The Selection of Proper Auxiliary Parameters in the Homotopy Analysis Method. A Case Study: Uniform Solutions of Undamped Duffing Equation

doi:10.4236/am.2011.26105

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Applied Mathematics, 2011, 2, 783-790

doi:10.4236/am.2011.26105 Published Online June 2011 (http://www.SciRP.org/journal/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: [email protected]

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 [5]. Some other numerical and approximate

methods follow [6-10].

Liao in [11] 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 [12]. 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 [15]

proposed the so-called homotopy perturbation method

(HPM). Unfortunately, Sajid and Hayat [16] 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 [17] 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 [18] also pointed out this fact. More

recently, Liang and Jeffrey in [19] pointed out that in

some cases the series solution given by the HPM and

VIM (another method proposed also by He in [20]) is

divergent at all points except that however de-

fines the initial condition.

=0t

We 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 [15], our findings and the exact

numerical ones. Finally conclusions follow in Section 5.

2. The Homotopy Analysis Method

Liao in [1] 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 [21] 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 [12]. To briefly describe the method

let’s consider the following nonlinear differential equa-

tion









=0, Nu frr



 (1)

with boundary conditions

,=0,

Bu r



 





 (2)

where is a nonlinear operator, is a boundary

operator, is an unknown analytic function and

NuB



the boundary of the domain . By this technique, we

construct a homotopy







,vrp from the cartesian set





0, 1 to which satisfies R













 

,=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

=0p1=p









 

,0== 0,

,1== 0.

HvLv u

HvNu 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





ur 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



,vt

=1p





=0p

 

utu tut



 (5)

where are defined by

ukp



.

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

 

d=0, 0=, 0=0.

uuuu Au





 (6)

The parameters



and

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 [2]



=, =.

11sin

Akx







 (7)

M. TURKYILMAZOGLU785

For the system (6) Liao in [22] suggested the

subsequent auxiliary linear operator



Lu u



(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 [15] obtained a

first-order approximation to the solution in subsequent

form



0=1

h=1

 

=cos cos

41 491











(10)

in which



is defined as

10764 10449





 



(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

2π

=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 ra

, He found that

(it should be emphasized that a simple

error occurred in [15] for the evaluation of

= 96rat 1.040 ra

Therefore, He in [15] concluded that for any value of



the maximal relative error is only 4 percent.

In place of the linear operator (8), He [23] proposed to

use



Lu u



 (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=h

sinsin 2

=cos ,

uA tt





 (14)

with



defined as

=43





 (15)

It can be immediately seen that the displacement

in (14) does not blow up for any physical value of



and the form of



obtained in (15) is much simpler

than Equation (11) obtained in [15]. 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



, choosing the initial approximation as 0=cuAos



with 2



(note that this selection of is nece- h

ssary 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



=31coscos3





 (16)

With the frequency 0



, the ratio between the

exact and approximate is the same as for homotopy

approach of He [23] 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

128 43



 (17)

leading to

01 2

256 189

128 43



 







whose large



limit yields a ratio 1.00623, so only 0.62

M. TURKYILMAZOGLU

786

percentage 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







4 221/31/32/32/3

1/3

4096409633601323808 38083 1433043

= ,

12 3

iiiAA ii

 





 

(18)

97 9









where



24263

4224263 84

= 26214473113652809659833

6486327681239041665369251317113.

AAA

AAAA







 

 



By this means, the homotopy method (3) yields the first-order solution













222222222

4224

2 cos18131822cos2

=cos .

410 9

tA AA

uA 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 [15]

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 [15] and

the ones obtained in section Section 3. We particularly

choose the parameters =1

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 [15].

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.

It 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

and



. Therefore, we do not need to

approximate the small



perturbation solutions or the

large



asymptotic solutions as in [2]. 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

[15], third is from approach of He [23], 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

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 [23] (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).

M. TURKYILMAZOGLU

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 [15] has

been reinvestigated further calculating the second-order

contribution that was missed in [15]. 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 [15]. 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%. In addition to this, the displacement

function is uniformly valid and does not exhibit any

singularities as compared to the solution in [15].

The homotopy technique proposed here can be safely

adopted for sets of fully coupled, highly nonlinear

equations governing other physical problems in science

and engineering. Analytical solutions obtained here also

provide a good scientific base for the validation of the

numerically computed values using different schemes in

the literature.

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