DENMDifferential Equations and Nonlinear Mechanics1687-41021687-4099Hindawi Publishing Corporation84209410.1155/2009/842094842094Research ArticleDirect Solution of nth-Order IVPs by Homotopy Analysis MethodBatainehA. Sami1NooraniM. S. M.2HashimI.2HayatTasawar1Department of MathematicsIrbid National UniversityIrbid 2600Jordaninu.edu.jo2Centre for Modelling and Data AnalysisSchool of Mathematical SciencesUniversiti Kebangsaan Malaysia (National University of Malaysia)43600 Bangi SelangorMalaysiaukm.my2009282009200903022009040620092009Copyright © 2009This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Direct solution of a class of nth-order initial value problems (IVPs) is considered based on the homotopy analysis method (HAM). The HAM solutions contain an auxiliary parameter which provides a convenient way of controlling the convergence region of the series solutions. The HAM gives approximate analytical solutions which are of comparable accuracy to the seven- and eight-order Runge-Kutta method (RK78).

1. Introduction

Higher-order initial value problems (IVPs) arise in mathematical models for problems in physics and engineering. Generally, second- and higher-order IVPs are more difficult to solve than first-order IVPs. It is possible to integrate a special nth-order IVP by reducing it to a first-order system and applying one of the established methods available for such system. However, it seems more natural to provide direct numerical methods for solving the nth-order IVPs.

It is the purpose of the present paper to present an alternative approach for the direct solution of nth-order IVPs based on the homotopy analysis method (HAM). The analytic homotopy analysis method (HAM), initially proposed by Professor Liao in his Ph.D. thesis , is a powerful method for solving both linear and nonlinear problems. (The interested reader can refer to the much-cited book  for a systematic and clear exposition on this method.) In recent years, this method has been successfully employed to solve many types of nonlinear problems in science and engineering . All of these successful applications verified the validity, effectiveness and flexibility of the HAM. More recently, Bataineh et al.  employed the standard HAM to solve some problems in engineering sciences. HAM yields a very rapid convergence of the solution series and in most cases, usually only a few iterations leading to very accurate solutions. Thus Liao's HAM is a universal one which can solve various kinds of nonlinear equations. Bataineh et al.  first presented a modified HAM called (MHAM) to solve systems of second-order BVPs. Another new approach in HAM was presented by Yabushita et al.  who applied HAM not only to the governing differential equations, but also to algebraic equation. We call this new variant of HAM as NHAM.

In this work, we consider a class of nth-order IVPs of the form x(n)(t)=f(t,x(t),x(t),,x(n-1)(t)), subject to the initial conditions x(t0)=x0,x(t0)=x0,,x(n-1)(t0)=x0(n-1), where f represents a continuous, real linear/nonlinear function, and x0, x0, , x0(n-1) are prescribed. Some of the more recent direct (purely) numerical methods for solving second-order IVPs were developed by Cash , Ramos and Vigo-Aguiar [28, 29]. Recently Yahaya et al.  applied the seminumeric multistage modified Adomian decomposition method to solve the nth-order IVPs (1.1)-(1.2). Very recently, Chowdhury and Hashim  demonstrated the applicability of the analytic homotopy-perturbation method for solving nth-order IVPs.

The aim of this paper is to apply HAM and NHAM for the first time to obtain approximate solutions of nth-order IVPs directly. We demonstrate the accuracy of the HAM and NHAM through some test examples. Numerical comparison will be made against the seven- and eight-order Runge-Kutta method (RK78).

2. Basic Ideas of HAM

To describe the basic ideas of the HAM, we consider the following differential equation: N[x(t)]=0, where N is a nonlinear operator, t denotes the independent variable, x(t) is an unknown function. By means of generalizing the traditional homotopy method, Liao  constructs the so-called zero-order deformation equation(1-q)L[ϕ(t;q)-x0(t)]=q{N[ϕ(t;q)]}, where q[0,1] is an embedding parameter, is a nonzero auxiliary function, L is an auxiliary linear operator, x0(t) is an initial guess of x(t) and ϕ(t;q) is an unknown function. It is important to note that one has great freedom to choose auxiliary objects such as and L in HAM. Obviously, when q=0 and q=1, both ϕ(t;0)=x0(t),ϕ(t;1)=x(t), hold. Thus as q increases from 0 to 1, the solution ϕ(t;q) varies from the initial guess x0(t) to the solution x(t). Expanding ϕ(t;q) in Taylor series with respect to q, one has ϕ(t;q)=x0(t)+m=1+xm(t)qm, where xm=1m!mϕ(t;q)qm|q=0. If the auxiliary linear operator, the initial guess, the auxiliary parameter , and the auxiliary function are so properly chosen, then the series (2.4) converges at q=1 and one has ϕ(t;1)=x0(t)+m=1+xm(t), which must be one of the solutions of the original nonlinear equation, as proved by Liao . If =-1, (2.2) becomes (1-q)L[ϕ(t;q)-x0(t)]+q{N[ϕ(t;q)]}=0, which is used mostly in the HPM .

According to (2.5), the governing equations can be deduced from the zero-order deformation equations (2.2). We define the vectors xi={x0(t),x1(t),,xi(t)}. Differentiating (2.2) m times with respect to the embedding parameter q and then setting q=0 and finally dividing them by m!, we have the so-called mth-order deformation equationL[xm(t)-χmxm-1(t)]=Rm(xm-1), where Rm(xm-1)=1(m-1)!m-1{N[ϕ(t;q)]}qm-1|q=0,χm={0,m1,1,m>1. It should be emphasized that xm(t) (m1) are governed by the linear equation (2.9) with the linear boundary conditions that come from the original problem, which can be easily solved by symbolic computation softwares such as Maple and Mathematica.

A new approach in the HAM was proposed by Yabushita et al. . We will call this method NHAM. Yabushita et al.  considered the following projectile problem: du  dt+f(t)u=0,dv  dt+f(t)v+1=0, where f(t)=u2+v2. The standard HAM applied to this problem yields a divergent solution on some part of the solution domain. In NHAM, the zeroth-order deformation equations were constructed for not only (2.11), but also for (2.12). This slight modification in the NHAM gives a more accurate solution.

3. Numerical Experiments

To illustrate the effectiveness of the HAM we will consider four examples of nth-order IVPs (1.1)-(1.2).

3.1. Example  1

We first consider the nonlinear second-order IVP x+x2=0, subject to the initial conditions x(0)=1,x(0)=2. The exact solution is x(t)=1+ln(1+2t). To solve (3.1)-(3.2) by means of HAM, we choose the initial approximation x0(t)=1+2t and the linear operator L[Φ(t;q)]=2ϕ(t;q)t2, with the property L[c1+tc2]=0, where ci (i=1,2) are constants of integration. Furthermore, (3.1) suggests that we define the nonlinear operator as N[ϕ(x;q)]=2ϕ(t;q)t2+ϕ2(t;q)t. Using the above definition, we construct the zeroth-order deformation equation as in (2.2) and the mth-order deformation equation for m1 is as in (2.9) with the initial conditions xm(0)=0,xm  (0)=0, where Rm(xm-1)=xm-1(t)+j=0m-1xj(t)xm-1-j(t), now the solution of the mth-order deformation for m1 is xm(t)=χmxm-1(t)+L-1Rm(xm-1). We now successively obtain x1(t)=2t2,x2(t)=2t2+22t2+832t3,x3(t)=2t2+42t2+22t2+1632t3+1633t3+43t4, Then the series solution expression can be written in the form x(t)=x0(t)+x1(t)+x2(t)+, and so forth. Hence, the series solution when =-1 is x(t)1+2t-2t2+83t3-4t4+325t5-, which converges to the closed-form solution (3.3).

3.2. Example  2

Consider the linear fourth-order IVP, x(4)=-5x-4x, subject to the initial conditions x(0)=1,x(0)=0,x(0)=0,x(0)=1. The exact solution is x(t)=43cost+13sint-13cos2t-16sin2t. To solve (3.14)-(3.15) by means of HAM, we choose the initial approximation x0(t)=1+16t3, and the linear operator L[Φ(t;q)]=4ϕ(t;q)t4, with the property L[c1+tc2+t2c3+t3c4]=0, where ci (i=1,2,3,4) are constants of integration. According to the zeroth-order deformation equation (2.2) and the mth-order deformation equation for m1 (2.9) with the initial conditions xm(0)=0,xm(0)=0,xm(0)=0,xm(0)=0, where Rm(xm-1)=xm-1(t)+5xm-1(t)+4xm-1(t), the solution of the mth-order deformation for m1 is the same as (3.10).

We now successively obtain x1(t)=16t4+124t5+11260t7,x2(t)=16t4+162t4+124t5+1362t6+11260t7+2950402t7+125202t8+190722t9+12494800t11, Then the series solution expression can be written in the form x(t)=x0(t)+x1(t)+x2(t)+, and so forth. Hence, the series solution when =-1 is x1(t)=-16t4-124t5-11260t7,x2(t)=136t6+51008t7+12520t8+19072t9+12494800t11,x3(t)=-52016t8-2572576t9-122680t10-1133056t11-17484400t12-125945920t13-120432412000t15,x4(t)=536288t10+251596672t11+1399168t12+515567552t13+190810720t14+1544864320t15+181729648000t16+1277880803200t17+1475176173472000t19, and so forth. Hence, the series solution is x(t)1+16t3-16t4-124t5+136t6+1240t7-1480t8-1772576t9+17181440t10+, which converges to the closed-form solution (3.16).

3.3. Example  3

Now consider the nonlinear fourth-order IVP, x(4)=-xx+x2, subject to the initial conditions x(0)=0,x(0)=1,x(0)=1,x(0)=1. The exact solution is x(t)=et-1. According to the HAM, the initial approximation is x0(t)=t+12t2+16t3, and the linear operator is (3.18) with the property (3.19) where ci (i=1,2,3,4) are constants of integration. According to the zeroth-order deformation equation (2.2) and the mth-order deformation equation (2.9) with the initial conditions (3.20) with Rm(xm-1)=xm-1+j=0m-1xj(t)xm-1-j(t)-j=0m-1xj(t)xm-1-j(t), the solution of the mth-order deformation for m1 is the same as (3.10).

We now successively obtain x1(t)=-124t4-1120t5-1270t6-12520t7-120160t8 Then the series solution expression can be written in the form x(t)=x0(t)+x1(t)+x2(t)+, and so forth. Hence, the series solution when =-1 is x1(t)=124t4+1120t5+1720t6+12520t7+120160t8,x2(t)=-15040t7-140320t8+1362880t9-1453600t10-1319958400t11-111404800t12-1148262400t13,x3(t)=1403200t10+11478400t11+43479001600t12+155598400t13+111981324800t14+5332691859200t15+29106748928000t16+82929640619008000t17+829533531142144000t18 and so forth.

Hence, the series solution is x(t)t+12t2+16t3+124t4+1120t5+1720t6+15040t7+140320t8+1362880t9+13628800t10+, which converges to the closed-form solution (3.28).

3.4. Example  4

Finally we consider the nonlinear Genesio equation  x(t)+ax(t)+bx(t)-f(x(t))=0, where f(x(t))=-cx(t)+x2(t), subject to the initial conditions x(0)=0.2,x(0)=-0.3,x(0)=0.1, where a,  b,  c are positive constants satisfying ab<c.

First we solve (3.35) by means of HAM. According to the HAM, the initial approximation is x0(t)=0.2-0.3t+0.05t2, and the linear operator is L[Φ(t;q)]=3ϕ(t;q)t3, with the property L[c1+tc2+t2c3]=0, where ci (i=1,2,3) are constants of integration. According to the zeroth-order deformation equation (2.2) and the mth-order deformation equation (2.9) with the initial conditions xm(0)=0,xm(0)=0,xm(0)=0, where Rm(xm-1)=xm-1(t)+axm-1(t)+bxm-1(t)+cxm-1(t)-j=0m-1xj(t)xm-1-j(t), the solution of the mth-order deformation for m1 is the same as (3.10).

We now successively obtain when a=1.2, b=2.92 and c=6, x1(t)=0.0673333t3-0.0578333t4+0.0031666t5+0.00025t6-0.0000119048t7,x2(t)=0.0673333t3+0.06733332t3-0.0578333t4-0.03763332t4+0.00316667t5-0.0008826672t5+0.00025t6-0.0016032t6-0.0000119048t7-0.001098732t7-0.00005928572t8+0.00001753972t9-0.00000003242t10-0.00000000322t11+0.00000000092t12, and so forth.

Now we use the new technique, namely NHAM, of Yabushita et al.  to solve (3.35). In this technique, we construct the zeroth-order deformation equations for not only (3.35) but also for (3.36) as follows: (1-q)L[ϕ(t;q)-x0(t)]=q[3ϕ(t;q)t3+a2ϕ(t;q)t2+bϕ(t;q)t-f(t;q)],(1-q)L[f(t;q)-f0(x(t))]=q[f(t;q)+c  ϕ(t;q)-ϕ2(t;q)], and the mth-order deformation equationL[xm(t)-χmxm-1(t)]=Rm(xm-1),L[fm(x(t))-χmfm-1(x(t))]=Rm(fm-1), with the initial conditions xm(0)=0,xm  (0)=0,xm(0)=0, where Rm(xm-1)=xm-1(t)+axm-1(t)+bxm-1(t)-fm-1(x(t)),Rm(fm-1)=cxm-1(t)-j=0m-1xj(t)xm-1-j(t). Again, we successively obtain when a=1.2, b=2.92 and c=6, f0(x(t))=-1.16+1.68t-0.19t2-0.03t3+0.0025t4,f1(x(t))=0,f2(x(t))=0.3770672t3-0.2834672t4-0.02372t5+0.009083332t6-0.0002333332t7-0.0000321422t8+0.000001190482t9,x1(t)=0.0673333t3-0.0578333t4+0.00316667t5+0.00025t6-0.0000119048t7,x2(t)=0.0673333t3-0.06733332t3-0.0578333t4-0.03763332t4+0.00316667t5-0.0008826672t5+0.00025t6-0.004745782t6-0.0000119048t7+0.0002511112t7+0.000011252t8-0.0000004822t9, and so forth. Then the series solution expression can be written in the form

x(t)=x0(t)+x1(t)+x2(t)+,f(t)=f0(t)+f1(t)+f2(t)+. The series solutions (3.12), (3.23), (3.32), (3.49) and (3.50) contain the auxiliary parameter . The validity of the method is based on such an assumption that the series (2.4) converges at q=1. It is the auxiliary parameter which ensures that this assumption can be satisfied. In general, by means of the so-called -curve, it is straightforward to choose a proper value of which ensures that the solution series is convergent. Figure 1 show the -curves obtained from the fifth-order HAM approximation solutions of (3.1), (3.14) and (3.26). From this figure, the valid regions of correspond to the line segments nearly parallel to the horizontal axis. Substituting the special choice =-1 into the series solutions (3.12), (3.23) and (3.32) yields the exact solution (3.3), (3.16) and (3.28). Also Figures 2 and 3 show the -curves obtained from the eleventh-order HAM and NHAM approximation solutions of (3.35) and (3.36). In Figure 4 we obtain numerical solution of the Genesio equation using the eleventh-order HAM and NHAM approximation. It is demonstrated that the HAM and NHAM solutions agree very well with the solutions obtained by the seven- and eight-order Runge-Kutta method (RK78). Moreover we conclude that the proposed algorithm given by NHAM is more stable than the classical HAM.

The -curve of x(0) given by (3.1), (3.14) and (3.26): fifth-order approximation of x(0).

The -curve of x(0) obtained from the eleventh-order HAM approximation solution of (3.35).

The -curve of x(0) and f(0) given by (3.49) and (3.50): eleventh-order NHAM approximation of (3.49) and (3.50).

The eleventh-order HAM and NHAM solutions (3.49) with =-0.7998 for HAM solution and =-0.7 for NHAM solution versus RK78 solution for the (3.35) when a=1.2, b=2.92 and c=6.

Remarks 1.

Equation (3.35) represented by Genesio  as a system includes a simple square part and three simple ordinary differential equations that depend on three positive real parameters. Bataineh et al.  discussed the behavior of this system in the interval t[0,2] by using HAM, so according to Figure 4 we conclude that the behavior of numerical solution (3.35) is more stable than the numerical solution obtained by  using the classical HAM.

4. Conclusions

In this paper, the homotopy analysis method HAM was applied to solve a class of linear and nonlinear nth-order IVPs and the Genesio equation. HAM provides us with a convenient way of controlling the convergence of approximation series, which is a fundamental qualitative difference in analysis between HAM and other methods. The illustrative examples suggest that HAM is a powerful method for nonlinear problems in science and engineering.

Acknowledgments

The authors would like to acknowledge the financial support received from the MOSTI Sciencefund Grants: 04-01-02-SF0177 and the SAGA Grant STGL-011-2006 (P24c).

LiaoS.-J.The proposed homotopy analysis techniques for the solution of nonlinear problems, Ph.D. dissertation1992Shanghai, ChinaShanghai Jiao Tong UniversityLiaoS.-J.Beyond Perturbation: Introduction to the Homotopy Analysis Method20042Boca Raton, Fla, USAChapman & Hall/CRCxii+322CRC Series: Modern Mechanics and MathematicsMR2058313LiaoS.-J.An approximate solution technique not depending on small parameters: a special exampleInternational Journal of Non-Linear Mechanics199530337138010.1016/0020-7462(94)00054-EMR1336915ZBL0837.76073LiaoS.-J.A kind of approximate solution technique which does not depend upon small parameters. II. An application in fluid mechanicsInternational Journal of Non-Linear Mechanics199732581582210.1016/S0020-7462(96)00101-1MR1459007LiaoS.-J.An explicit, totally analytic approximate solution for Blasius' viscous flow problemsInternational Journal of Non-Linear Mechanics199934475977810.1016/S0020-7462(98)00056-0MR1688603LiaoS.-J.On the homotopy analysis method for nonlinear problemsApplied Mathematics and Computation2004147249951310.1016/S0096-3003(02)00790-7MR2012589ZBL1086.35005LiaoS.-J.sjliao@sjtu.edu.cnPopI.popi@math.ubbcluj.roExplicit analytic solution for similarity boundary layer equationsInternational Journal of Heat and Mass Transfer20044717585EID2-s2.0-014190785310.1016/S0017-9310(03)00405-8ZBL1045.76008LiaoS.-J.Comparison between the homotopy analysis method and homotopy perturbation methodApplied Mathematics and Computation200516921186119410.1016/j.amc.2004.10.058MR2174713ZBL1082.65534LiaoS.-J.A new branch of solutions of boundary-layer flows over an impermeable stretched plateInternational Journal of Heat and Mass Transfer200548122529253910.1016/j.ijheatmasstransfer.2005.01.005EID2-s2.0-17944369480AyubM.RasheedA.HayatT.Exact flow of a third grade fluid past a porous plate using homotopy analysis methodInternational Journal of Engineering Science200341182091210310.1016/S0020-7225(03)00207-6MR1994304HayatT.KhanM.AsgharS.Homotopy analysis of MHD flows of an Oldroyd 8-constant fluidActa Mechanica20041683-421323210.1007/s00707-004-0085-2EID2-s2.0-2442526691ZBL1063.76108HayatT.KhanM.Homotopy solutions for a generalized second-grade fluid past a porous plateNonlinear Dynamics200542439540510.1007/s11071-005-7346-zMR2190665ZBL1094.76005TanY.AbbasbandyS.Homotopy analysis method for quadratic Riccati differential equationCommunications in Nonlinear Science and Numerical Simulation200813353954610.1016/j.cnsns.2006.06.006EID2-s2.0-34848880798ZBL1132.34305AbbasbandyS.The application of homotopy analysis method to nonlinear equations arising in heat transferPhysics Letters A2006360110911310.1016/j.physleta.2006.07.065MR2288118AbbasbandyS.The application of homotopy analysis method to solve a generalized Hirota-Satsuma coupled KdV equationPhysics Letters A20061516AbbasbandyS.Approximate solution for the nonlinear model of diffusion and reaction in porous catalysts by means of the homotopy analysis methodChemical Engineering Journal20081362-314415010.1016/j.cej.2007.03.022EID2-s2.0-38649139314AbbasbandyS.LiaoS.-J.A new modification of false position method based on homotopy analysis methodApplied Mathematics and Mechanics200829222322810.1007/s10483-008-0209-zMR2391551BatainehA. S.NooraniM. S. M.HashimI.Modified homotopy analysis method for solving systems of second-order BVPsCommunications in Nonlinear Science and Numerical Simulation200914243044210.1016/j.cnsns.2007.09.012MR2458820BatainehA. S.NooraniM. S. M.HashimI.Solving systems of ODEs by homotopy analysis methodCommunications in Nonlinear Science and Numerical Simulation200813102060207010.1016/j.cnsns.2007.05.026MR2417577BatainehA. S.NooraniM. S. M.HashimI.Solutions of time-dependent Emden-Fowler type equations by homotopy analysis methodPhysics Letters A20073711-2728210.1016/j.physleta.2007.05.094EID2-s2.0-35348973440BatainehA. S.NooraniM. S. M.HashimI.The homotopy analysis method for Cauchy reaction-diffusion problemsPhysics Letters A2008372561361810.1016/j.physleta.2007.07.069MR2378731BatainehA. S.NooraniM. S. M.HashimI.Series solution of the multispecies Lotka-Volterra equations by means of the homotopy analysis methodDifferential Equations & Nonlinear Mechanics2008200814816787MR242509310.1155/2008/816787ZBL1160.34302BatainehA. S.NooraniM. S. M.HashimI.Approximate analytical solutions of systems of PDEs by homotopy analysis methodComputers & Mathematics with Applications2008551229132923MR2401440ZBL1142.65423BatainehA. S.NooraniM. S. M.HashimI.Homotopy analysis method for singular IVPs of Emden-Fowler typeCommunications in Nonlinear Science and Numerical Simulation20091441121113110.1016/j.cnsns.2008.02.004MR2468944HashimI.AbdulazizO.MomaniS.Homotopy analysis method for fractional IVPsCommunications in Nonlinear Science and Numerical Simulation200914367468410.1016/j.cnsns.2007.09.014MR2449879YabushitaK.YamashitaM.TsuboiK.An analytic solution of projectile motion with the quadratic resistance law using the homotopy analysis methodJournal of Physics A200740298403841610.1088/1751-8113/40/29/015MR2371242CashJ. R.A variable step Runge-Kutta-Nyström integrator for reversible systems of second order initial value problemsSIAM Journal on Scientific Computing200526396397810.1137/S030601727MR2126121ZBL1121.65335RamosH.Vigo-AguiarJ.Variable-stepsize Chebyshev-type methods for the integration of second-order I.V.P.'sJournal of Computational and Applied Mathematics2007204110211310.1016/j.cam.2006.04.032MR2320339ZBL1117.65106RamosH.Vigo-AguiarJ.Variable stepsize Störmer-Cowell methodsMathematical and Computer Modelling2005427-883784610.1016/j.mcm.2005.09.011MR2178513ZBL1092.65060YahayaF.HashimI.IsmailE. S.ZulkifleA. K.Direct solutions of nth order initial value problems in decomposition seriesInternational Journal of Nonlinear Sciences and Numerical Simulation200783385392EID2-s2.0-34548575151ChowdhuryM. S. H.HashimI.Direct solutions of nth-order initial value problems by homotopy-perturbation methodInternational Journal of Computer Mathematics. In press10.1080/00207160802172224HeJ.-H.Homotopy perturbation method: a new nonlinear analytical techniqueApplied Mathematics and Computation20031351737910.1016/S0096-3003(01)00312-5MR1934316ZBL1030.34013GenesioR.TesiA.Harmonic balance methods for the analysis of chaotic dynamics in nonlinear systemsAutomatica199228353154810.1016/0005-1098(92)90177-HEID2-s2.0-0026866475ZBL0765.93030