 Applied Mathematics, 2011, 2, 783-790 doi:10.4236/am.2011.26105 Published Online June 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM The Selection of Proper Auxiliary Parameters in the Homotopy Analysis Method. A Case Study: Uniform Solutions of Undamped Duffing Equation Mustafa Turkyilmazoglu Mathematics Department, University of Hacettepe, An kara, Turkey E-mail: turkyilm@hotmail.com Received March 20, 2011; revised May 9, 2011; accepted May 12, 2011 Abstract The present paper is concerned with two novel approximate analytic solutions of the undamped Duffing equation. Instead of the traditional perturbation or asymptotic methods, a homotopy technique is employed, which does not require a small perturbation parameter or a large parameter for an asymptotic expansion. It is shown that proper choices of an auxiliary linear operator and also an initial approximation during the imple-mentation of the homotopy analysis method can yield uniformly valid and accurate solutions. The obtained explicit analytical expressions for the solution predict the displacement, frequency and period of the oscilla-tions much more accurate than the previously known asymptotic or perturbation formulas. Keywords: Homotopy Analysis Method, Auxiliary Linear Operator, Initial Approximation, Duffing Oscillator, Uniform Solution 1. Introduction The importance of nonlinear Duffing equation has been widely understood among the scientists since it plays a key role in some important practical phenomena such as periodic orbit extraction, non uniformity caused by an infinite domain, nonlinear mechanical oscillators, predic- tion of diseases, and so on. Since the Duffing equation engulfs a strong nonli- nearity within itself, it’s full analytical solution has not been achieved so far. Therefore, considerable attention has been directed towards the study of strongly nonlinear oscillators and several methods have been used to find approximate solutions to nonlinear oscillators. Most of the methodologies involve the introduction of a small parameter into the equation, which might be artificial, and then expansion of the solution through the per- turbation series around this parameter. However, the solutions obtained within this approach may not be uniform, restricting the applicability of such perturbation methods [1,2]. To overcome the limitations of the perturbative techniques, many novel techniques have been proposed in recent years. For example, modified Lindstedt-Poincare method [3,4] and variational iteration methods . Some other numerical and approximate methods follow [6-10]. Liao in  proposed a new technique which is based on the homotopy concept in topology, named the ho- motopy analysis method (HAM). Unlike the aforemen- tioned traditional perturbation methods, this technique does not require a small perturbation parameter in the equation. In this method, according to the homotopy technique, a homotopy with an imbedding parameter is constructed, and the imbedding parameter is considered as a small parameter. Thus the original nonlinear pro- blem is converted into an infinite number of linear problems without using the perturbation techniques, see the book by Liao . Different from other methods, the HAM provides a simple way to control and adjust the convergence region of solution series by means of an auxiliary parameter [13,14]. Following the introduction of HAM by Liao, He in  proposed the so-called homotopy perturbation method (HPM). Unfortunately, Sajid and Hayat  pointed out that the so-called homotopy perturbation method has nothing new except its new name, because the HPM is only a special case of the homotopy analysis method (HAM) when so that all results given by the HPM can be obtained by the HAM as a special case. Moreover, as a special case of the HAM, the HPM can not give con- M. TURKYILMAZOGLU 784 vergent series solution for strongly nonlinear problems. Abbasbandy in  gave a simple example to show that, like perturbation approximations, the results given by the HPM are divergent when the physical parameters become large. Among many other authors VanGorder and Vajravelu in  also pointed out this fact. More recently, Liang and Jeffrey in  pointed out that in some cases the series solution given by the HPM and VIM (another method proposed also by He in ) is divergent at all points except that however de- fines the initial condition. =0tWe in the present paper use the homotopy analysis technique for the solution of the nonlinear Duffing equation. Better auxiliary linear operators and initial approximations are the essential target to be used here within the homotopy concept so that highly accurate solutions have been obtained. The proposed linear operators together with the homotopy analysis method provide formulas for the displacement, frequency and period of the oscillations, which are valid for all the existing parameters and also are more accurate than those already available in the literature. The following strategy is adopted in the rest of the paper. In Section 2 the idea of homotopy analysis method is laid out. Application of the method is implemented in Section 3, in which, first the homotopy analysis method is revisited and later using different auxiliary parameters and initial approximations, analytic expressions are derived yielding better results. Section4 presents compari- sons of the period and displacement between those in , our findings and the exact numerical ones. Finally conclusions follow in Section 5. 2. The Homotopy Analysis Method Liao in  proposed a new kind of analytic technique for nonlinear problems, namely the homotopy analysis method. This method is based on the homotopy and has several advantages. To underline, firstly its validity does not depend upon whether or not nonlinear equations under consideration contain small or large parameters, hence it can solve more of strongly nonlinear equations than the perturbation techniques. Secondly, it provides us with a great freedom to select proper auxiliary linear operators and initial guesses so that uniformly valid approximations can be obtained. Thirdly, it gives a family of approximations which are convergent in a larger region. Liao successfully applied the homotopy analysis method to solve some nonlinear problems in mechanics. For example, Liao in  gave a purely analytic solution of 2D Blasius’s viscous flow over a semi-infinite flat plate, which is uniformly valid in the whole physical region. Further examples are provided within the reference . To briefly describe the method let’s consider the following nonlinear differential equa- tion =0, Nu frr (1) with boundary conditions ,=0, uBu rn  (2) where is a nonlinear operator, is a boundary operator, is an unknown analytic function and NuB is the boundary of the domain . By this technique, we construct a homotopy ,vrp from the cartesian set 0, 1 to which satisfies R 0,=1 =0HvppLvuphNufr ,(3) where [0,1]p is an embedding parameter, 0 is an initial approximation of Equation (1) that satisfies the boundary condition (2) and is a constant that can be adjusted to speed the convergence. It is clear from Equation (3) that for and respectively the followings hold uh=0p1=p 0,0== 0,,1== 0.HvLv uHvNu fr (4) Hence, it can be deduced from (4) that the deformation process of from zero to unity is just that of the solution from p0ur to . Further, considering as a parameter, the solution to system (1-2) can be naturally expressed taking into account a Taylor expansion of the solution at and later imposing the expansion at ur,vt=1ppp,=0p 0=1=kkutu tut (5) where are defined by ku=01=!kpuukp. 3. Application to Duffing Equation Duffing equation describes many kinds of nonlinear oscillatory systems in physics, mechanics and engi- neering. The unforced, undamped Duffing equation that we consider consists of the nonlinear initial value system  232d=0, 0=, 0=0.duuuu Aut (6) The parameters  and A in (6) measure respec- tively the nonlinearity and amplitude of the displacement ut. Actually the period of the solution can be for- mulated analytically as in  2π220224d=, =.2111sinxATkAAkx (7) Copyright © 2011 SciRes. AM M. TURKYILMAZOGLU785 For the system (6) Liao in  suggested the subsequent auxiliary linear operator 22d=duLu ut (8) with an assumption of the initial solution of the form 0=cos,uA t (9) such that  is a nonzero unknown constant with the property of . Considering the homotopy in- troduced in (3) with and imposing the constraint that no secular terms are allowed, He in  obtained a first-order approximation to the solution in subsequent form 0=1h=1 23223=cos cos41 491AAut3,t (10) in which  is defined as 2210764 10449=.1824AAA  (11) It should be remarked here that the solution given in (10) has a non removable singularity for =1. Carrying out a further step, it is easy to obtain the next-order solution 2 in (5) which can later be com- bined with (10) to form a second-order solution. He then obtained the period of the approximate solution by u2π=T, (12) with  given as in (11). He, later claimed that the solution he achieved by the homotopy technique with the properly chosen auxiliary parameters is of high accuracy and it is valid for a very large values of . Denoting the ratio of exact period (7) to approximate one (12) in the limit of  tending to infinity as rat, He found that (it should be emphasized that a simple error occurred in  for the evaluation of = 96rat 1.040 rat). Therefore, He in  concluded that for any value of  the maximal relative error is only 4 percent. In place of the linear operator (8), He  proposed to use 222d=duLu ut (13) with the same assumption of the initial solution as in (9) and . If the occurrence of secular terms in the solution is eliminated in the first iteration, the solution to Duffing Equation (6) can be written at the first-order in the form 1=h32sinsin 2=cos ,16AtuA tt (14) with  defined as 21=432.A (15) It can be immediately seen that the displacement in (14) does not blow up for any physical value of u and the form of  obtained in (15) is much simpler than Equation (11) obtained in . With this frequency, the ratio between the exact and approximate period can be computed as 1.0222 in the limit . Thus (15) represents a better estimate and only 2 percent relative error occurs, even for infinite value of . A comparison between (10-11) and (14-15) will be performed later in section Section 4. It is a well-known fact that in the homotopy method there is a great freedom for the choice of auxiliary linear operator and initial approximate solution. We prove, in what follows, that with different proper selections, a better solution than (10-11) can be obtained. 3.1. A New Proposed Homotopy 1 Making a substitution =t permits the frequency  to appear explicitly in the Duffing Equation (6). This is actually a coupled technogy of homotopy analysis me- thod with the Lindstedt Poincare method. Afterwards, keeping the linear operator as in (8) with replaced by t, choosing the initial approximation as 0=cuAos with 21=hA (note that this selection of is nece- hssary to keep the consistency in the limit ). If the occurrence of secular terms in the solution is eliminated in the first iteration, the leading-order 0 is found to be the same as (15) and the solution to the transformed Duffing equation can be written at the first-order in the form 1=31coscos332uA. (16) With the frequency 0=, the ratio between the exact and approximate is the same as for homotopy approach of He  in the limit . To get a better estimate and hence improve the frequency, we again set the coefficient of cos to disappear so that secular terms in the next solution will be eliminated. This results in 2123=128 43AwA, (17) leading to 201 2256 189==128 43AA  whose large  limit yields a ratio 1.00623, so only 0.62 Copyright © 2011 SciRes. AM M. TURKYILMAZOGLU Copyright © 2011 SciRes. AM 786 tpercentage relative error is observed, which is much better than the above approaches. 3.2. A New Proposed Homotopy 2 If we now select the linear operator as in (13) but assume the initial approximation to the solution as 0=cosuA, with  denoting the frequency of the solution different from that of linear solution , they can be determined at the first iteration by restricting the appearance of secular terms at the next level. Within this approach  and  are found to be 4 221/31/32/32/31/34096409633601323808 38083 1433043= ,12 3iiiAA ii   (18) 222297 9=,1AA where 242634224263 84= 26214473113652809659833 6486327681239041665369251317113.AAAAAAAA   By this means, the homotopy method (3) yields the first-order solution 22222222242242 cos18131822cos2=cos .410 9AtA AAuA t  t (19) The forms of the solution in (19) and of the frequency in (18) are more complicated when compared with those of (10-11) and (14-15). However, after a straightforward evaluation it is found that in the limit = 0.997314rat, resulting in only 0.26 percent maximal relative error, which is best of all among those presented in  and above by our two approaches. 4. Discussion of the Results In this section, to show the advantages of our homotopy approaches, a comparison is made between the full nu- merical simulation results of (6), those of He in  and the ones obtained in section Section 3. We particularly choose the parameters =1A and =1 in what follows. For these specific values, it is straightforward to compute the correspondences of the periods, which are tabulated in Table 1. As compared to the exact period, our homotopy technique is seen to give extremely more accurate results than the homotopy perturbation em- ployed in . We next present numerical simulation results of the displacement function. Figure 1 shows the exact nume- rical and second-order approximate solution of He, wher- eas Figures 2-4 demonstrate the exact numerical and first-order solutions from the homotopy approaches. It is observed from Figure 1 related to the second-order homotopy perturbation method of He that the uniformity of the approximate solution gets weakened as in- creases. On the other hand, our first-order approach in the homotopy analysis gives extremely accurate solu- tions when compared with the exact solution. Particu- larly as seen from Figure 4, a much higher accuracy can be observed from the approach 3. Thus, the analytic approximations as displayed in Figures 2-4 are suffi- ciently reliable and can be used in the analysis of Duff- ing oscillator in further research. tIt should be emphasized that only the first-order app- roximate solutions that we obtained in Section 3 reveal ex- cellent agreement with the exact numerical solutions. Addition of higher approximations from the homotopy technique would yield more remarkable agreement. It is furthermore worthwhile to state that the homotopy solu- tions obtained here are valid for all the values of the parameters A and . Therefore, we do not need to approximate the small  perturbation solutions or the large  asymptotic solutions as in . In fact, our Table 1. Illustrating the periods for the Duffing Equation (6) for A = 1 and ε = 1. First is the exact, second is from He in , third is from approach of He , fourth is from approach 1 and the fifth is from approach 2. s 4.76802 4.73233 4.74964 4.78166 4.76877 M. TURKYILMAZOGLU Copyright © 2011 SciRes. AM 787 Figure 1. Solution of the Duffing Equation (6): straight curve from the numerical solution and dashed curve from the second-order solution of He (10). Figure 2. Solution of the Duffing Equation (6): straight curve from the numerical solution and dashed curve from the first-order solution of approach of He  (14). M. TURKYILMAZOGLU 788 Figure 3. Solution of the Duffing Equation (6): straight curve from the numerical solution and dashed curve from the first-order solution of approach 1 (16). Figure 4. Solution of the Duffing Equation (6): straight curve from the numerical solution and dashed curve from the first-order solution of approach 2 (19). Copyright © 2011 SciRes. AM M. TURKYILMAZOGLU Copyright © 2011 SciRes. AM 789 homotopy approaches are able to estimate the frequency and period of the oscillations in a better way. Moreover, unlike the solution of He, no singularity appears in the displacement solution. Thus it can be concluded that the homotopy method presented here can safely handle the situations of high nonlinearity occurring in the Duffing problems. 5. Concluding Remarks In this paper the nonlinear problem of Duffing equation has been considered with the homotopy analysis tech- nique. As compared to the perturbation methods, the homotopy treatment employed here does not require any small parameter, yielding explicit formulas for the quantities of physical desire, such as the displacement, frequency and the period of the oscillations. Four homotopy approaches have been pursued in this analysis. First, the homotopy method of He in  has been reinvestigated further calculating the second-order contribution that was missed in . Taking the advantage of free selection of the linear operator and the initial approximation to the solution, two more approa- ches have been proposed here. The homotopy methods with these choices are proven to generate analytic approximations that are more accurate than the result presented in . Moreover, the approximate solutions obtained by these techniques are valid not only for small parameters but also very large parameters that are in- volved in the degree of the nonlinearity. The purely explicit analytical formulas obtained for the frequency and period agree excellently with the exact values even for infinitely large parameters, the discrepancy of the approximate period with respect to the exact one is as low as 0.26%. 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