A Nonlinear Model Arising in the Buckling Analysis and its New AnalyticApproximate SolutionYasir Khana and Waleed Al-Hayanib,c

a Department of Mathematics, Zhejiang University, Hangzhou 310027, Chinab Departamento de Matematicas, Escuela Politecnica Superior, Universidad Carlos III de

Madrid, Avenida de la Universidad, 30, 28911 Leganes, Madrid, Spainc Department of Mathematics, College of Computer Science and Mathematics, University of

Mosul, Mosul-Iraq

Reprint requests to Y. K.; E-mail: yasirmath@yahoo.com

Z. Naturforsch. 68a, 355 – 361 (2013) / DOI: 10.5560/ZNA.2013-0011Received July 3, 2012 / revised November 26, 2012 / published online April 10, 2013

An analytical nonlinear buckling model where the rod is assumed to be an inextensible columnand prismatic is studied. The dimensionless parameters reduce the constitutive equation to a nonlin-ear ordinary differential equation which is solved using the Adomian decomposition method (ADM)through Green’s function technique. The nonlinear terms can be easily handled by the use of Ado-mian polynomials. The ADM technique allows us to obtain an approximate solution in a series form.Results are presented graphically to study the efficiency and accuracy of the method. To the author’sknowledge, the current paper represents a new approach to the solution of the buckling of the rodproblem. The fact that ADM solves nonlinear problems without using perturbations and small pa-rameters can be judged as a lucid benefit of this technique over the other methods.

Key words: Adomian Decomposition Method; Adomian Polynomials; Green’s Function; BucklingPhenomena.

1. Introduction

Buckling phenomena are widely used in wave prop-agation in nanostructures, nanobeams, nanoarches,nanorings, nanoplates, and nanoshells [1 – 6]. For ex-ample, buckling drastically cooperate the structural in-tegrity of nanostructures. Two types of analysis areused: One of small deflections and the other of largedeflections. Mostly, the analysis is done of small de-flections because the evolution of nanostructures aftera buckling behaviour can not be predicted in the caseof large deflection. Recently, a significant number ofnonlinear differential equations arising in the mathe-matical buckling model have been proposed [7 – 13].These models have been used to explain different phe-nomena. One of these models is mentioned for the non-local elasticity theory [13]. The original idea of stud-ied this model is based on Eringen’s nonlocal elastic-ity and Timoshenko’s beam model [9, 10]. The samemodel was then re-examined and re-solved by Xuet al. [13] for a buckling response. The complete di-mensional governing equation can be found in the orig-

inal manuscript of Xu et al. [13]. Here we present andanalyze the corresponding nondimensional governingequation and boundary conditions which can be writ-ten as

θ′′ =−µ

2 sinθ + µ2δ(cosθθ

′′− sinθθ′2)+ µ

2χ

2

· cos−3θ(θ′′+3tanθθ

′2)−δ µ2χ

2

· cos−1θ

[θ

(4) +2tanθθ′θ′′′

+(1+2tan2

θ)

θ′2

θ′′+ tanθθ

′′2],

θ(0) = 0 , θ′(1) = 0 , θ(1) = α .

(1)

Various kinds of solution methods [13 – 15] were usedto handle the buckling analysis. One of these meth-ods is the Adomian decomposition method (ADM)proposed by Adomian [16] and further developed bymany eminent researchers [17 – 26]. ADM is very wellsuited to physical problems since it does not requireunnecessary linearization, discretization or other re-strictive methods and assumptions which may changethe problem to be solved, sometimes seriously. The

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356 Y. Khan and W. Al-Hayani · Buckling Analysis and New Analytic Approximate Solution

Fig. 1 (colour online). Analytic aproximate solutions φ5: line,φ4: circle. Parameters: µ = 0.1, δ = 0, α = 60, χ = 0 (red),0.1 (blue), and 0.2 (black).

basic motivation of the present study is to proposea new approach to develop an approximate solutionfor the buckling phenomena equations. Inspired andmotivated by the ongoing research in this area, weapply the ADM with the Green function technique forsolving the governing problem. The ADM is mucheasier to implement as compared with the homotopyperturbation method (HPM) where huge complexi-ties are involved. To the best of our knowledge, itseems to me that no attempt is available in the liter-ature with the help of ADM through the Green func-tion technique to solve a governing nonlinear model.The fact that ADM solves nonlinear problems withoutusing perturbation theory [27 – 35] can be consideredas a clear advantage of this technique over the pertur-bation method.

2. Description of the Method

In the beginning of the 1980’s, Adomian [16] pro-posed a new and fruitful method (hereafter called theAdomian decomposition method or ADM) for solv-ing linear and nonlinear (algebraic, differential, par-tial differential, integral, etc.) equations. It has beenshown that this method yields a rapid convergence ofthe solution series to linear and nonlinear deterministicand stochastic equations. In order to elucidate the solu-tion procedure of the ADM through the Green function

Fig. 2 (colour online). Analytic aproximate solutions φ5: line,φ4: circle. Parameters: µ = 0.05, δ = 0, α = 120, χ = 0 (red),0.1 (blue), and 0.2 (black).

technique, we consider the general nonlinear differen-tial equation

θ′′(x)+g(x,θ) = f (x) , a≤ x≤ b ,

θ(a) = α , θ(b) = β , α,β ∈ R ,(2)

where θ = θ(x), g(x,θ) is a linear or nonlinear func-tion of θ , and f (x) is a continuous function defined inthe interval. We are seeking for the solution θ satisfy-ing (2) and assume that (2) has an unique solution.

Applying the decomposition method as in [16], (2)can be written as

Lθ = f (x)−Nθ , (3)

where L = d2

dx2 is the linear operator and Nθ = g(x,θ)is the nonlinear operator. Consequently,

θ = h(x)+∫ b

aG(x,ξ )

{f (ξ )−Nθ

}dξ , (4)

where h(x) is the solution of Lθ = 0 with the boundaryconditions, and G(x,ξ ) is the Green function given by

G(x,ξ ) =

{g1(x,ξ ) if a≤ ξ ≤ x≤ b ,

g2(x,ξ ) if a≤ x≤ ξ ≤ b .(5)

The Adomian technique consists in approximating thesolution of (4) as an infinite series

Y. Khan and W. Al-Hayani · Buckling Analysis and New Analytic Approximate Solution 357

Fig. 3 (colour online). Analytic aproximate solutions φ4: line,φ3: circle. Parameters: µ = 0.1, δ = 0, χ = 0.

θ =∞

∑n=0

θn , (6)

and decomposing the nonlinear operator Nθ as

Nθ =∞

∑n=0

An , (7)

where An are polynomials of θ0, . . . ,θn (called Ado-mian’s polynomials [16]) given by

An =1n!

dn

dλ n

[N

(∞

∑i=0

λiyi

)]λ=0

,

n = 0,1,2, . . . .

(8)

The proofs of the convergence of the series ∑∞n=0 θn

and ∑∞n=0 An are given in [17]. Substituting (6) and (7)

into (4) yields

∞

∑n=0

θn = h(x)+∫ b

aG(x,ξ )

{f (ξ )−

∞

∑n=0

An

}dξ . (9)

Thus, we can identify

θ0 = h(x)+∫ b

aG(x,ξ ) f (ξ )dξ ,

θn+1 =−∫ b

aG(x,ξ )An dξ , n = 0,1,2, . . . .

(10)

Fig. 4 (colour online). Analytic aproximate solutions φ4−α:line, φ3−α: circle. Parameters: µ = 0.1, χ = 0.

Now all components of θ can be calculated oncethe An are given. We then define the n-term ap-proximant to the solution θ by φn[θ ] = ∑

n−1i=0 θi with

limn→∞ φn[θ ] = θ .

3. Numerical Application

In this section, we apply the Adomian decompo-sition method through the Green function techniquefor finding the approximate solution of the studiedmodel.

Case 1. In this case, we use the assumption

sinθ ≈ θ , cosθ ≈ 1 , tanθ ≈ θ , (11)

then (1) becomes

θ′′ =−µ

2θ + µ

2δ(θ′′−θθ

′2)+ µ2χ

2

·(θ′′+3θθ

′2)−δ µ2χ

2[

θ(4)

+2θθ′θ′′′+

(1+2θ

2)θ′2

θ′′+θθ

′′2].

(12)

Expanding and collecting the terms with the same co-efficients, we get

358 Y. Khan and W. Al-Hayani · Buckling Analysis and New Analytic Approximate Solution

Fig. 5 (colour online). Analytic aproximate solutions φ4: line,φ3: circle. Parameters: µ = 0.1, δ = 0, χ = 0, 0.1, 0.2.

θ′′ =− µ2

(1−µ2δ −µ2χ2)θ − µ2

(1−µ2δ −µ2χ2)

·[(

δ −3χ2)

θθ′2 +δ χ

2(

θ(4) +2θθ

′θ′′′)

+δ χ2 (

θ′2

θ′′+2θ

2θ′2

θ′′+θθ

′′2)] .

(13)

In view of (3), (13) can be written as

Lθ =− µ2

(1−µ2δ −µ2χ2)θ − µ2

(1−µ2δ −µ2χ2)

·[(

δ −3χ2)N1θ +δ χ

2(

θ(4) +2N2θ

)+δ χ

2 (N3θ +2N4θ +N5θ)],

(14)

where L = d2

dx2 is the linear operator and

N1θ = θθ′2 , N2θ = θθ

′θ′′′ , N3θ = θ

′2θ′′ ,

N4θ = θ2θ′2

θ′′ , N5θ = θθ

′′2 (15)

are the nonlinear operators.Consequently,

Fig. 6 (colour online). Analytic aproximate solutions φ4 −α: line, φ3 − α: circle. Parameters: µ = 0.1, δ = 0, χ =0, 0.1, 0.2.

θ = ax− µ2

(1−µ2δ −µ2χ2)

∫ 1

0G(x,ξ )θ(ξ )dξ

− µ2

(1−µ2δ −µ2χ2)

∫ 1

0G(x,ξ )

{(δ −3ξ

2)·N1θ +δ χ

2(

θ(4) +2N2θ

)+δ χ

2(N3θ

+2N4θ +N5θ)}

dξ ,

(16)

where G(x,ξ ) is the Green function given by

G(x,ξ ) =

{(x−1)ξ if 0≤ ξ ≤ x≤ 1 ,

(ξ −1)x if 0≤ x≤ ξ ≤ 1 .(17)

Firstly, we set

N1θ = θθ′2 = A1,n , N2θ = θθ

′θ′′′ = A2,n ,

N3θ = θ′2

θ′′ = A3,n ,

N4θ = θ2θ′2

θ′′ = A4,n ,

N5θ = θθ′′2 = A5,n .

(18)

Substituting (6) and (18) in (16), the iterations are thendetermined in the following recursive way:

Y. Khan and W. Al-Hayani · Buckling Analysis and New Analytic Approximate Solution 359

Fig. 7 (colour online). Analytic aproximate solutions φ5: line,φ4: circle. Parameters: µ = 0.1, δ = 0.05, χ = 0, α =30, 60, 90, 120.

θ0 = ax ,

θn+1 =− µ2

(1−µ2δ −µ2χ2)

∫ 1

0G(x,ξ )θn(ξ )dξ

− µ2

(1−µ2δ −µ2χ2)

∫ 1

0G(x,ξ ){(

δ −3χ2)A1,n +δ χ

2(

θ(4)n +2A2,n

)+δ χ

2 (A3,n +2A4,n +A5,n)}

dξ ,

n = 0,1,2, . . . .

(19)

Fig. 8 (colour online). Analytic aproximate solutions φ4: line,φ3: circle. Parameters: µ = 0.1, δ = 1, 2, 3, χ = 0.

Fig. 9 (colour online). Analytic aproximate solutions φ4−α:line, φ3−α: circle. Parameters: µ = 0.1, δ = 1, 2, 3, χ = 0.

That is, we use the functional iteration with analyticalintegration to compute θn(x). To obtain the sequence{θn(x)}∞

n=0, we also calculate φn(x) in ordinary form,i. e., φn(x) = ∑

n−1i=0 θi(x).

Case 2. In this case, we choose χ = 0, and we will notconsider the assumptions defined in (11):

θ′′ =−µ

2 sinθ + µ2δ(cosθθ

′′− sinθθ′2)

+ µ2χ

2 cos−3θ(θ′′+3tanθθ

′2)−δ µ

2χ

2 cos−1θ

[θ

(4) +2tanθθ′θ′′′

+(1+2tan2

θ)

θ′2

θ′′+ tanθθ

′′2],

(20)

subject to the same boundary conditions defined in (1),

θ′′ =−µ

2 sinθ + µ2δ(cosθθ

′′− sinθθ′2)+ µ

2χ

2

·(cos−3

θθ′′+3cos−4

θ sinθθ′2)−δ µ

2χ

2

·[

cos−1θθ

(4) +2cos−2θ sinθθ

′θ′′′

+ cos−1θ(1+2tan2

θ)

θ′2

θ′′

+ cos−2θ sinθθ

′′2].

(21)

Applying the ADM as in [16], (21) can be written as

360 Y. Khan and W. Al-Hayani · Buckling Analysis and New Analytic Approximate Solution

Lθ =−µ2N1θ + µ

2δ (N2θ −N3θ)

+ µ2χ

2(N4θ +3N5θ)

−δ µ2χ

2 [N6θ +2N7θ +N8θ +N9θ ] ,

(22)

where L = d2

dx2 is the linear operator and the nonlinearterm can be decomposed as

N1θ = sinθ , N2θ = cosθθ′′ , N3θ = sinθθ

′2 ,

N4θ = cos−3θθ′′ , N5θ = cos−4

θ sinθθ′2 ,

N6θ = cos−1θθ

(4) , N7θ = cos−2θ sinθθ

′θ′′′ ,

N8θ = cos−1θ(1+2tan2

θ)

θ′2

θ′′ ,

N9θ = cos−2θ sinθθ

′′2 .

(23)

From (22), we have

θ = ax+∫ 1

0G(x,ξ )

{−µ

2N1θ + µ2δ (N2θ −N3θ)

+ µ2χ

2(N4θ +3N5θ)−δ µ2χ

2[N6θ

+2N7θ +N8θ +N9θ

]}dξ .

(24)

The first few components of the Adomian polynomials,for example, are given by

N1θ = sinθ = B1,n , N2θ = cosθθ′′ = B2,n ,

N3θ = sinθθ′2 = B3,n ,

N4θ = cos−3θθ′′ = B4,n ,

N5θ = cos−4θ sinθθ

′2 = B5,n ,

N6θ = cos−1θθ

(4) = B6,n ,

N7θ = cos−2θ sinθθ

′θ′′′ = B7,n ,

N8θ = cos−1θ(1+2tan2

θ)

θ′2

θ′′′ = B8,n ,

N9θ = cos−2θ sinθθ

′′2 = B9,n .

(25)

It is clear from (24), that the recursive relation is

θ0 = αx ,

θn+1 =∫ 1

0G(x,ξ )

{−µ

2B1,n + µ2δ (B2,n−B3,n)

+ µ2χ

2 (B4,n +3B5,n)−δ µ2χ

2[B6,n

+2B7,n +B8,n +B9,n

]}dξ ,

n = 0,1,2, . . . .

(26)

4. Results and Discussion

Equation (1) subject to the boundary conditions issolved analytically using the Adomian decompositionmethod through the Green function technique for twodifferent cases. The results in the form of the differ-ent physical parameters µ , δ , χ , and α for two dif-ferent cases are shown in Figures 1 – 9. From Fig-ures 1 and 2, one can easily observe that the increas-ing value of free end slope α and dimensionless pa-rameter χ reduces the buckling load, while the buck-ling load increases curvedly with respect to differentvalues of α and χ = 0 in Figures 3 to 4. Figures 5to 6 demonstate the similar effect for different valuesof χ . The effect of α for Case 2 is shown in Fig-ure 7. It presents a quite opposite behaviour to Fig-ures 1 to 2, while the Figures 8 and 9 show simi-lar behaviour for the second case as discussed in Fig-ures 3 – 6. The buckling response becomse more note-worthy as the parameters µ , δ , χ , and α become largerand the magnitude of the post-buckling load remainspermanent.

5. Conclusion

We have derived an analytic-approximate solutionof a nonlinear buckling model. This particular prob-lem has received a great deal of interest both fromthe analysis and numerical communities. However, webelieve that this is the first time that an ADM so-lution through Green’s function has been presented.The ADM procedure is straightforward to implementand provides only with a few terms a reliable analytic-approximate solution. It also avoids the difficulties andmassive computational work as compared to other an-alytical and numerical methods. The method is ap-plied here in a direct manner without the use of lin-earization, transformation, discretization or other re-strictive assumptions. The analytic-approximate solu-tion obtained by ADM are proven to be convergentand uniformly valid. This study shows that ADMcoupled with the Green function technique suits forother dynamics models arising in applied sciences andengineering.

Y. Khan and W. Al-Hayani · Buckling Analysis and New Analytic Approximate Solution 361

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