Solving systems of fractional differential equations by

Physics Letters A 372 (2008) 451–459 www.elsevier.com/locate/pla

Solving systems of fractional differential equations by homotopy-perturbation method O. Abdulaziz a , I. Hashim a,∗ , S. Momani b a School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia b Department of Mathematics, Mutah University, PO Box 7, Al-Karak, Jordan

Received 3 May 2007; received in revised form 10 June 2007; accepted 25 July 2007 Available online 1 August 2007 Communicated by A.R. Bishop

Abstract In this Letter, approximate analytical solutions of systems of Fractional Differential Equations (FDEs) are derived by the HomotopyPerturbation Method (HPM). The fractional derivatives are described in the Caputo sense. The solutions are obtained in the form of rapidly convergent infinite series with easily computable terms. Numerical results reveal that HPM is very effective and simple for obtaining approximate solutions of nonlinear systems of FDEs. © 2007 Elsevier B.V. All rights reserved. Keywords: Homotopy-perturbation method; Fractional differential equations; Caputo fractional derivative

1. Introduction In the last decades, fractional calculus has found diverse applications in various scientific and technological fields [1,2], such as thermal engineering, acoustics, electromagnetism, control, robotics, viscoelasticity, diffusion, edge detection, turbulence, signal processing and many other physical processes. Fractional Differential Equations (FDEs) have also been applied in modelling many physical and engineering problems. Finding accurate and efficient methods for solving FDEs has been an active research undertaking. Exact solutions of most of the FDEs cannot be found easily, thus analytical and numerical methods must be used. Some of the numerical methods for solving FDEs were presented in [3–7]. He [8] was the first to apply the analytic Variational Iteration Method (VIM) to solve FDEs. Drˇagˇanescu [9] and Momani and Odibat [10–12] applied VIM to some complicated FDEs. The Adomian Decomposition Method (ADM) was shown to be applicable to linear and nonlinear FDEs [10,13–21]. However, one of the disadvantages of ADM is the inherent difficulty in calculating the Adomian polynomials (cf. [22–24]). Yet another promising analytic technique for nonlinear problems is called the homotopy-perturbation method (HPM), first proposed by He [25,26]. The HPM, a coupling of the traditional perturbation method and homotopy in topology, deforms continuously a difficult problem to a simple problem which is easily solved. The HPM does not require a small parameter in an equation and the perturbation equation can be easily constructed by a homotopy in topology. Momani and Odibat established the application of HPM to solve fractional quadratic Riccati differential equation, linear and nonlinear partial FDEs in [27,28] and [29], respectively. Very recently, Wang solved fractional KdV–Burgers and fractional KdV equations using HPM in [30] and [31], respectively. Many real-life applications are modelled by systems of FDEs which can be written in the form:

* Corresponding author.

E-mail address: [email protected] (I. Hashim). 0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.07.059

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O. Abdulaziz et al. / Physics Letters A 372 (2008) 451–459

D α1 y1 (t) = f1 (t, y1 , y2 , . . . , yn ),

(1)

D y2 (t) = f2 (t, y1 , y2 , . . . , yn ), .. .

(2)

D αn yn (t) = fn (t, y1 , y2 , . . . , yn ),

(3)

α2

subject to the following initial conditions: yk (0) = ck ,

k = 1, 2, . . . , n,

(4)

where is the fractional derivative of yi of order αi (0 < αi 1) and fi are arbitrary linear or nonlinear functions. Linear and nonlinear systems of FDEs were solved using the ADM in [32] and [33,34], respectively. In [21], Momani and Odibat presented a numerical comparison between the ADM and VIM for solving systems of linear and nonlinear systems of FDEs. Very recently, Ertürk and Momani [35] applied the differential transform method to systems of FDEs. In this Letter, we are interested in extending the applicability of HPM to systems of FDEs (1)–(3). To demonstrate the effectiveness of the HPM algorithm, several numerical experiments of linear and nonlinear systems of FDEs shall be presented. D αi

2. Preliminaries and notations In this section, we give some definitions and properties of the fractional calculus [2]. Definition 1. A real function h(t), t > 0, is said to be in the space Cμ , μ ∈ R, if there exists a real number p > μ, such that h(t) = t p h1 (t), where h1 (t) ∈ C(0, ∞), and it is said to be in the space Cμn if and only if h(n) ∈ Cμ , n ∈ N . Definition 2. The Riemann–Liouville fractional integral operator (J α ) of order α 0, of a function h ∈ Cμ , μ −1, is defined as 1 J h(t) = (α)

t (t − τ )α−1 h(τ ) dτ

α

(α > 0),

0

J 0 h(t) = h(t),

(5)

where (z) is the well-known gamma function. Some of the properties of the operator J α , which we will need here, are as follows: For h ∈ Cμ , μ −1, α, β 0 and γ −1: (1) J α J β h(t) = J α+β h(t), (2) J α J β h(t) = J β J α h(t), (γ +1) α+γ . (3) J α t γ = (α+γ +1) t Definition 3. The fractional derivative (D α ) of h(t) in the Caputo sense is defined as 1 D h(t) = (n − α)

t (t − τ )n−α−1 h(n) (τ ) dτ,

α

(6)

0

n . for n − 1 < α n, n ∈ N , t > 0, h ∈ C−1 The following are two basic properties of the Caputo fractional derivative [36]: n , n ∈ N . Then D α h, 0 α n is well defined and D α h ∈ C . (1) Let h ∈ C−1 −1 (2) Let n − 1 < α n, n ∈ N and h ∈ Cμn , μ −1. Then

n−1 tk J α D α h(t) = h(t) − h(k) (0+ ) . k!

(7)

k=0

3. HPM for system of FDEs In view of the HPM [25,26], we construct the following homotopy: D αi yi = pfi (t, y1 , y2 , . . . , yn ),

(8)


453

where i = 1, 2, . . . , n and p is an embedding parameter which changes from zero to unity. If p = 0, Eq. (8) becomes the linear equation D αi yi = 0,

(9)

and when p = 1, the homotopy (8) turns out to be the original system given in (1)–(3). Using the parameter p, we expand the solution of the system (1)–(3) in the following form: yi (t) = yi0 + pyi1 + p 2 yi2 + p 3 yi3 + · · · .

(10)

Substituting (10) into (8) and collecting the terms with the same powers of p, we obtain a series of linear equations of the form p0 :

D αi yi0 = 0,

(11)

p1 :

D αi yi1 = fi1 (t, y10 , y20 , . . . , yn0 ),

(12)

p2 :

D αi yi2 = fi2 (t, y10 , y20 , . . . , yn0 , y11 , y21 , . . . , yn1 ),

(13)

D yi3 = fi3 (t, y10 , y20 , . . . , yn0 , y11 , y11 , . . . , yn1 , y12 , y22 , . . . , yn2 ),

(14)

3

p :

αi

etc., where the functions fi1 , fi2 , . . . , satisfy the following equation: fi t, y10 + py11 + p 2 y12 + · · · , . . . , yn0 + pyn1 + p 2 yn2 + · · · = fi1 (t, y10 , y20 , . . . , yn0 ) + pfi2 (t, y10 , y20 , . . . , yn0 , y11 , y21 , . . . , yn1 ) + p 2 fi3 (t, y10 , y20 , . . . , yn0 , y11 , y11 , . . . , yn1 , y12 , y22 , . . . , yn2 ) + · · · . It is obvious that these linear equations can be easily solved by applying the operator J αi , i.e., the inverse of the operator D αi , which HPM solution can be determined. That is, by setting p = 1 is defined by (5). Hence, the components yik (k = 0, 1, 2, . . .) of the in (10) we can entirely determine the HPM series solutions, yi (t) = ∞ k=0 yik (t). The convergence of the series was discussed in [37] and the asymptotic behavior of the series was illustrated in [38,39]. For later numerical computations we will approximate the HPM series solution, yi (t) = ∞ y (t), by the following N -term truncated series: k=0 ik φiN (t) =

N−1

yik (t).

(15)

k=0

4. Numerical experiments To demonstrate the effectiveness of the HPM algorithm discussed above, several examples of linear and nonlinear systems of FDEs will be studied. 4.1. Problem 1 We consider the following linear system of FDEs: D α x(t) = x(t) + y(t),

(16)

D y(t) = −x(t) + y(t),

(17)

β

subject to the initial conditions x(0) = 0,

y(0) = 1.

(18)

According to (8) we construct the following homotopy: D α x = p(x + y),

(19)

D y = p(−x + y).

(20)

β

Substituting (10) into (19)–(20) and collecting terms of the same powers of p yields the following two sets of linear equations: p0 :

D α x0 = 0,

p1 :

D α x 1 = x 0 + y0 ,

p2 :

D α x 2 = x 1 + y1 ,

454


Fig. 1. The 8-term HPM approximate solutions for system (16)–(18) when (a) α = β = 1 and (b) α = 0.7 and β = 0.9.

p3 :

D α x 3 = x 2 + y2 , .. .

p0 :

D β y0 = 0,

p1 :

D β y1 = −x0 + y0 ,

p2 :

D β y2 = −x1 + y1 ,

p3 :

D β y3 = −x2 + y2 , .. .

Consequently, by applying the operators J α and J β to the above sets of linear equations, the first few terms of the HPM series solution for the system (16)–(17) are obtained as follows: x0 = x(0) = 0, tα , x1 = (α + 1)

y0 = y(0) = 1, tβ y1 = , (β + 1)

x2 =

t 2α t α+β + , (2α + 1) (α + β + 1)

x3 =

t 3α t α+2β + , (3α + 1) (α + 2β + 1)

y3 =

t 3β 2t 2β+α t β+2α − − , (2β + 1) (2β + α + 1) (β + 2α + 1)

y2 =

t 2β t α+β − , (2β + 1) (α + β + 1)

etc. Hence, the HPM series solutions are be given by x(t) =

tα t 2α t α+β t 3α t α+2β + + + + + ···, (α + 1) (2α + 1) (α + β + 1) (3α + 1) (α + 2β + 1)

y(t) = 1 +

t 2β t α+β t 3β 2t 2β+α t β+2α tβ + − + − − + ···. (β + 1) (2β + 1) (α + β + 1) (2β + 1) (2β + α + 1) (β + 2α + 1)

(21) (22)

In Fig. 1 we plot the solution given by the 8-term HPM solutions (21)–(22) for different values of α and β. We note that the approximate solution obtained by HPM is in good agreement with the results obtained in [21] by using the VIM and ADM. 4.2. Problem 2 Now let us consider the following nonlinear system of FDEs: D 1.3 x(t) = x(t) + y(t)2 ,

(23)

y(t) = x(t) + 5y(t),

(24)

D

2.4

where the initial conditions are given by x(0) = 0,

x (0) = 1,

y(0) = 0,

y (0) = y (0) = 1.

(25)


455

Fig. 2. The 4-term HPM approximate solutions, φ4 (t) and ϕ4 (t), for the system (23)–(25).

In view of (8), substituting (10) into (8) and equating the terms with the same powers of p, the linear equations (11)–(14) for system (23)–(24) can be reduced to the following two sets of linear equations: p0 :

D 1.3 x0 = 0,

D 2.4 y0 = 0,

p1 :

D 1.3 x1 = x0 + y02 ,

p2 :

D 1.3 x2 = x1 + 2y0 y1 ,

p3 :

D 1.3 x3 = x2 + 2y0 y2 + y12 , .. .

D 2.4 y1 = x0 + 5y0 , D 2.4 y2 = x1 + 5y1 , D 2.4 y3 = x2 + 5y2 ,

Applying the operators J 1.3 and J 2.4 to the above linear equations and according to (7) we can choose y0 to be the simplest term in the initial conditions (25) and the remaining part of the initial conditions to y1 [15]. Hence, the first four terms of the HPM approximate series solution for the system (23)–(24) are obtained as follows: x0 = x(0) = 0, x1 = x (0)t + J

y0 = y(0) = 0, x0 + y02 = t,

1.3

23

x2 =

t 10 ( 33 10 )

34

y2 =

,

33

x3 =

2t 10 ( 43 10 )

+

( 44 10 )

+

5t 10 ( 54 10 )

43

( 46 10 )

+

t2 , 2

44

6t 10

36

t 10

y1 = t + ,

53

6t 10 ( 53 10 )

+

47

6t 10 ( 63 10 )

y3 =

,

2t 10 ( 57 10 )

58

+

30t 10 ( 68 10 )

68

+

25t 10 ( 78 10 )

.

Hence, the 4-term HPM approximate solutions, φ4 (t) and ϕ4 (t), of the series solutions x(t) and y(t), respectively, are given by 23

φ4 (t) t +

t 10 ( 33 10 )

33

+

2t 10 ( 43 10 )

34

36

+

t 10 ( 46 10 )

43

+

44

6t 10 ( 53 10 )

47

53

+

6t 10 ( 63 10 ) 58

(26)

, 68

t2 6t 10 5t 10 2t 10 30t 10 25t 10 ϕ4 (t) t + + 44 + 54 + 57 + 68 + 78 , 2 ( 10 ) ( 10 ) ( 10 ) ( 10 ) ( 10 )

(27)

which are shown in Fig. 2. 4.3. Problem 3 Let us consider the following nonlinear system of FDEs: D α x(t) = 2y 2 ,

(28)

D y(t) = tx,

(29)

D z(t) = yz,

(30)

β γ

where 0 < α, β, γ 1 and the initial conditions are as follows: x(0) = 0,

y(0) = 1,

z(0) = 1.

(31)

456


In view of Eq. (8), and following the same steps as in the last two examples, we obtain p0 :

D α x0 = 0,

p1 :

D α x1 = 2y02 ,

p2 :

D α x2 = 4y0 y1 ,

p3 :

D α x3 = 4y0 y2 + 2y12 , .. .

p0 :

D β y0 = 0,

p1 :

D β y1 = tx0 ,

p2 :

D β y2 = tx1 ,

p3 :

D β y3 = tx2 , .. .

p0 :

D γ z0 = 0,

p1 :

D γ z1 = y0 z0 ,

p2 :

D γ z2 = y0 z1 + y1 z0 ,

p3 :

D γ z3 = y0 z2 + y1 z1 + y2 z0 , .. .

Applying the operators J α , J β and J γ to the above three sets of linear equations, the approximate HPM series solutions for the system (28)–(30) when α = β = γ = 1 are obtained as follows: 2t 4 4t 7 4t 10 + + + ···, 3 21 105 4t 9 2t 3 t 6 + + + ···, y(t) = 1 + 3 9 189 t 2 t 3 5t 4 7t 5 61t 6 221t 7 1481t 8 + + + + + ···. z(t) = 1 + t + + + 2 6 24 40 720 5040 40320 If we take α = 0.5, β = 0.4 and γ = 0.3, then the first few terms of the HPM approximate series solution can be given by √ 4 t 12t 12/5 x(t) = √ + + ···, π ( 17 5 ) x(t) = 2t +

y(t) = 1 + z(t) = 1 +

3t 19/10 ( 29 10 ) t 3/10 ( 13 10 )

+

+

153t 19/5 5( 24 5 ) t 3/5

( 85 )

+

+ ···,

t 9/10 ( 19 10 )

+

(32) (33) (34)

(35) (36)

t 6/5 ( 11 5 )

t 9/5 4t 3/2 t 21/10 + √ + 14 + 31 + · · · . 3 π ( 5 ) ( 5 )

(37)

We note that the HPM approximate solutions (32)–(34) and (35)–(37) are the same solutions obtained by the differential transform method, [35]. Fig. 3(a) and (b) show the first eight terms of the HPM approximate solutions (32)–(34) and (35)–(37) for the system (28)–(30) with α = β = γ = 1 and α = 0.5, β = 0.4 and γ = 0.3, respectively. We note that the HPM approximate solutions (32)–(34) and (35)–(37) are the same solutions obtained by the differential transform method in [35]. Graphical results are also in very good agreement with results of Momani and Odibat [21]. 4.4. Problem 4 Let us consider the following nonlinear system of FDEs: D α x(t) = x 2 + y,

(38)

D y(t) = y cos x,

(39)

β

where α, β ∈ (0, 1), and subject to the following initial conditions: x(0) = 0,

y(0) = 1.

(40)


457

Fig. 3. The 8-term HPM approximate solutions for the system (28), (29), (31) with (a) α = β = γ = 1 and (b) α = 0.5, β = 0.4, γ = 0.3.

Using Taylor series expansion, the nonlinear term cos x in (39) can be expressed as follows [40]: x2 . 2 Hence, we can approximate the system (38)–(39) by the following: cos x ≈ 1 −

(41)

D α x(t) = x 2 + y, (42) 2 x y . D β y(t) = y − (43) 2 In view of (8), substituting (10) into (8) and equating the terms with the same powers of p, the linear equations (11)–(14) for the system (42)–(43) can be reduced to the following two sets of linear equations: p0 :

D α x0 = 0,

p1 :

D α x1 = y0 + x02 ,

p2 :

D α x2 = y1 + 2x0 x1 ,

p3 :

D α x3 = y2 + 2x0 x2 + x12 , .. .

p0 :

D β y0 = 0,

p1 :

D β y1 = y 0 −

p2 :

D β y 2 = y 1 − y0 x 0 x 1 −

p3 :

y0 x02 , 2

y1 x02 , 2 y2 x02 y0 x12 D β y 3 = y 2 − y0 x 0 x 2 − − y1 x 0 x 1 − , 2 2 .. .

Applying the fractional integration operator J α , the HPM approximate series solution can be given by t α+β t α+2β (2α + 1)t 3α tα + ···, + + + (α + 1) (α + β + 1) (α + 2β + 1) (α + 1)2 (3α + 1) tβ t 2β t 3β (2α + 1)t 2α+β y(t) = 1 + + ···. + + − (β + 1) (2β + 1) (3β + 1) (α + 1)2 (2α + β + 1) In Fig. 4 we plot the 6-term HPM series solutions (44)–(45) for the system (38)–(40) for the case α = 0.5 and β = 0.3. x(t) =

(44) (45)

4.5. Problem 5 Let us consider the following nonlinear system of fractional integro-differential equations: t D α x(t) = x(t) K1 − γ1 y(t) − t−T0

e−(t−s) y(s) ds ,

K1 > 0,

(46)

458


Fig. 4. T (45)–(46) for α = 0.5 and β = 0.3.

t

D y(t) = y(t) −K2 + γ2 x(t) + α

e

−(t−s)

x(s) ds ,

K2 > 0,

(47)

t−T0

where 0 < α 1, and subject to the following initial conditions: y(0) = N2 .

x(0) = N1 ,

(48)

This problem was considered in [41] via ADM. Let us take D α x(t) = u(t) ⇒ x(t) = x(0) + J α u(t) , D α y(t) = v(t) ⇒ y(t) = y(0) + J α v(t) .

(49)

So we have the following system of four integral equations: x(t) = x(0) + J α u(t) , t

(51)

e−(t−s) y(s) ds ,

u(t) = x(t) K1 − γ1 y(t) − y(t) = y(0) + J α v(t) ,

(50)

(52)

t−T0

t

v(t) = y(t) −K2 + γ2 x(t) +

(53)

e−(t−s) x(s) ds .

(54)

t−T0

Substituting from (50)–(51) into (51)–(54), we obtain x(t) = N1 + J α u(t) ,

t α −(t−s) u(t) = K1 N1 + J u(t) − x(t) γ1 y(t) + e y(s) ds , y(t) = N2 + J

α

v(t) ,

(55) (56)

t−T0

v(t) = −K2 N2 + J α v(t) + y(t) γ2 x(t) +

t

e−(t−s) x(s) ds .

(57) (58)

t−T0

Applying the same steps as in the previous examples, the 3-term HPM approximate solutions for the system (46)–(47) can be given by x(t) = N1 + N1 K1 − γ1 N2 − N2 1 − e−T0

t 2α tα + K12 N1 , (α + 1) (2α + 1) t 2α tα + K22 N2 , y(t) = N2 − N2 K2 − γ2 N1 − N1 1 − e−T0 (α + 1) (2α + 1) and when α = 1, then we have n1 (t) = N1 + N1 K1 − γ1 N2 − N2 1 − e−T0 t + 0.5K12 N1 t 2 ,


459

n2 (t) = N2 − N2 K2 − γ2 N1 − N1 1 − e−T0 t + 0.5K22 N2 t 2 . 5. Conclusions In this work, the HPM algorithm for approximate analytical solutions of linear and nonlinear systems of fractional differential equations with initial conditions was presented. This HPM yields a very rapid convergence of the solution series in most cases, usually only a few iterations leading to very accurate solutions. Some examples of systems of FDEs were solved using HPM to illustrate the efficiency and accuracy of the method. Acknowledgement The financial support received from the Academy of Sciences Malaysia under the SAGA grant No. P24c (STGL-011-2006) is gratefully acknowledged. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41]

R. Hilfer (Ed.), Applications of Fractional Calculus in Physics, Academic Press, Orlando, 1999. I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999. K. Diethelm, Electron. Trans. Numer. Anal. 5 (1997) 1. K. Diethelm, N.J. Ford, J. Math. Anal. Appl. 265 (2002) 229. K. Diethelm, N.J. Ford, A.D. Freed, Nonlinear Dynam. 29 (2002) 3. K. Diethelm, G. Walz, Numer. Algorithms 16 (1997) 231. P. Kumar, O.P. Agrawal, Signal Process. 86 (2006) 2602. J.H. He, Comput. Methods Appl. Mech. Eng. 167 (1998) 57. G.E. Drˇagˇanescu, J. Math. Phys. 47 (8) (2006), Art. No. 082902. S. Momani, Z. Odibat, Chaos Solitons Fractals 31 (2007) 1248. S. Momani, Z. Odibat, Phys. Lett. A 355 (2006) 271. Z. Odibat, S. Momani, Int. J. Nonlinear Sci. Numer. Simul. 7 (1) (2006) 27. H.L. Arora, F.I. Abdelwahid, Appl. Math. Lett. 6 (1993) 21. A.J. George, A. Chakrabarti, Appl. Math. Lett. 8 (1995) 91. N.T. Shawagfeh, Appl. Math. Comput. 131 (2002) 517. S.S. Ray, R.K. Bera, J. Appl. Math. 4 (2004) 331. S.S. Ray, R.K. Bera, Appl. Math. Comput. 167 (2005) 561. O. Abdulaziz, I. Hashim, M.S.H. Chowdhury, A.K. Zulkifle, Far East J. Appl. Math. 28 (1) (2007) 95. O. Abdulaziz, I. Hashim, E.S. Ismail, Far East J. Appl. Math., in press. S. Momani, Appl. Math. Comput. 182 (2006) 761. S. Momani, Z. Odibat, J. Comput. Appl. Math. 207 (1) (2007) 96. N. Bildik, A. Konuralp, Int. J. Nonlinear Sci. Numer. Simul. 7 (1) (2006) 65. A. Ghorbani, J. Saberi-Nadjafi, Int. J. Nonlinear Sci. Numer. Simul. 8 (2) (2007) 229. E. Yusufoglu, Int. J. Nonlinear Sci. Numer. Simul. 8 (2) (2007) 153. J.H. He, Comput. Methods Appl. Mech. Eng. 178 (1999) 257. J.H. He, Int. J. Nonlinear Mech. 35 (2000) 37. Z. Odibat, S. Momani, Chaos Solitons Fractals 36 (2008) 167. S. Momani, Z. Odibat, Comput. Math. Appl., doi: 10.1016/j.camwa.2006.12.037. S. Momani, Z. Odibat, Phys. Lett. A 365 (2007) 345. Q. Wang, Chaos Solitons Fractals 35 (2008) 843. Q. Wang, Appl. Math. Comput. 190 (2) (2007) 1795. V. Daftardar-Gejji, H. Jafari, J. Math. Anal. Appl. 301 (2005) 508. S. Momani, K. Al-Khaled, Appl. Math. Comput. 162 (2005) 1351. H. Jafari, V. Daftardar-Gejji, J. Comput. Appl. Math. 196 (2006) 644. V.S. Ertürk, S. Momani, J. Comput. Appl. Math., doi: 10.1016/j.cam.2007.03.029. R. Gorenflo, F. Mainardi, Fractional calculus: Integral and differential equations of fractional order, in: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, Springer Verlag, New York, 1997. J.H. He, Non-Perturbative Methods for Strongly Nonlinear Problems, Dissertation, de-Verlag im Internet GmbH, Berlin, 2006. J.H. He, Int. J. Mod. Phys. B 20 (2006) 1. J.H. He, Int. J. Mod. Phys. B 20 (2006) 1141. L.N. Zhang, J.H. He, Math. Probl. Eng. Volume 2006, Article ID 83878, Pages 1–6. S. Momani, R. Qaralleh, Comput. Math. Appl. 52 (2006) 459.