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
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
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
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
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
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
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
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
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
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
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
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
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
