 at 119905 = 1 for Model 2 Theconvergence of approximate orders to exact solutions is givenin Table 2 for Model 1 and Figure 5 for Model 2The residualshave been plotted for Model 1 in Figure 2 and for Model 2in Figure 5 We have concluded that the results obtained byOHAM are identical to the results obtained by ADM HAMand exact OHAM converges rapidly with increasing of theorder of approximation
5 Conclusion
In this work we have seen the effectiveness of OHAM[12ndash16] to differential-difference equations By applying thebasic idea of OHAM to differential-difference equations wefound it simpler in applicability more convenient to con-trol convergence and involved less computational overheadTherefore OHAM shows its validity and great potential forthe differential-difference equations arising in science andengineering
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Theauthors are thankful to the anonymous reviewers for theirconstructive and valuable suggestions
References
[1] D Levi and R Yamilov ldquoConditions for the existence of highersymmetries of evolutionary equations on the latticerdquo Journal ofMathematical Physics vol 38 no 12 pp 6648ndash6674 1997
[2] R I Yamilov ldquoConstruction scheme for discreteMiura transfor-mationsrdquo Journal of Physics A Mathematical and General vol27 no 20 pp 6839ndash6851 1994
[3] I Yu Cherdantsev and R I Yamilov ldquoMaster symmetries fordifferential-difference equations of the Volterra typerdquo PhysicaD Nonlinear Phenomena vol 87 no 1ndash4 pp 140ndash144 1995
[4] A B Shabat and R I Yamilov ldquoSymmetries of nonlinearlatticesrdquo Leningrad Mathematical Journal vol 2 p 377 1991
[5] D-J Zhang ldquoSingular solutions in Casoratian form for twodifferential-difference equationsrdquo Chaos Solitons and Fractalsvol 23 no 4 pp 1333ndash1350 2005
[6] K Narita ldquoSoliton solution for a highly nonlinear difference-differential equationrdquo Chaos Solitons and Fractals vol 3 no 3pp 279ndash283 1993
[7] Z Wang and H-Q Zhang ldquoNew exact solutions to somedifference differential equationsrdquoChinese Physics vol 15 no 10pp 2210ndash2215 2006
[8] W Zhen and H Zhang ldquoA symbolic computational methodfor constructing exact solutions to difference-differential equa-tionsrdquoAppliedMathematics andComputation vol 178 no 2 pp431ndash440 2006
[9] L Zou Z Zong and G H Dong ldquoGeneralizing homotopyanalysis method to solve Lotka-Volterra equationrdquo Computersamp Mathematics with Applications vol 56 no 9 pp 2289ndash22932008
[10] Z Wang L Zou and H Zhang ldquoApplying homotopy analysismethod for solving differential-difference equationrdquo PhysicsLetters A General Atomic and Solid State Physics vol 369 no1-2 pp 77ndash84 2007
[11] Z Wang L Zou and Z Zong ldquoAdomian decomposition andPade approximate for solving differential-difference equationrdquoApplied Mathematics and Computation vol 218 no 4 pp 1371ndash1378 2011
[12] N Herisanu and VMarinca ldquoExplicit analytical approximationto large-amplitude non-linear oscillations of a uniform can-tilever beam carrying an intermediate lumped mass and rotaryinertiardquoMeccanica vol 45 no 6 pp 847ndash855 2010
[13] V Marinca and N Herisanu ldquoApplication of optimal homotopyasymptotic method for solving nonlinear equations arising inheat transferrdquo International Communications in Heat and MassTransfer vol 35 no 6 pp 710ndash715 2008
[14] V Marinca N Herisanu C Bota and B Marinca ldquoAn optimalhomotopy asymptotic method applied to the steady flow of afourth-grade fluid past a porous platerdquo Applied MathematicsLetters vol 22 no 2 pp 245ndash251 2009
[15] V Marinca N Herisanu and I Nemes ldquoA new analyticapproach to nonlinear vibration of an electrical machinerdquoProceedings of the RomanianAcademy vol 9 pp 229ndash236 2008
[16] V Marinca and N Herisanu ldquoDetermination of periodicsolutions for the motion of a particle on a rotating parabola bymeans of the optimal homotopy asymptotic methodrdquo Journal ofSound and Vibration vol 329 no 9 pp 1450ndash1459 2010
[17] M Idrees S Islam and A M Tirmizi ldquoApplication of optimalhomotopy asymptotic method of the Korteqag-de-Varies equa-tionrdquo Computers amp Mathematics with Applications vol 63 pp695ndash707 2012
[18] HUllah S IslamM Idrees andMArif ldquoSolution of boundarylayer problems with heat transfer by optimal homotopy asymp-totic methodrdquo Abstract and Applied Analysis vol 2013 ArticleID 324869 10 pages 2013
[19] R Nawaz H Ullah S Islam and M Idrees ldquoApplication ofoptimal homotopy asymptotic method to Burger equationsrdquoJournal of Applied Mathematics vol 2013 Article ID 387478 8pages 2013
[20] H Ullah S Islam M Idrees and R Nawaz ldquoApplicationof optimal homotopy asymptotic method to doubly wavesolutions of the coupled Drinfelrsquod-Sokolov-Wilson EquationsrdquoMathematical Problems in Engineering vol 2013 Article ID362816 8 pages 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 3
If 119877(119905 119862119894) = 0 then V
119899(119905 119862119894) is the exact solution of
the problem Generally it does not happen especially innonlinear problems
For the determinations of auxiliary constants 119862119894 119894 =
1 2 3 there are different methods like Galerkinrsquos methodRitz method least squares method and collocation methodOne can apply the method of least squares as follows
119869 (119862119894) = int119887
119886
intΩ
1198772 (119905 119862119894) 119889119899 119889119905 (17)
where 119886 and 119887 are two values depending on the nature of thegiven problem
The auxiliary constants 119862119894can be optimally calculated as
120597119869
1205971198621
=120597119869
1205971198622
=120597119869
1205971198623
sdot sdot sdot120597119869
120597119862119898
= 0 (18)
The119898th order approximate solution can be obtained by theseobtained constants
The convergence of OHAM is directly proportional to thenumber of optimal constants 119862
119894
3 Application of OHAM toDifference-Differential Equations
To show the validity and effectiveness of OHAM formulationto difference-differential equation we apply it to Volteraequations in different form
Model 1 We consider Voltera equation of the form
120597V119899(119905)
120597119905= V119899(119905) (V119899minus1
(119905) minus V119899+1
(119905)) 119899 ≻ 0 119905 isin [0 1]
(19)
with initial conditions
V119899(0) = 119899 (20)
The exact solution of (31) is given by
V119899(119905) =
119899
1 + 119899119905 (21)
and we take
V119899(119905) = V
1198990(119905) + 119901V
1198991(119905) + 1199012V
1198992(119905) + 1199013V
1198993(119905)
V119899+1
(119905) = V(119899+1)0
(119905) + 119901V(119899+1)1
(119905)
+ 1199012V(119899+1)2
(119905) + 1199013V(119899+1)3
(119905)
V119899minus1
(119905) = V(119899minus1)0
(119905) + 119901V(119899minus1)1
(119905)
+ 1199012V(119899minus1)2
(119905) + 1199013V(119899minus1)3
(119905)
(22)
According to (1) we have
L =120597V119899(119905)
120597119905 119891 (119905) = 0
N = minusV119899(119905) (V119899minus1
(119905) minus V119899+1
(119905))
(23)
Using (19) (20) and (22) into (5) and using the methoddiscussed in Section 2 lead to the following
Zeroth Order Problem Consider120597V1198990(119905)
120597119905= 0 (24)
with
V1198990(0) =119899 (25)
from which we obtain
V1198990(119905) =119899 (26)
First Order Problem One has120597V1198991(119905)
120597119905= (1 + 119862
1)120597V1198990(119905)
120597119905
+ 1198621(V(119899+1)0
(119905) minus V(119899minus1)0
(119905)) V1198990(119905)
(27)
with
V1198991(0) = 0 (28)
Its solution is
V1198991(119905) = 2119862
1119899119905 (29)
Second Order Problem Consider120597V1198992(119905)
120597119905= (1 + 119862
1)120597V1198991(119905)
120597119905+ 1198622
120597V1198990(119905)
120597119905
+ 1198621(V(119899+1)0
(119905) minus V(119899minus1)0
(119905)) V1198991(119905)
+ 1198621(V(119899+1)1
(119905) minus V(119899minus1)1
(119905)) V1198990(119905)
+ 1198622(V(119899+1)0
(119905) minus V(119899minus1)0
(119905)) V1198990(119905)
(30)
with
V1198992(0) = 0 (31)
V1198992(119905) = 2 (119862
1119899119905 + 1198622
1119899119905 + 119862
2119899119905 + 21198622
11198991199052) (32)
Third Order Problem One has120597V1198993(119905)
120597119905= (1 + 119862
1)120597V1198992(119905)
120597119905+ 1198622
120597V1198991(119905)
120597119905+ 1198623
120597V1198990(119905)
120597119905
+ 1198621(V(119899+1)0
(119905) minus V(119899minus1)0
(119905)) V1198992(119905)
+ 1198621(V(119899+1)1
(119905) minus V(119899minus1)1
(119905)) V1198991(119905)
+ 1198622(V(119899+1)0
(119905) minus V(119899minus1)0
(119905)) V1198991(119905)
+ 1198621(V(119899+1)2
(119905) minus V(119899minus1)2
(119905)) V1198990
+ 1198622(V(119899+1)1
(119905) minus V(119899minus1)1
(119905)) V1198990(119905)
+ 1198623(V(119899+1)0
(119905) minus V(119899minus1)0
(119905)) V1198990(119905)
(33)
4 Abstract and Applied Analysis
Table 1 Comparisons of OHAM exact and ADM results atdifferent values of 119899 at 119905 = 1
119899 OHAM Exact ADM Abs error1 0333333 0333333 0333333 717103 times 10minus7
2 0666665 0666665 0666665 143421 times 10minus6
3 0999998 0999998 0999998 215131 times 10minus6
4 133333 133333 133333 286841 times 10minus6
5 166666 166666 166666 358551 times 10minus6
6 2 2 2 430262 times 10minus6
7 233333 233333 233333 501972 times 10minus6
8 266666 266666 266666 573682 times 10minus6
9 299999 299999 299999 645392 times 10minus6
10 333333 333333 333333 717103 times 10minus6
Table 2 Comparisons of first second and third order and exactsolutions at different values of 119899 at 119905 = 1
119899 First order Second order Third order Exact sol1 0527142 0858619 0333333 03333332 105428 171724 0666665 06666653 158143 257586 0999998 09999984 210857 343447 133333 1333335 263571 429309 166666 1666666 316285 515171 2 27 368999 601033 233333 2333338 421713 686895 266666 2666669 474428 772757 299999 29999910 527142 858619 333333 333333
with
V1198993(0) = 0 (34)
V1198993(119905) =
2
3(31198621119899119905 + 61198622
1119899119905 + 31198623
1119899119905 + 3119862
2119899119905 + 3119862
3119899119905
+12119862211198991199052 + 121198623
11198991199052 + 6119862
111986221198991199052 + 41198623
11198991199053)
(35)
Adding (26) (29) (32) and (35) we obtain
V119899(119905) = V
1198990(119905) + V
1198991(119905 1198621) + V1198992(119905 1198621 1198622)
+ V1198993(119905 1198621 1198622 1198623)
(36)
V119899(119905) = 119899 + 2119862
1119899119905 + 2 (119862
1119899119905 + 1198622
1119899119905 + 119862
2119899119905 + 21198622
11198991199052)
+2
3(31198621119899119905 + 61198622
1119899119905 + 31198623
1119899119905 + 3119862
2119899119905 + 3119862
3119899119905
+12119862211198991199052 + 121198623
11198991199052 + 6119862
111986221198991199052 + 41198623
11198991199053) (37)
For the computation of the constants1198621 1198622 and119862
3applying
the method of least square mentioned in (16)ndash(18) we get
1198621= minus023642908394894221
1198622= 02344713680405294
1198623= minus04015040325725273 for 119899 = 1
(38)
2 4 6 8 1000
05
10
15
20
25
30
ADMExactOHAM
n
n
Figure 1 Comparison ofOHAM exact andADMsolutions at 119905 = 1
2 4 6 8 10minus6
minus5
minus4
minus3
minus2
minus1
0Re
sidual
Residual
n
Figure 2 Residual of (19) at 119905 = 1
Putting these values in (37) we obtained the approximatesolution of the form
V119899(119905) = 119899 minus 0472858119899119905 + 2 (005119899119905 + 01117971198991199052)
+2
3(minus0914641119899119905 + 01795761198991199052 minus 005286431198991199053)
(39)
The exact solution of the problem (31) is [11]
V119899(119905) =
119899
1 + 119899119905 (40)
The Adomian solution with Pade approximation of (31) isgiven by [11]
V119899(119905) =
119899
1 + 119899119905 (41)
Abstract and Applied Analysis 5
Model 2 Now consider another form of Volterra equation
120597V119899(119905)
120597119905= V119899(119905) (V119899minus1
(119905) minus V119899+1
(119905))
+ V119899(119905) (V119899minus2
(119905) minus V119899+2
(119905)) 119899 ≻ 0 119905 isin [0 1] (42)
with initial conditions
V119899(0) = 119899 (43)
The exact solution of (42) is given by
V119899(119905) =
119899
1 + 6119905 (44)
We take
V119899(119905) = V
1198990(119905) + 119901V
1198991(119905) + 1199012V
1198992(119905)
+ 1199013V1198993(119905)
V119899+1
(119905) = V(119899+1)0
(119905) + 119901V(119899+1)1
(119905)
+ 1199012V(119899+1)2
(119905) + 1199013V(119899+1)3
(119905)
V119899minus1
(119905) = V(119899minus1)0
(119905) + 119901V(119899minus1)1
(119905)
+ 1199012V(119899minus1)2
(119905) + 1199013V(119899minus1)3
(119905)
V119899+2
(119905) = V(119899+2)0
(119905) + 119901V(119899+2)1
(119905)
+ 1199012V(119899+2)2
(119905) + 1199013V(119899+2)3
(119905)
V119899minus2
(119905) = V(119899minus2)0
(119905) + 119901V(119899minus2)1
(119905)
+ 1199012V(119899minus2)2
(119905) + 1199013V(119899minus2)3
(119905)
(45)
According to (1) we have
L =120597V119899(119905)
120597119905 119891 (119905) = 0
N = minusV119899(119905) (V119899minus1
(119905) minus V119899+1
(119905))
minus V119899(119905) (V119899minus2
(119905) minus V119899+2
(119905))
(46)
Using (42) (43) and (45) into (5) and using the methoddiscussed in Section 2 lead to the following
Zeroth Order Problem Consider
120597V1198990(119905)
120597119905= 0 (47)
with
V1198990(0) = 119899 (48)
from which we obtain
V1198990(119905) = 119899 (49)
First Order Problem One has the following
120597V1198991(119905)
120597119905= (1 + 119862
1)120597V1198990(119905)
120597119905
+ 1198621(V(119899+2)0
(119905) minus 1198621V(119899minus2)0
(119905)) V1198990(119905)
+ 1198621(V(119899+1)0
(119905) minus 1198621V(119899minus1)0
(119905)) V1198990(119905)
(50)
with
V1198991(0) = 0 (51)
Its solution is
V1198991(119905) = 6119862
1119899119905 (52)
Second Order Problem Consider
120597V1198992(119905)
120597119905= (1 + 119862
1)120597V1198991(119905)
120597119905+ 1198622
120597V1198990(119905)
120597119905
+ 1198621(V(119899+2)1
(119905) minus V(119899minus2)1
(119905)) V1198990(119905)
+ 1198622(V(119899+2)0
(119905) minus V(119899minus2)0
(119905)) V1198990
+ 1198621(V(119899+2)0
(119905) minus V(119899minus2)0
(119905)) V1198991(119905)
+ 1198621(V(119899+1)0
(119905) minus V(119899minus1)0
(119905)) V1198991(119905)
+ 1198622(V(119899+1)0
(119905) minus V(119899minus1)0
(119905)) V1198990(119905)
+ 1198621(V(119899+1)1
(119905) minus V(119899minus1)1
(119905)) V1198990(119905)
(53)
with
V1198992(0) = 0 (54)
V1198992(119905) = 6119899119905 (119862
1+ 1198622+ 11986221(1 + 6119905)) (55)
Third Order Problem One has
120597V1198993(119905)
120597119905= (1 + 119862
1)120597V1198992(119905)
120597119905+ 1198622
120597V1198991(119905)
120597119905+ 1198623
120597V1198990(119905)
120597119905
+ 1198621(V(119899+2)2
(119905) minus V(119899minus2)2
(119905)) V1198990(119905)
+ 1198622(V(119899+2)1
(119905) minus V(119899minus2)1
(119905)) V1198990(119905)
+ 1198623(V(119899+2)0
(119905) minus V(119899minus2)0
(119905)) V1198990(119905)
+ 1198621(V(119899+2)1
(119905) minus V(119899minus2)1
(119905)) V1198991(119905)
6 Abstract and Applied Analysis
+ 1198622(V(119899+2)0
(119905) minus V(119899minus2)0
(119905)) V1198991(119905)
+ 1198621(V(119899+2)0
(119905) minus V(119899minus2)0
(119905)) V1198992(119905)
+ 1198621(V(119899+1)2
(119905) minus V(119899minus1)2
(119905)) V1198990(119905)
+ 1198621(V(119899+1)0
(119905) minus V(119899minus1)0
(119905)) V1198992(119905)
+ 1198621(V(119899+1)1
(119905) minus V(119899minus1)1
(119905)) V1198991(119905)
+ 1198622(V(119899+1)0
(119905) minus V(119899minus1)0
(119905)) V1198991(119905)
+ 1198621(V(119899+1)2
(119905) minus V(119899minus1)2
(119905)) V1198990(119905)
+ 1198622(V(119899+1)1
minus V(119899minus1)1
) V1198990(119899 119905)
+ 1198623(V(119899+1)0
(119899 119905) minus V(119899minus1)0
(119899 119905)) V1198990(119905)
(56)
with
V1198993(0) = 0 (57)
V1198993(119905) = 6119899119905 (119862
2+ 1198623+ 211986221(1 + 6119905)
+11986231(1 + 6119905)2 + 119862
1(1 + 2119862
2(1 + 6119905)))
(58)
Adding (49) (52) (55) and (58) we obtain
V119899(119905) = V
1198990(119905) + V
1198991(119905 1198621) + V1198992(119905 1198621 1198622)
+ V1198993(119905 1198621 1198622 1198623)
V1198993(119905) = 119899 + 6119862
1119899119905 + 6119899119905 (119862
1+ 1198622+ 11986221(1 + 6119905))
+ 6119899119905 (1198622+ 1198623+ 211986221(1 + 6119905) + 1198623
1(1 + 6119905)2
+ 11986231(1 + 6119905)2 + 119862
1(1 + 2119862
2(1 + 6119905)))
(59)
For the computation of the constants11986211198622 and119862
3applying
the method of least square mentioned in (16)ndash(18) we get
1198621= minus023640805995815892
1198622= minus0039496242739004025
1198623= minus0012135699742215487 for 119899 = 1
(60)
The exact solution of problem (42) is [9]
V119899(119905) =
119899
1 + 6119905 (61)
The HAM solution of (42) is given by [9]
V119899(119905) = (119899 + 119905) + ℎ (119899 + 1) 119905 + 3ℎ1199052 + 119905ℎ (1 + ℎ) (1 + 6119899)
+ (36ℎ2119899 + 6ℎ2 + 3ℎ) 1199052 + 18ℎ21199053(62)
2 4 6 8 1000
02
04
06
08
10
12
14
HAMExactOHAM
n
n
Figure 3 Comparisons of OHAM HAM and exact solutions at 119905 =1 ℎ = minus1
0 2 4 6 8 1000
05
10
15
20
25
ExactThird
SecondFirst
n
n
Figure 4 Comparison of first second third of OHAM HAM andexact solutions at 119905 = 1 ℎ = minus1
0 2 4 6 8 10minus7
minus6
minus5
minus4
minus3
minus2
minus1
0
Resid
ual
Residual
n
Figure 5 Residual of (42) at 119905 = 1
Abstract and Applied Analysis 7
4 Results and Discussions
The formulation presented in Section 2 provides highly accu-rate solutions for the problems demonstrated in Section 3We have used Mathematica 7 for most of our computationalwork Table 1 and Figure 1 give the comparisons of OHAMwith ADM and exact solutions for Model 1 Also the absoluteerrors at different values of 119899 at 119905 = 1 are given (Figure 4)Figure 3 shows the comparisons ofOHAMresults withHAMexact at different values of 119899 at 119905 = 1 for Model 2 Theconvergence of approximate orders to exact solutions is givenin Table 2 for Model 1 and Figure 5 for Model 2The residualshave been plotted for Model 1 in Figure 2 and for Model 2in Figure 5 We have concluded that the results obtained byOHAM are identical to the results obtained by ADM HAMand exact OHAM converges rapidly with increasing of theorder of approximation
5 Conclusion
In this work we have seen the effectiveness of OHAM[12ndash16] to differential-difference equations By applying thebasic idea of OHAM to differential-difference equations wefound it simpler in applicability more convenient to con-trol convergence and involved less computational overheadTherefore OHAM shows its validity and great potential forthe differential-difference equations arising in science andengineering
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Theauthors are thankful to the anonymous reviewers for theirconstructive and valuable suggestions
References
[1] D Levi and R Yamilov ldquoConditions for the existence of highersymmetries of evolutionary equations on the latticerdquo Journal ofMathematical Physics vol 38 no 12 pp 6648ndash6674 1997
[2] R I Yamilov ldquoConstruction scheme for discreteMiura transfor-mationsrdquo Journal of Physics A Mathematical and General vol27 no 20 pp 6839ndash6851 1994
[3] I Yu Cherdantsev and R I Yamilov ldquoMaster symmetries fordifferential-difference equations of the Volterra typerdquo PhysicaD Nonlinear Phenomena vol 87 no 1ndash4 pp 140ndash144 1995
[4] A B Shabat and R I Yamilov ldquoSymmetries of nonlinearlatticesrdquo Leningrad Mathematical Journal vol 2 p 377 1991
[5] D-J Zhang ldquoSingular solutions in Casoratian form for twodifferential-difference equationsrdquo Chaos Solitons and Fractalsvol 23 no 4 pp 1333ndash1350 2005
[6] K Narita ldquoSoliton solution for a highly nonlinear difference-differential equationrdquo Chaos Solitons and Fractals vol 3 no 3pp 279ndash283 1993
[7] Z Wang and H-Q Zhang ldquoNew exact solutions to somedifference differential equationsrdquoChinese Physics vol 15 no 10pp 2210ndash2215 2006
[8] W Zhen and H Zhang ldquoA symbolic computational methodfor constructing exact solutions to difference-differential equa-tionsrdquoAppliedMathematics andComputation vol 178 no 2 pp431ndash440 2006
[9] L Zou Z Zong and G H Dong ldquoGeneralizing homotopyanalysis method to solve Lotka-Volterra equationrdquo Computersamp Mathematics with Applications vol 56 no 9 pp 2289ndash22932008
[10] Z Wang L Zou and H Zhang ldquoApplying homotopy analysismethod for solving differential-difference equationrdquo PhysicsLetters A General Atomic and Solid State Physics vol 369 no1-2 pp 77ndash84 2007
[11] Z Wang L Zou and Z Zong ldquoAdomian decomposition andPade approximate for solving differential-difference equationrdquoApplied Mathematics and Computation vol 218 no 4 pp 1371ndash1378 2011
[12] N Herisanu and VMarinca ldquoExplicit analytical approximationto large-amplitude non-linear oscillations of a uniform can-tilever beam carrying an intermediate lumped mass and rotaryinertiardquoMeccanica vol 45 no 6 pp 847ndash855 2010
[13] V Marinca and N Herisanu ldquoApplication of optimal homotopyasymptotic method for solving nonlinear equations arising inheat transferrdquo International Communications in Heat and MassTransfer vol 35 no 6 pp 710ndash715 2008
[14] V Marinca N Herisanu C Bota and B Marinca ldquoAn optimalhomotopy asymptotic method applied to the steady flow of afourth-grade fluid past a porous platerdquo Applied MathematicsLetters vol 22 no 2 pp 245ndash251 2009
[15] V Marinca N Herisanu and I Nemes ldquoA new analyticapproach to nonlinear vibration of an electrical machinerdquoProceedings of the RomanianAcademy vol 9 pp 229ndash236 2008
[16] V Marinca and N Herisanu ldquoDetermination of periodicsolutions for the motion of a particle on a rotating parabola bymeans of the optimal homotopy asymptotic methodrdquo Journal ofSound and Vibration vol 329 no 9 pp 1450ndash1459 2010
[17] M Idrees S Islam and A M Tirmizi ldquoApplication of optimalhomotopy asymptotic method of the Korteqag-de-Varies equa-tionrdquo Computers amp Mathematics with Applications vol 63 pp695ndash707 2012
[18] HUllah S IslamM Idrees andMArif ldquoSolution of boundarylayer problems with heat transfer by optimal homotopy asymp-totic methodrdquo Abstract and Applied Analysis vol 2013 ArticleID 324869 10 pages 2013
[19] R Nawaz H Ullah S Islam and M Idrees ldquoApplication ofoptimal homotopy asymptotic method to Burger equationsrdquoJournal of Applied Mathematics vol 2013 Article ID 387478 8pages 2013
[20] H Ullah S Islam M Idrees and R Nawaz ldquoApplicationof optimal homotopy asymptotic method to doubly wavesolutions of the coupled Drinfelrsquod-Sokolov-Wilson EquationsrdquoMathematical Problems in Engineering vol 2013 Article ID362816 8 pages 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Abstract and Applied Analysis
Table 1 Comparisons of OHAM exact and ADM results atdifferent values of 119899 at 119905 = 1
119899 OHAM Exact ADM Abs error1 0333333 0333333 0333333 717103 times 10minus7
2 0666665 0666665 0666665 143421 times 10minus6
3 0999998 0999998 0999998 215131 times 10minus6
4 133333 133333 133333 286841 times 10minus6
5 166666 166666 166666 358551 times 10minus6
6 2 2 2 430262 times 10minus6
7 233333 233333 233333 501972 times 10minus6
8 266666 266666 266666 573682 times 10minus6
9 299999 299999 299999 645392 times 10minus6
10 333333 333333 333333 717103 times 10minus6
Table 2 Comparisons of first second and third order and exactsolutions at different values of 119899 at 119905 = 1
119899 First order Second order Third order Exact sol1 0527142 0858619 0333333 03333332 105428 171724 0666665 06666653 158143 257586 0999998 09999984 210857 343447 133333 1333335 263571 429309 166666 1666666 316285 515171 2 27 368999 601033 233333 2333338 421713 686895 266666 2666669 474428 772757 299999 29999910 527142 858619 333333 333333
with
V1198993(0) = 0 (34)
V1198993(119905) =
2
3(31198621119899119905 + 61198622
1119899119905 + 31198623
1119899119905 + 3119862
2119899119905 + 3119862
3119899119905
+12119862211198991199052 + 121198623
11198991199052 + 6119862
111986221198991199052 + 41198623
11198991199053)
(35)
Adding (26) (29) (32) and (35) we obtain
V119899(119905) = V
1198990(119905) + V
1198991(119905 1198621) + V1198992(119905 1198621 1198622)
+ V1198993(119905 1198621 1198622 1198623)
(36)
V119899(119905) = 119899 + 2119862
1119899119905 + 2 (119862
1119899119905 + 1198622
1119899119905 + 119862
2119899119905 + 21198622
11198991199052)
+2
3(31198621119899119905 + 61198622
1119899119905 + 31198623
1119899119905 + 3119862
2119899119905 + 3119862
3119899119905
+12119862211198991199052 + 121198623
11198991199052 + 6119862
111986221198991199052 + 41198623
11198991199053) (37)
For the computation of the constants1198621 1198622 and119862
3applying
the method of least square mentioned in (16)ndash(18) we get
1198621= minus023642908394894221
1198622= 02344713680405294
1198623= minus04015040325725273 for 119899 = 1
(38)
2 4 6 8 1000
05
10
15
20
25
30
ADMExactOHAM
n
n
Figure 1 Comparison ofOHAM exact andADMsolutions at 119905 = 1
2 4 6 8 10minus6
minus5
minus4
minus3
minus2
minus1
0Re
sidual
Residual
n
Figure 2 Residual of (19) at 119905 = 1
Putting these values in (37) we obtained the approximatesolution of the form
V119899(119905) = 119899 minus 0472858119899119905 + 2 (005119899119905 + 01117971198991199052)
+2
3(minus0914641119899119905 + 01795761198991199052 minus 005286431198991199053)
(39)
The exact solution of the problem (31) is [11]
V119899(119905) =
119899
1 + 119899119905 (40)
The Adomian solution with Pade approximation of (31) isgiven by [11]
V119899(119905) =
119899
1 + 119899119905 (41)
Abstract and Applied Analysis 5
Model 2 Now consider another form of Volterra equation
120597V119899(119905)
120597119905= V119899(119905) (V119899minus1
(119905) minus V119899+1
(119905))
+ V119899(119905) (V119899minus2
(119905) minus V119899+2
(119905)) 119899 ≻ 0 119905 isin [0 1] (42)
with initial conditions
V119899(0) = 119899 (43)
The exact solution of (42) is given by
V119899(119905) =
119899
1 + 6119905 (44)
We take
V119899(119905) = V
1198990(119905) + 119901V
1198991(119905) + 1199012V
1198992(119905)
+ 1199013V1198993(119905)
V119899+1
(119905) = V(119899+1)0
(119905) + 119901V(119899+1)1
(119905)
+ 1199012V(119899+1)2
(119905) + 1199013V(119899+1)3
(119905)
V119899minus1
(119905) = V(119899minus1)0
(119905) + 119901V(119899minus1)1
(119905)
+ 1199012V(119899minus1)2
(119905) + 1199013V(119899minus1)3
(119905)
V119899+2
(119905) = V(119899+2)0
(119905) + 119901V(119899+2)1
(119905)
+ 1199012V(119899+2)2
(119905) + 1199013V(119899+2)3
(119905)
V119899minus2
(119905) = V(119899minus2)0
(119905) + 119901V(119899minus2)1
(119905)
+ 1199012V(119899minus2)2
(119905) + 1199013V(119899minus2)3
(119905)
(45)
According to (1) we have
L =120597V119899(119905)
120597119905 119891 (119905) = 0
N = minusV119899(119905) (V119899minus1
(119905) minus V119899+1
(119905))
minus V119899(119905) (V119899minus2
(119905) minus V119899+2
(119905))
(46)
Using (42) (43) and (45) into (5) and using the methoddiscussed in Section 2 lead to the following
Zeroth Order Problem Consider
120597V1198990(119905)
120597119905= 0 (47)
with
V1198990(0) = 119899 (48)
from which we obtain
V1198990(119905) = 119899 (49)
First Order Problem One has the following
120597V1198991(119905)
120597119905= (1 + 119862
1)120597V1198990(119905)
120597119905
+ 1198621(V(119899+2)0
(119905) minus 1198621V(119899minus2)0
(119905)) V1198990(119905)
+ 1198621(V(119899+1)0
(119905) minus 1198621V(119899minus1)0
(119905)) V1198990(119905)
(50)
with
V1198991(0) = 0 (51)
Its solution is
V1198991(119905) = 6119862
1119899119905 (52)
Second Order Problem Consider
120597V1198992(119905)
120597119905= (1 + 119862
1)120597V1198991(119905)
120597119905+ 1198622
120597V1198990(119905)
120597119905
+ 1198621(V(119899+2)1
(119905) minus V(119899minus2)1
(119905)) V1198990(119905)
+ 1198622(V(119899+2)0
(119905) minus V(119899minus2)0
(119905)) V1198990
+ 1198621(V(119899+2)0
(119905) minus V(119899minus2)0
(119905)) V1198991(119905)
+ 1198621(V(119899+1)0
(119905) minus V(119899minus1)0
(119905)) V1198991(119905)
+ 1198622(V(119899+1)0
(119905) minus V(119899minus1)0
(119905)) V1198990(119905)
+ 1198621(V(119899+1)1
(119905) minus V(119899minus1)1
(119905)) V1198990(119905)
(53)
with
V1198992(0) = 0 (54)
V1198992(119905) = 6119899119905 (119862
1+ 1198622+ 11986221(1 + 6119905)) (55)
Third Order Problem One has
120597V1198993(119905)
120597119905= (1 + 119862
1)120597V1198992(119905)
120597119905+ 1198622
120597V1198991(119905)
120597119905+ 1198623
120597V1198990(119905)
120597119905
+ 1198621(V(119899+2)2
(119905) minus V(119899minus2)2
(119905)) V1198990(119905)
+ 1198622(V(119899+2)1
(119905) minus V(119899minus2)1
(119905)) V1198990(119905)
+ 1198623(V(119899+2)0
(119905) minus V(119899minus2)0
(119905)) V1198990(119905)
+ 1198621(V(119899+2)1
(119905) minus V(119899minus2)1
(119905)) V1198991(119905)
6 Abstract and Applied Analysis
+ 1198622(V(119899+2)0
(119905) minus V(119899minus2)0
(119905)) V1198991(119905)
+ 1198621(V(119899+2)0
(119905) minus V(119899minus2)0
(119905)) V1198992(119905)
+ 1198621(V(119899+1)2
(119905) minus V(119899minus1)2
(119905)) V1198990(119905)
+ 1198621(V(119899+1)0
(119905) minus V(119899minus1)0
(119905)) V1198992(119905)
+ 1198621(V(119899+1)1
(119905) minus V(119899minus1)1
(119905)) V1198991(119905)
+ 1198622(V(119899+1)0
(119905) minus V(119899minus1)0
(119905)) V1198991(119905)
+ 1198621(V(119899+1)2
(119905) minus V(119899minus1)2
(119905)) V1198990(119905)
+ 1198622(V(119899+1)1
minus V(119899minus1)1
) V1198990(119899 119905)
+ 1198623(V(119899+1)0
(119899 119905) minus V(119899minus1)0
(119899 119905)) V1198990(119905)
(56)
with
V1198993(0) = 0 (57)
V1198993(119905) = 6119899119905 (119862
2+ 1198623+ 211986221(1 + 6119905)
+11986231(1 + 6119905)2 + 119862
1(1 + 2119862
2(1 + 6119905)))
(58)
Adding (49) (52) (55) and (58) we obtain
V119899(119905) = V
1198990(119905) + V
1198991(119905 1198621) + V1198992(119905 1198621 1198622)
+ V1198993(119905 1198621 1198622 1198623)
V1198993(119905) = 119899 + 6119862
1119899119905 + 6119899119905 (119862
1+ 1198622+ 11986221(1 + 6119905))
+ 6119899119905 (1198622+ 1198623+ 211986221(1 + 6119905) + 1198623
1(1 + 6119905)2
+ 11986231(1 + 6119905)2 + 119862
1(1 + 2119862
2(1 + 6119905)))
(59)
For the computation of the constants11986211198622 and119862
3applying
the method of least square mentioned in (16)ndash(18) we get
1198621= minus023640805995815892
1198622= minus0039496242739004025
1198623= minus0012135699742215487 for 119899 = 1
(60)
The exact solution of problem (42) is [9]
V119899(119905) =
119899
1 + 6119905 (61)
The HAM solution of (42) is given by [9]
V119899(119905) = (119899 + 119905) + ℎ (119899 + 1) 119905 + 3ℎ1199052 + 119905ℎ (1 + ℎ) (1 + 6119899)
+ (36ℎ2119899 + 6ℎ2 + 3ℎ) 1199052 + 18ℎ21199053(62)
2 4 6 8 1000
02
04
06
08
10
12
14
HAMExactOHAM
n
n
Figure 3 Comparisons of OHAM HAM and exact solutions at 119905 =1 ℎ = minus1
0 2 4 6 8 1000
05
10
15
20
25
ExactThird
SecondFirst
n
n
Figure 4 Comparison of first second third of OHAM HAM andexact solutions at 119905 = 1 ℎ = minus1
0 2 4 6 8 10minus7
minus6
minus5
minus4
minus3
minus2
minus1
0
Resid
ual
Residual
n
Figure 5 Residual of (42) at 119905 = 1
Abstract and Applied Analysis 7
4 Results and Discussions
The formulation presented in Section 2 provides highly accu-rate solutions for the problems demonstrated in Section 3We have used Mathematica 7 for most of our computationalwork Table 1 and Figure 1 give the comparisons of OHAMwith ADM and exact solutions for Model 1 Also the absoluteerrors at different values of 119899 at 119905 = 1 are given (Figure 4)Figure 3 shows the comparisons ofOHAMresults withHAMexact at different values of 119899 at 119905 = 1 for Model 2 Theconvergence of approximate orders to exact solutions is givenin Table 2 for Model 1 and Figure 5 for Model 2The residualshave been plotted for Model 1 in Figure 2 and for Model 2in Figure 5 We have concluded that the results obtained byOHAM are identical to the results obtained by ADM HAMand exact OHAM converges rapidly with increasing of theorder of approximation
5 Conclusion
In this work we have seen the effectiveness of OHAM[12ndash16] to differential-difference equations By applying thebasic idea of OHAM to differential-difference equations wefound it simpler in applicability more convenient to con-trol convergence and involved less computational overheadTherefore OHAM shows its validity and great potential forthe differential-difference equations arising in science andengineering
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Theauthors are thankful to the anonymous reviewers for theirconstructive and valuable suggestions
References
[1] D Levi and R Yamilov ldquoConditions for the existence of highersymmetries of evolutionary equations on the latticerdquo Journal ofMathematical Physics vol 38 no 12 pp 6648ndash6674 1997
[2] R I Yamilov ldquoConstruction scheme for discreteMiura transfor-mationsrdquo Journal of Physics A Mathematical and General vol27 no 20 pp 6839ndash6851 1994
[3] I Yu Cherdantsev and R I Yamilov ldquoMaster symmetries fordifferential-difference equations of the Volterra typerdquo PhysicaD Nonlinear Phenomena vol 87 no 1ndash4 pp 140ndash144 1995
[4] A B Shabat and R I Yamilov ldquoSymmetries of nonlinearlatticesrdquo Leningrad Mathematical Journal vol 2 p 377 1991
[5] D-J Zhang ldquoSingular solutions in Casoratian form for twodifferential-difference equationsrdquo Chaos Solitons and Fractalsvol 23 no 4 pp 1333ndash1350 2005
[6] K Narita ldquoSoliton solution for a highly nonlinear difference-differential equationrdquo Chaos Solitons and Fractals vol 3 no 3pp 279ndash283 1993
[7] Z Wang and H-Q Zhang ldquoNew exact solutions to somedifference differential equationsrdquoChinese Physics vol 15 no 10pp 2210ndash2215 2006
[8] W Zhen and H Zhang ldquoA symbolic computational methodfor constructing exact solutions to difference-differential equa-tionsrdquoAppliedMathematics andComputation vol 178 no 2 pp431ndash440 2006
[9] L Zou Z Zong and G H Dong ldquoGeneralizing homotopyanalysis method to solve Lotka-Volterra equationrdquo Computersamp Mathematics with Applications vol 56 no 9 pp 2289ndash22932008
[10] Z Wang L Zou and H Zhang ldquoApplying homotopy analysismethod for solving differential-difference equationrdquo PhysicsLetters A General Atomic and Solid State Physics vol 369 no1-2 pp 77ndash84 2007
[11] Z Wang L Zou and Z Zong ldquoAdomian decomposition andPade approximate for solving differential-difference equationrdquoApplied Mathematics and Computation vol 218 no 4 pp 1371ndash1378 2011
[12] N Herisanu and VMarinca ldquoExplicit analytical approximationto large-amplitude non-linear oscillations of a uniform can-tilever beam carrying an intermediate lumped mass and rotaryinertiardquoMeccanica vol 45 no 6 pp 847ndash855 2010
[13] V Marinca and N Herisanu ldquoApplication of optimal homotopyasymptotic method for solving nonlinear equations arising inheat transferrdquo International Communications in Heat and MassTransfer vol 35 no 6 pp 710ndash715 2008
[14] V Marinca N Herisanu C Bota and B Marinca ldquoAn optimalhomotopy asymptotic method applied to the steady flow of afourth-grade fluid past a porous platerdquo Applied MathematicsLetters vol 22 no 2 pp 245ndash251 2009
[15] V Marinca N Herisanu and I Nemes ldquoA new analyticapproach to nonlinear vibration of an electrical machinerdquoProceedings of the RomanianAcademy vol 9 pp 229ndash236 2008
[16] V Marinca and N Herisanu ldquoDetermination of periodicsolutions for the motion of a particle on a rotating parabola bymeans of the optimal homotopy asymptotic methodrdquo Journal ofSound and Vibration vol 329 no 9 pp 1450ndash1459 2010
[17] M Idrees S Islam and A M Tirmizi ldquoApplication of optimalhomotopy asymptotic method of the Korteqag-de-Varies equa-tionrdquo Computers amp Mathematics with Applications vol 63 pp695ndash707 2012
[18] HUllah S IslamM Idrees andMArif ldquoSolution of boundarylayer problems with heat transfer by optimal homotopy asymp-totic methodrdquo Abstract and Applied Analysis vol 2013 ArticleID 324869 10 pages 2013
[19] R Nawaz H Ullah S Islam and M Idrees ldquoApplication ofoptimal homotopy asymptotic method to Burger equationsrdquoJournal of Applied Mathematics vol 2013 Article ID 387478 8pages 2013
[20] H Ullah S Islam M Idrees and R Nawaz ldquoApplicationof optimal homotopy asymptotic method to doubly wavesolutions of the coupled Drinfelrsquod-Sokolov-Wilson EquationsrdquoMathematical Problems in Engineering vol 2013 Article ID362816 8 pages 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 5
Model 2 Now consider another form of Volterra equation
120597V119899(119905)
120597119905= V119899(119905) (V119899minus1
(119905) minus V119899+1
(119905))
+ V119899(119905) (V119899minus2
(119905) minus V119899+2
(119905)) 119899 ≻ 0 119905 isin [0 1] (42)
with initial conditions
V119899(0) = 119899 (43)
The exact solution of (42) is given by
V119899(119905) =
119899
1 + 6119905 (44)
We take
V119899(119905) = V
1198990(119905) + 119901V
1198991(119905) + 1199012V
1198992(119905)
+ 1199013V1198993(119905)
V119899+1
(119905) = V(119899+1)0
(119905) + 119901V(119899+1)1
(119905)
+ 1199012V(119899+1)2
(119905) + 1199013V(119899+1)3
(119905)
V119899minus1
(119905) = V(119899minus1)0
(119905) + 119901V(119899minus1)1
(119905)
+ 1199012V(119899minus1)2
(119905) + 1199013V(119899minus1)3
(119905)
V119899+2
(119905) = V(119899+2)0
(119905) + 119901V(119899+2)1
(119905)
+ 1199012V(119899+2)2
(119905) + 1199013V(119899+2)3
(119905)
V119899minus2
(119905) = V(119899minus2)0
(119905) + 119901V(119899minus2)1
(119905)
+ 1199012V(119899minus2)2
(119905) + 1199013V(119899minus2)3
(119905)
(45)
According to (1) we have
L =120597V119899(119905)
120597119905 119891 (119905) = 0
N = minusV119899(119905) (V119899minus1
(119905) minus V119899+1
(119905))
minus V119899(119905) (V119899minus2
(119905) minus V119899+2
(119905))
(46)
Using (42) (43) and (45) into (5) and using the methoddiscussed in Section 2 lead to the following
Zeroth Order Problem Consider
120597V1198990(119905)
120597119905= 0 (47)
with
V1198990(0) = 119899 (48)
from which we obtain
V1198990(119905) = 119899 (49)
First Order Problem One has the following
120597V1198991(119905)
120597119905= (1 + 119862
1)120597V1198990(119905)
120597119905
+ 1198621(V(119899+2)0
(119905) minus 1198621V(119899minus2)0
(119905)) V1198990(119905)
+ 1198621(V(119899+1)0
(119905) minus 1198621V(119899minus1)0
(119905)) V1198990(119905)
(50)
with
V1198991(0) = 0 (51)
Its solution is
V1198991(119905) = 6119862
1119899119905 (52)
Second Order Problem Consider
120597V1198992(119905)
120597119905= (1 + 119862
1)120597V1198991(119905)
120597119905+ 1198622
120597V1198990(119905)
120597119905
+ 1198621(V(119899+2)1
(119905) minus V(119899minus2)1
(119905)) V1198990(119905)
+ 1198622(V(119899+2)0
(119905) minus V(119899minus2)0
(119905)) V1198990
+ 1198621(V(119899+2)0
(119905) minus V(119899minus2)0
(119905)) V1198991(119905)
+ 1198621(V(119899+1)0
(119905) minus V(119899minus1)0
(119905)) V1198991(119905)
+ 1198622(V(119899+1)0
(119905) minus V(119899minus1)0
(119905)) V1198990(119905)
+ 1198621(V(119899+1)1
(119905) minus V(119899minus1)1
(119905)) V1198990(119905)
(53)
with
V1198992(0) = 0 (54)
V1198992(119905) = 6119899119905 (119862
1+ 1198622+ 11986221(1 + 6119905)) (55)
Third Order Problem One has
120597V1198993(119905)
120597119905= (1 + 119862
1)120597V1198992(119905)
120597119905+ 1198622
120597V1198991(119905)
120597119905+ 1198623
120597V1198990(119905)
120597119905
+ 1198621(V(119899+2)2
(119905) minus V(119899minus2)2
(119905)) V1198990(119905)
+ 1198622(V(119899+2)1
(119905) minus V(119899minus2)1
(119905)) V1198990(119905)
+ 1198623(V(119899+2)0
(119905) minus V(119899minus2)0
(119905)) V1198990(119905)
+ 1198621(V(119899+2)1
(119905) minus V(119899minus2)1
(119905)) V1198991(119905)
6 Abstract and Applied Analysis
+ 1198622(V(119899+2)0
(119905) minus V(119899minus2)0
(119905)) V1198991(119905)
+ 1198621(V(119899+2)0
(119905) minus V(119899minus2)0
(119905)) V1198992(119905)
+ 1198621(V(119899+1)2
(119905) minus V(119899minus1)2
(119905)) V1198990(119905)
+ 1198621(V(119899+1)0
(119905) minus V(119899minus1)0
(119905)) V1198992(119905)
+ 1198621(V(119899+1)1
(119905) minus V(119899minus1)1
(119905)) V1198991(119905)
+ 1198622(V(119899+1)0
(119905) minus V(119899minus1)0
(119905)) V1198991(119905)
+ 1198621(V(119899+1)2
(119905) minus V(119899minus1)2
(119905)) V1198990(119905)
+ 1198622(V(119899+1)1
minus V(119899minus1)1
) V1198990(119899 119905)
+ 1198623(V(119899+1)0
(119899 119905) minus V(119899minus1)0
(119899 119905)) V1198990(119905)
(56)
with
V1198993(0) = 0 (57)
V1198993(119905) = 6119899119905 (119862
2+ 1198623+ 211986221(1 + 6119905)
+11986231(1 + 6119905)2 + 119862
1(1 + 2119862
2(1 + 6119905)))
(58)
Adding (49) (52) (55) and (58) we obtain
V119899(119905) = V
1198990(119905) + V
1198991(119905 1198621) + V1198992(119905 1198621 1198622)
+ V1198993(119905 1198621 1198622 1198623)
V1198993(119905) = 119899 + 6119862
1119899119905 + 6119899119905 (119862
1+ 1198622+ 11986221(1 + 6119905))
+ 6119899119905 (1198622+ 1198623+ 211986221(1 + 6119905) + 1198623
1(1 + 6119905)2
+ 11986231(1 + 6119905)2 + 119862
1(1 + 2119862
2(1 + 6119905)))
(59)
For the computation of the constants11986211198622 and119862
3applying
the method of least square mentioned in (16)ndash(18) we get
1198621= minus023640805995815892
1198622= minus0039496242739004025
1198623= minus0012135699742215487 for 119899 = 1
(60)
The exact solution of problem (42) is [9]
V119899(119905) =
119899
1 + 6119905 (61)
The HAM solution of (42) is given by [9]
V119899(119905) = (119899 + 119905) + ℎ (119899 + 1) 119905 + 3ℎ1199052 + 119905ℎ (1 + ℎ) (1 + 6119899)
+ (36ℎ2119899 + 6ℎ2 + 3ℎ) 1199052 + 18ℎ21199053(62)
2 4 6 8 1000
02
04
06
08
10
12
14
HAMExactOHAM
n
n
Figure 3 Comparisons of OHAM HAM and exact solutions at 119905 =1 ℎ = minus1
0 2 4 6 8 1000
05
10
15
20
25
ExactThird
SecondFirst
n
n
Figure 4 Comparison of first second third of OHAM HAM andexact solutions at 119905 = 1 ℎ = minus1
0 2 4 6 8 10minus7
minus6
minus5
minus4
minus3
minus2
minus1
0
Resid
ual
Residual
n
Figure 5 Residual of (42) at 119905 = 1
Abstract and Applied Analysis 7
4 Results and Discussions
The formulation presented in Section 2 provides highly accu-rate solutions for the problems demonstrated in Section 3We have used Mathematica 7 for most of our computationalwork Table 1 and Figure 1 give the comparisons of OHAMwith ADM and exact solutions for Model 1 Also the absoluteerrors at different values of 119899 at 119905 = 1 are given (Figure 4)Figure 3 shows the comparisons ofOHAMresults withHAMexact at different values of 119899 at 119905 = 1 for Model 2 Theconvergence of approximate orders to exact solutions is givenin Table 2 for Model 1 and Figure 5 for Model 2The residualshave been plotted for Model 1 in Figure 2 and for Model 2in Figure 5 We have concluded that the results obtained byOHAM are identical to the results obtained by ADM HAMand exact OHAM converges rapidly with increasing of theorder of approximation
5 Conclusion
In this work we have seen the effectiveness of OHAM[12ndash16] to differential-difference equations By applying thebasic idea of OHAM to differential-difference equations wefound it simpler in applicability more convenient to con-trol convergence and involved less computational overheadTherefore OHAM shows its validity and great potential forthe differential-difference equations arising in science andengineering
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Theauthors are thankful to the anonymous reviewers for theirconstructive and valuable suggestions
References
[1] D Levi and R Yamilov ldquoConditions for the existence of highersymmetries of evolutionary equations on the latticerdquo Journal ofMathematical Physics vol 38 no 12 pp 6648ndash6674 1997
[2] R I Yamilov ldquoConstruction scheme for discreteMiura transfor-mationsrdquo Journal of Physics A Mathematical and General vol27 no 20 pp 6839ndash6851 1994
[3] I Yu Cherdantsev and R I Yamilov ldquoMaster symmetries fordifferential-difference equations of the Volterra typerdquo PhysicaD Nonlinear Phenomena vol 87 no 1ndash4 pp 140ndash144 1995
[4] A B Shabat and R I Yamilov ldquoSymmetries of nonlinearlatticesrdquo Leningrad Mathematical Journal vol 2 p 377 1991
[5] D-J Zhang ldquoSingular solutions in Casoratian form for twodifferential-difference equationsrdquo Chaos Solitons and Fractalsvol 23 no 4 pp 1333ndash1350 2005
[6] K Narita ldquoSoliton solution for a highly nonlinear difference-differential equationrdquo Chaos Solitons and Fractals vol 3 no 3pp 279ndash283 1993
[7] Z Wang and H-Q Zhang ldquoNew exact solutions to somedifference differential equationsrdquoChinese Physics vol 15 no 10pp 2210ndash2215 2006
[8] W Zhen and H Zhang ldquoA symbolic computational methodfor constructing exact solutions to difference-differential equa-tionsrdquoAppliedMathematics andComputation vol 178 no 2 pp431ndash440 2006
[9] L Zou Z Zong and G H Dong ldquoGeneralizing homotopyanalysis method to solve Lotka-Volterra equationrdquo Computersamp Mathematics with Applications vol 56 no 9 pp 2289ndash22932008
[10] Z Wang L Zou and H Zhang ldquoApplying homotopy analysismethod for solving differential-difference equationrdquo PhysicsLetters A General Atomic and Solid State Physics vol 369 no1-2 pp 77ndash84 2007
[11] Z Wang L Zou and Z Zong ldquoAdomian decomposition andPade approximate for solving differential-difference equationrdquoApplied Mathematics and Computation vol 218 no 4 pp 1371ndash1378 2011
[12] N Herisanu and VMarinca ldquoExplicit analytical approximationto large-amplitude non-linear oscillations of a uniform can-tilever beam carrying an intermediate lumped mass and rotaryinertiardquoMeccanica vol 45 no 6 pp 847ndash855 2010
[13] V Marinca and N Herisanu ldquoApplication of optimal homotopyasymptotic method for solving nonlinear equations arising inheat transferrdquo International Communications in Heat and MassTransfer vol 35 no 6 pp 710ndash715 2008
[14] V Marinca N Herisanu C Bota and B Marinca ldquoAn optimalhomotopy asymptotic method applied to the steady flow of afourth-grade fluid past a porous platerdquo Applied MathematicsLetters vol 22 no 2 pp 245ndash251 2009
[15] V Marinca N Herisanu and I Nemes ldquoA new analyticapproach to nonlinear vibration of an electrical machinerdquoProceedings of the RomanianAcademy vol 9 pp 229ndash236 2008
[16] V Marinca and N Herisanu ldquoDetermination of periodicsolutions for the motion of a particle on a rotating parabola bymeans of the optimal homotopy asymptotic methodrdquo Journal ofSound and Vibration vol 329 no 9 pp 1450ndash1459 2010
[17] M Idrees S Islam and A M Tirmizi ldquoApplication of optimalhomotopy asymptotic method of the Korteqag-de-Varies equa-tionrdquo Computers amp Mathematics with Applications vol 63 pp695ndash707 2012
[18] HUllah S IslamM Idrees andMArif ldquoSolution of boundarylayer problems with heat transfer by optimal homotopy asymp-totic methodrdquo Abstract and Applied Analysis vol 2013 ArticleID 324869 10 pages 2013
[19] R Nawaz H Ullah S Islam and M Idrees ldquoApplication ofoptimal homotopy asymptotic method to Burger equationsrdquoJournal of Applied Mathematics vol 2013 Article ID 387478 8pages 2013
[20] H Ullah S Islam M Idrees and R Nawaz ldquoApplicationof optimal homotopy asymptotic method to doubly wavesolutions of the coupled Drinfelrsquod-Sokolov-Wilson EquationsrdquoMathematical Problems in Engineering vol 2013 Article ID362816 8 pages 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Abstract and Applied Analysis
+ 1198622(V(119899+2)0
(119905) minus V(119899minus2)0
(119905)) V1198991(119905)
+ 1198621(V(119899+2)0
(119905) minus V(119899minus2)0
(119905)) V1198992(119905)
+ 1198621(V(119899+1)2
(119905) minus V(119899minus1)2
(119905)) V1198990(119905)
+ 1198621(V(119899+1)0
(119905) minus V(119899minus1)0
(119905)) V1198992(119905)
+ 1198621(V(119899+1)1
(119905) minus V(119899minus1)1
(119905)) V1198991(119905)
+ 1198622(V(119899+1)0
(119905) minus V(119899minus1)0
(119905)) V1198991(119905)
+ 1198621(V(119899+1)2
(119905) minus V(119899minus1)2
(119905)) V1198990(119905)
+ 1198622(V(119899+1)1
minus V(119899minus1)1
) V1198990(119899 119905)
+ 1198623(V(119899+1)0
(119899 119905) minus V(119899minus1)0
(119899 119905)) V1198990(119905)
(56)
with
V1198993(0) = 0 (57)
V1198993(119905) = 6119899119905 (119862
2+ 1198623+ 211986221(1 + 6119905)
+11986231(1 + 6119905)2 + 119862
1(1 + 2119862
2(1 + 6119905)))
(58)
Adding (49) (52) (55) and (58) we obtain
V119899(119905) = V
1198990(119905) + V
1198991(119905 1198621) + V1198992(119905 1198621 1198622)
+ V1198993(119905 1198621 1198622 1198623)
V1198993(119905) = 119899 + 6119862
1119899119905 + 6119899119905 (119862
1+ 1198622+ 11986221(1 + 6119905))
+ 6119899119905 (1198622+ 1198623+ 211986221(1 + 6119905) + 1198623
1(1 + 6119905)2
+ 11986231(1 + 6119905)2 + 119862
1(1 + 2119862
2(1 + 6119905)))
(59)
For the computation of the constants11986211198622 and119862
3applying
the method of least square mentioned in (16)ndash(18) we get
1198621= minus023640805995815892
1198622= minus0039496242739004025
1198623= minus0012135699742215487 for 119899 = 1
(60)
The exact solution of problem (42) is [9]
V119899(119905) =
119899
1 + 6119905 (61)
The HAM solution of (42) is given by [9]
V119899(119905) = (119899 + 119905) + ℎ (119899 + 1) 119905 + 3ℎ1199052 + 119905ℎ (1 + ℎ) (1 + 6119899)
+ (36ℎ2119899 + 6ℎ2 + 3ℎ) 1199052 + 18ℎ21199053(62)
2 4 6 8 1000
02
04
06
08
10
12
14
HAMExactOHAM
n
n
Figure 3 Comparisons of OHAM HAM and exact solutions at 119905 =1 ℎ = minus1
0 2 4 6 8 1000
05
10
15
20
25
ExactThird
SecondFirst
n
n
Figure 4 Comparison of first second third of OHAM HAM andexact solutions at 119905 = 1 ℎ = minus1
0 2 4 6 8 10minus7
minus6
minus5
minus4
minus3
minus2
minus1
0
Resid
ual
Residual
n
Figure 5 Residual of (42) at 119905 = 1
Abstract and Applied Analysis 7
4 Results and Discussions
The formulation presented in Section 2 provides highly accu-rate solutions for the problems demonstrated in Section 3We have used Mathematica 7 for most of our computationalwork Table 1 and Figure 1 give the comparisons of OHAMwith ADM and exact solutions for Model 1 Also the absoluteerrors at different values of 119899 at 119905 = 1 are given (Figure 4)Figure 3 shows the comparisons ofOHAMresults withHAMexact at different values of 119899 at 119905 = 1 for Model 2 Theconvergence of approximate orders to exact solutions is givenin Table 2 for Model 1 and Figure 5 for Model 2The residualshave been plotted for Model 1 in Figure 2 and for Model 2in Figure 5 We have concluded that the results obtained byOHAM are identical to the results obtained by ADM HAMand exact OHAM converges rapidly with increasing of theorder of approximation
5 Conclusion
In this work we have seen the effectiveness of OHAM[12ndash16] to differential-difference equations By applying thebasic idea of OHAM to differential-difference equations wefound it simpler in applicability more convenient to con-trol convergence and involved less computational overheadTherefore OHAM shows its validity and great potential forthe differential-difference equations arising in science andengineering
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Theauthors are thankful to the anonymous reviewers for theirconstructive and valuable suggestions
References
[1] D Levi and R Yamilov ldquoConditions for the existence of highersymmetries of evolutionary equations on the latticerdquo Journal ofMathematical Physics vol 38 no 12 pp 6648ndash6674 1997
[2] R I Yamilov ldquoConstruction scheme for discreteMiura transfor-mationsrdquo Journal of Physics A Mathematical and General vol27 no 20 pp 6839ndash6851 1994
[3] I Yu Cherdantsev and R I Yamilov ldquoMaster symmetries fordifferential-difference equations of the Volterra typerdquo PhysicaD Nonlinear Phenomena vol 87 no 1ndash4 pp 140ndash144 1995
[4] A B Shabat and R I Yamilov ldquoSymmetries of nonlinearlatticesrdquo Leningrad Mathematical Journal vol 2 p 377 1991
[5] D-J Zhang ldquoSingular solutions in Casoratian form for twodifferential-difference equationsrdquo Chaos Solitons and Fractalsvol 23 no 4 pp 1333ndash1350 2005
[6] K Narita ldquoSoliton solution for a highly nonlinear difference-differential equationrdquo Chaos Solitons and Fractals vol 3 no 3pp 279ndash283 1993
[7] Z Wang and H-Q Zhang ldquoNew exact solutions to somedifference differential equationsrdquoChinese Physics vol 15 no 10pp 2210ndash2215 2006
[8] W Zhen and H Zhang ldquoA symbolic computational methodfor constructing exact solutions to difference-differential equa-tionsrdquoAppliedMathematics andComputation vol 178 no 2 pp431ndash440 2006
[9] L Zou Z Zong and G H Dong ldquoGeneralizing homotopyanalysis method to solve Lotka-Volterra equationrdquo Computersamp Mathematics with Applications vol 56 no 9 pp 2289ndash22932008
[10] Z Wang L Zou and H Zhang ldquoApplying homotopy analysismethod for solving differential-difference equationrdquo PhysicsLetters A General Atomic and Solid State Physics vol 369 no1-2 pp 77ndash84 2007
[11] Z Wang L Zou and Z Zong ldquoAdomian decomposition andPade approximate for solving differential-difference equationrdquoApplied Mathematics and Computation vol 218 no 4 pp 1371ndash1378 2011
[12] N Herisanu and VMarinca ldquoExplicit analytical approximationto large-amplitude non-linear oscillations of a uniform can-tilever beam carrying an intermediate lumped mass and rotaryinertiardquoMeccanica vol 45 no 6 pp 847ndash855 2010
[13] V Marinca and N Herisanu ldquoApplication of optimal homotopyasymptotic method for solving nonlinear equations arising inheat transferrdquo International Communications in Heat and MassTransfer vol 35 no 6 pp 710ndash715 2008
[14] V Marinca N Herisanu C Bota and B Marinca ldquoAn optimalhomotopy asymptotic method applied to the steady flow of afourth-grade fluid past a porous platerdquo Applied MathematicsLetters vol 22 no 2 pp 245ndash251 2009
[15] V Marinca N Herisanu and I Nemes ldquoA new analyticapproach to nonlinear vibration of an electrical machinerdquoProceedings of the RomanianAcademy vol 9 pp 229ndash236 2008
[16] V Marinca and N Herisanu ldquoDetermination of periodicsolutions for the motion of a particle on a rotating parabola bymeans of the optimal homotopy asymptotic methodrdquo Journal ofSound and Vibration vol 329 no 9 pp 1450ndash1459 2010
[17] M Idrees S Islam and A M Tirmizi ldquoApplication of optimalhomotopy asymptotic method of the Korteqag-de-Varies equa-tionrdquo Computers amp Mathematics with Applications vol 63 pp695ndash707 2012
[18] HUllah S IslamM Idrees andMArif ldquoSolution of boundarylayer problems with heat transfer by optimal homotopy asymp-totic methodrdquo Abstract and Applied Analysis vol 2013 ArticleID 324869 10 pages 2013
[19] R Nawaz H Ullah S Islam and M Idrees ldquoApplication ofoptimal homotopy asymptotic method to Burger equationsrdquoJournal of Applied Mathematics vol 2013 Article ID 387478 8pages 2013
[20] H Ullah S Islam M Idrees and R Nawaz ldquoApplicationof optimal homotopy asymptotic method to doubly wavesolutions of the coupled Drinfelrsquod-Sokolov-Wilson EquationsrdquoMathematical Problems in Engineering vol 2013 Article ID362816 8 pages 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 7
4 Results and Discussions
The formulation presented in Section 2 provides highly accu-rate solutions for the problems demonstrated in Section 3We have used Mathematica 7 for most of our computationalwork Table 1 and Figure 1 give the comparisons of OHAMwith ADM and exact solutions for Model 1 Also the absoluteerrors at different values of 119899 at 119905 = 1 are given (Figure 4)Figure 3 shows the comparisons ofOHAMresults withHAMexact at different values of 119899 at 119905 = 1 for Model 2 Theconvergence of approximate orders to exact solutions is givenin Table 2 for Model 1 and Figure 5 for Model 2The residualshave been plotted for Model 1 in Figure 2 and for Model 2in Figure 5 We have concluded that the results obtained byOHAM are identical to the results obtained by ADM HAMand exact OHAM converges rapidly with increasing of theorder of approximation
5 Conclusion
In this work we have seen the effectiveness of OHAM[12ndash16] to differential-difference equations By applying thebasic idea of OHAM to differential-difference equations wefound it simpler in applicability more convenient to con-trol convergence and involved less computational overheadTherefore OHAM shows its validity and great potential forthe differential-difference equations arising in science andengineering
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
Theauthors are thankful to the anonymous reviewers for theirconstructive and valuable suggestions
References
[1] D Levi and R Yamilov ldquoConditions for the existence of highersymmetries of evolutionary equations on the latticerdquo Journal ofMathematical Physics vol 38 no 12 pp 6648ndash6674 1997
[2] R I Yamilov ldquoConstruction scheme for discreteMiura transfor-mationsrdquo Journal of Physics A Mathematical and General vol27 no 20 pp 6839ndash6851 1994
[3] I Yu Cherdantsev and R I Yamilov ldquoMaster symmetries fordifferential-difference equations of the Volterra typerdquo PhysicaD Nonlinear Phenomena vol 87 no 1ndash4 pp 140ndash144 1995
[4] A B Shabat and R I Yamilov ldquoSymmetries of nonlinearlatticesrdquo Leningrad Mathematical Journal vol 2 p 377 1991
[5] D-J Zhang ldquoSingular solutions in Casoratian form for twodifferential-difference equationsrdquo Chaos Solitons and Fractalsvol 23 no 4 pp 1333ndash1350 2005
[6] K Narita ldquoSoliton solution for a highly nonlinear difference-differential equationrdquo Chaos Solitons and Fractals vol 3 no 3pp 279ndash283 1993
[7] Z Wang and H-Q Zhang ldquoNew exact solutions to somedifference differential equationsrdquoChinese Physics vol 15 no 10pp 2210ndash2215 2006
[8] W Zhen and H Zhang ldquoA symbolic computational methodfor constructing exact solutions to difference-differential equa-tionsrdquoAppliedMathematics andComputation vol 178 no 2 pp431ndash440 2006
[9] L Zou Z Zong and G H Dong ldquoGeneralizing homotopyanalysis method to solve Lotka-Volterra equationrdquo Computersamp Mathematics with Applications vol 56 no 9 pp 2289ndash22932008
[10] Z Wang L Zou and H Zhang ldquoApplying homotopy analysismethod for solving differential-difference equationrdquo PhysicsLetters A General Atomic and Solid State Physics vol 369 no1-2 pp 77ndash84 2007
[11] Z Wang L Zou and Z Zong ldquoAdomian decomposition andPade approximate for solving differential-difference equationrdquoApplied Mathematics and Computation vol 218 no 4 pp 1371ndash1378 2011
[12] N Herisanu and VMarinca ldquoExplicit analytical approximationto large-amplitude non-linear oscillations of a uniform can-tilever beam carrying an intermediate lumped mass and rotaryinertiardquoMeccanica vol 45 no 6 pp 847ndash855 2010
[13] V Marinca and N Herisanu ldquoApplication of optimal homotopyasymptotic method for solving nonlinear equations arising inheat transferrdquo International Communications in Heat and MassTransfer vol 35 no 6 pp 710ndash715 2008
[14] V Marinca N Herisanu C Bota and B Marinca ldquoAn optimalhomotopy asymptotic method applied to the steady flow of afourth-grade fluid past a porous platerdquo Applied MathematicsLetters vol 22 no 2 pp 245ndash251 2009
[15] V Marinca N Herisanu and I Nemes ldquoA new analyticapproach to nonlinear vibration of an electrical machinerdquoProceedings of the RomanianAcademy vol 9 pp 229ndash236 2008
[16] V Marinca and N Herisanu ldquoDetermination of periodicsolutions for the motion of a particle on a rotating parabola bymeans of the optimal homotopy asymptotic methodrdquo Journal ofSound and Vibration vol 329 no 9 pp 1450ndash1459 2010
[17] M Idrees S Islam and A M Tirmizi ldquoApplication of optimalhomotopy asymptotic method of the Korteqag-de-Varies equa-tionrdquo Computers amp Mathematics with Applications vol 63 pp695ndash707 2012
[18] HUllah S IslamM Idrees andMArif ldquoSolution of boundarylayer problems with heat transfer by optimal homotopy asymp-totic methodrdquo Abstract and Applied Analysis vol 2013 ArticleID 324869 10 pages 2013
[19] R Nawaz H Ullah S Islam and M Idrees ldquoApplication ofoptimal homotopy asymptotic method to Burger equationsrdquoJournal of Applied Mathematics vol 2013 Article ID 387478 8pages 2013
[20] H Ullah S Islam M Idrees and R Nawaz ldquoApplicationof optimal homotopy asymptotic method to doubly wavesolutions of the coupled Drinfelrsquod-Sokolov-Wilson EquationsrdquoMathematical Problems in Engineering vol 2013 Article ID362816 8 pages 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of