The homotopy analysis method to solve the Burgers–Huxley equation

Nonlinear Analysis: Real World Applications 10 (2009) 589–600 www.elsevier.com/locate/nonrwa

The homotopy analysis method to solve the Burgers–Huxley equation A. Molabahrami a,b,∗ , F. Khani a,c a Department of Mathematics, Ilam University, PO Box 69315516, Ilam, Iran b Department of Mathematics, Maragheh University, PO Box 533, Maragheh, Iran c Bakhtar Institute of Higher Education, PO Box 696, Ilam, Iran

Received 21 September 2007; accepted 11 October 2007

Abstract In this paper, an analytical technique, namely the homotopy analysis method (HAM) is applied to obtain an approximate analytical solution of the Burgers–Huxley equation. This paper introduces the two theorems which provide us with a simple and convenient way to apply the HAM to the nonlinear PDEs with the power-law nonlinearity. The homotopy analysis method contains the auxiliary parameter h¯ , which provides us with a simple way to adjust and control the convergence region of solution series. c 2007 Elsevier Ltd. All rights reserved.

Keywords: Homotopy analysis method; Burgers–Huxley equation; Burgers–Fisher equation; Power-law nonlinearity

1. Introduction Nonlinear phenomena play a crucial role in applied mathematics and physics. The theory of nonlinear problems has recently undergone many studies. The Burgers–Huxley equations arise from the mathematical modelling of many scientific phenomena. The tanh method developed for years, presented by Malfliet [1–3], is one of the most direct and effective algebraic methods for finding exact solutions of nonlinear evolution equations. The KdV equation, the modified KdV equation, the coupled Schrodinger–KdV equation, the KdV–Burgers equation, the double sine-Gordon equation, (2 + 1) dimensional sine-Gordon equation and many other equations were handled effectively by using the standard tanh method. Perturbation method is one of the well-known methods to solve nonlinear problems; it is based on the existence of small/large parameters, the so-called perturbation quantity [4,5]. Many nonlinear problems do not contain such perturbation quantity, and we can use non-perturbation methods, such as the artificial small parameter method [6], the δ-expansion method [7], the Adomian’s decomposition method [8] and the homotopy perturbation method (HPM) [9–12]. However, neither perturbation nor nonperturbation method can provide us with a simple way to adjust and ∗ Corresponding author at: Department of Mathematics, Ilam University, PO Box 69315516, Ilam, Iran. Tel.: +98 914 143 6187; fax: +98 841 222 7022. E-mail addresses: a m [email protected] (A. Molabahrami), farzad [email protected] (F. Khani).

c 2007 Elsevier Ltd. All rights reserved. 1468-1218/$ - see front matter doi:10.1016/j.nonrwa.2007.10.014

590

A. Molabahrami, F. Khani / Nonlinear Analysis: Real World Applications 10 (2009) 589–600

control the convergence region and rate of given approximate series. First Liao in 1992 employed the basic ideas of the homotopy in topology to propose a general analytical method for nonlinear problems, namely homotopy analysis method (HAM) [13–18]. After this, many types of nonlinear problems were solved with HAM by others [19–23]. In this paper, the basic idea of the HAM is introduced and then its application for Burgers–Huxley equation is studied. Also, we introduce the two theorems which provide us with a simple and convenient way to apply the HAM to the nonlinear PDEs with the power-law nonlinearity. In this paper, we shall apply HAM to find the approximate analytical solution of the Burgers–Huxley equation. Comparisons with the exact solution shall be performed. Consider the generalized Burgers–Huxley equation u t + αu δ u x − λu x x = βu(1 − u δ )(ηu δ − γ ),

(1)

where α, β, λ, η, γ and δ are parameters, δ is a positive integer. The values α = 0 and δ = 1 reduce the Eq. (1) to the Huxley equation which describes nerve pulse propagation in nerve fibres and wall motion in liquid crystals. The values η = 0 and γ = −1 reduce the Eq. (1) to the generalized Burgers–Fisher equation u t + αu δ u x − λu x x = βu(1 − u δ ).

(2)

In this paper, the Eq. (1) is solved on the infinite line −∞ < x < +∞ together with the initial condition u(x, 0) = f (x),

−∞ < x < +∞

(3)

by using the HAM. 2. Homotopy analysis method (HAM) In this section the basic idea of the homotopy analysis method [13] is introduced. To show the basic idea, let us consider the following nonlinear equation in a general form: N [u(r, t)] = 0, where N is a nonlinear operator, u(r, t) is an unknown function and r and t denote spatial and temporal independent variables, respectively. For simplicity, we ignore all boundary or initial conditions, which can be treated in the similar way. By means of generalizing the traditional homotopy method; more details about homotopy technique and its applications are found in [24–28]. Liao [13] constructs the so-called zero-order deformation equation (1 − q)L[φ(r, t; q) − u 0 (r, t)] = q h¯ H (r, t)N [φ(r, t; q)],

(4)

where q ∈ [0, 1] is the embedding parameter, h¯ 6= 0 is a nonzero auxiliary parameter, H (r, t) is an auxiliary function, L is an auxiliary linear operator, u 0 (r, t) is an initial guess of u(r, t), φ(r, t; q) is an unknown function. It is important that we have great freedom to choose auxiliary things in HAM. Obviously, when q = 0 and q = 1, it holds φ(r, t; 0) = u 0 (r, t)

and φ(r, t, 1) = u(r, t).

Thus as q increases from 0 to 1, the solution φ(r, t; q) varies from the initial guess u 0 (r, t) to the solution u(r, t). Expanding φ(r, t; q) in Taylor series with respect to q, we have φ(r, t; q) = u 0 (r, t) +

+∞ X

u m (r, t)q m ,

(5)

m=1

where u m (r, t) =

1 ∂ m φ(r, t; q) . m! ∂q m q=0

(6)

A. Molabahrami, F. Khani / Nonlinear Analysis: Real World Applications 10 (2009) 589–600

591

If the auxiliary linear operator, the initial guess, the auxiliary parameter and the auxiliary function are so properly chosen, the series (5) converges at q = 1, we have +∞ X

u(r, t) = u 0 (r, t) +

u m (r, t)

m=1

which must be one of solutions of the original nonlinear equation, as proved by Liao [13]. As h¯ = −1 and H (r, t) = 1, Eq. (4) becomes (1 − q)L[φ(r, t; q) − u 0 (r, t)] + q N [φ(r, t; q)] = 0 which is used mostly in the homotopy perturbation method (HPM), whereas the solution obtained directly, without using Taylor series. The comparison between HAM and HPM can be found in [29–32]. As H (r, t) = 1, Eq. (4) becomes (1 − q)L[φ(r, t; q) − u 0 (r, t)] = q h¯ N [φ(r, t; q)]

(7)

which is used in the HAM when it is not introduced the set of base functions. According to definition (6), the governing equation can be deduced from the zero-order deformation equation (4). Define the vector uEn = {u 0 (r, t), u 1 (r, t), . . . , u n (r, t)}. Differentiating Eq. (4) 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 equation L[u m (r, t) − χm u m−1 (r, t)] = h¯ H (r, t)Rm (E u m−1 , r, t),

(8)

1 ∂ m−1 N [φ(r, t; q)] Rm (E u m−1 , r, t) = (m − 1)! ∂q m−1 q=0

(9)

where

and χm =



0, 1,

m≤1 m ≥ 2.

Substituting (5) into the (9), we have " # +∞ X 1 ∂ m−1 n Rm (E u m−1 , r, t) = N u (r, t)q n (m − 1)! ∂q m−1 n=0

.

(10)

q=0

It should be emphasized that u m (r, t) for m ≥ 1 is governed by the linear equation (8) with the linear boundary conditions that come from the original problem, which can be easily solved by symbolic computation software such as Mathematica and Maple. 3. The HAM solution In this section, we obtain the series pattern solutions of Burgers–Huxley equation. The solution given by HAM can be represented by many different base functions. Now, we introduce the two theorems which provide us with a simple and convenient way to apply the HAM to the nonlinear PDEs with the power-law nonlinearity. P 1 dm (φ k ) n Theorem 1. Let Dm (φ k ) denote the m! where φ = +∞ n=0 u n q , then dq m q=0

Dm (φ k ) =

m X r1 =0

u m−r1

r1 X r2 =0

u r1 −r2

r2 X r3 =0

u r2 −r3 . . .

rX k−3 rk−2 =0

u rk−3 −rk−2

rX k−2 rk−1 =0

u rk−2 −rk−1 u rk−1 ,

592

A. Molabahrami, F. Khani / Nonlinear Analysis: Real World Applications 10 (2009) 589–600

and m X

Dm (φ k+1 ) =

u m− j D j (φ k ) =

m X

u j Dm− j (φ k ).

j=0

j=0

Proof. The proof is by induction on the k. Clearly, for k = 2, we have Dm (φ 2 ) =

m X

u m− j u j .

j=0

Put φ k+1 = φ k φ, by using the Leibnitz’s rule for higher derivatives of products, we have m   X m (φ k+1 )(m) = (φ k )( j) φ (m− j) . j j=0

Thus Dm (φ

k+1

! m   1 X m k ( j) (m− j) )= (φ ) φ m! j=0 j

q=0

m    1 X m j!D j (φ k ) (m − j)!u m− j = m! j=0 j

=

m X

u m− j D j (φ k ).

j=0

It is clear that m m X X u m− j D j (φ k ) = u j Dm− j (φ k ). j=0

j=0



This ends the proof.

Corollary 1. From Theorem 1, we have Dm (φ k−1 φx ) =

m X

u m−r1

r1 =0

Theorem 2. Let s = +∞ X

r1 X

u r1 −r2

r2 =0

P+∞

n=0 u n .

r2 X r3 =0

u r2 −r3 . . .

rX k−3 rk−2 =0

u rk−3 −rk−2

rX k−2

Then, recall the assumptions in Theorem 1, we have

Dm (φ k ) = s k .

m=0

Proof. The proof is by induction on the k. From Theorem 1, for k = 2, we have ! +∞ +∞ X m X X 2 Dm (φ ) = u m− j u j . m=0

m=0

j=0

Thus +∞ X

Dm (φ 2 ) =

+∞ X m X

u m− j u j

m=0 j=0

m=0

=

+∞ X +∞ X j=0 m= j

u rk−2 −rk−1 (u rk−1 )x .

rk−1 =0

u m− j u j

A. Molabahrami, F. Khani / Nonlinear Analysis: Real World Applications 10 (2009) 589–600 +∞ X

=

uj

j=0

+∞ X

593

u m− j

m= j

= s2. Put φ k+1 = φ k φ, from Theorem 1, we have +∞ X

Dm (φ

k+1

)=

m=0

+∞ X m X m=0

! u m− j D j (φ ) . k

j=0

Thus +∞ X

Dm (φ k+1 ) =

m=0

+∞ X m X

u m− j D j (φ k )

m=0 j=0

=

+∞ X +∞ X

u m− j D j (φ k )

j=0 m= j

=

+∞ X

D j (φ k )

j=0

+∞ X

u m− j

m= j

= sk s = s k+1 . 

This ends the proof.

Corollary 2. According to the Theorem 2, we have +∞ X

Dm (φ k−1 φx ) = s k−1 sx .

m=0

From initial condition (3), it is reasonable to express solution u(x, t) by set of base functions {gn (x)en (t)|n ≥ 0},

(11)

in the form u(x, t) =

+∞ X

an gn (x)en (t),

(12)

n=0

where an is a coefficient, gn (x) as a coefficient is a function with respect to x and en (t) can be chosen as follows en (t) = t n .

(13)

This provides us with the so-called rule of solution expression u(x, t). Under the rule of solution expression denoted by (12) and from Eq. (1), it is straightforward to choose ∂φ(x, t, q) + γ2 (t)φ(x, t, q) ∂t as auxiliary linear operator, where γ1 (t) 6= 0 and γ2 (t) are functions to be determined as follows: L[φ(x, t, q)] = γ1 (t)

γ1 (t) = 1,

(14)

(15)

γ2 (t) = 0. Thus, in this work, the auxiliary linear operator is L[φ(x, t, q)] =

∂φ(x, t, q) , ∂t

(16)

594

A. Molabahrami, F. Khani / Nonlinear Analysis: Real World Applications 10 (2009) 589–600

which has the property L[C] = 0, where C is coefficient. According to Eq. (1), we define the nonlinear operator N [φ(x, t, q)] =

∂ 2 φ(x, t, q) ∂φ(x, t, q) ∂φ(x, t, q) + α (φ(x, t, q))δ −λ ∂t ∂x ∂x2

δ δ − βφ(x, t, q) 1 − φ (x, t, q) ηφ (x, t, q) − γ .

(17)

From Eqs. (10) and (17) and Theorem 1, we have   Rm [E u m−1 (x, t)] = (u m−1 )t − λ (u m−1 )x x + βγ u m−1 + Dm−1 αu δ u x + βηu 2δ+1 − β(η + γ )u δ+1 .

(18)

Let u ∗m (x, t) denote a special solution of the equation L[u ∗m (x, t)] = h¯ H (x, t)Rm [E u m−1 (x, t)].

(19)

Now, the solution of the mth-order deformation equation (8) with initial condition u m (x, 0) = 0, for m ≥ 1 becomes u m (x, t) = χm u m−1 (x, t) + u ∗m (x, t) − u ∗m (x, 0).

(20)

According to (3) and the rule of solution expression (12), it is straightforward that the initial approximation should be in the form u 0 (x, t) = (1 + g(t)) f (x) + h(t),

(21)

where g(0) = h(0) = 0. The functions g(t) and h(t) are chosen subject to obey the rule of solution expression (12). According to the rule of solution expression denoted by (12) and from Eq. (20) to obey the rule of coefficient ergodicity the auxiliary function H (x, t) is chosen. 3.1. Convergence theorem In this subsection, we prove that, if the solution series given by HAM is convergent, it must be an exact solution of the considered nonlinear problem. Theorem 3. If the series u 0 (x, t) +

+∞ X

u m (x, t)

m=1

converges, where u m (x, t) is governed by the Eq. (20) under the definition (18), it must be an exact solution of Eq. (1) with initial condition (3). Proof. If the series is convergent, we can write s=

+∞ X

um

m=0

and it holds lim u n = 0.

n→+∞

Then, using (8) and (16), we have h¯ H (x, t)

+∞ X m=1

Rm [E u m−1 (x, t)] = lim

n→+∞

n X m=1

L[u m (x, t) − χm u m−1 (x, t)]

A. Molabahrami, F. Khani / Nonlinear Analysis: Real World Applications 10 (2009) 589–600

( lim

= L

n→+∞

n X

595

) [u m (x, t) − χm u m−1 (x, t)]

m=1

= L[ lim u n (x, t)] n→+∞

= 0, which gives, since h¯ 6= 0 and H (x, t) 6= 0, +∞ X

Rm [E u m−1 (x, t)] = 0.

m=1

Substituting (18) in the above expression, recall the Theorem 2 and Corollary 2 and simplifying it, we obtain +∞ X

Rm [E u m−1 (x, t)] =

m=1

+∞ n X

(u m−1 )t − λ (u m−1 )x x + βγ u m−1

m=1

 o + Dm−1 αu δ u x + βηu 2δ+1 − β(η + γ )u δ+1 = st − λsx x + βγ s +

+∞ X

  Dm αu δ u x + βηu 2δ+1 − β(η + γ )u δ+1

m=0 δ

= st − λsx x + βγ s + αs sx + βηs 2δ+1 − β(η + γ )s δ+1 = st + αs δ sx − λsx x − βs(1 − s δ )(ηs δ − γ ) = 0. From initial condition u m (x, 0) = 0 and (21), it holds s(x, 0) = f (x). So, s(x, t)satisfies Eqs. (1) and (3), and therefore is an exact solution of the Burgers–Huxley equation (1) with initial condition (3). This ends the proof.  To investigate the influence of h¯ on the convergence of the solution series given by the HAM, we first plot the so-called h¯ -curves of u tt (0, 0) and u ttt (0, 0). According to the h¯ -curves, it is easy to discover the valid region of h¯ , which corresponds to the line segment nearly parallel to the horizontal axis.

3.2. The HAM solutions for Burgers–Huxley equation

In this subsection, we obtain the solution of the Eq. (1) with initial condition (3), for δ = 2, by HAM in the form of (12). Under the rule of solution expression denoted by (12) and according to initial condition (3), it is straightforward to choose the u 0 (x, t) as the initial approximation of u(x, t), as follows u 0 (x, t) = f (x). Under the rule of solution expression denoted by (12) and from Eq. (20), the auxiliary function should be H (x, t) = 1. Now, from Eq. (20), we can successively obtain   u 1 (x, t) = h¯ βγ f (x) − β(γ + η) f 3 (x) + βη f 5 (x) + α f 2 (x) f 0 (x) − λ f 00 (x) t,

596

A. Molabahrami, F. Khani / Nonlinear Analysis: Real World Applications 10 (2009) 589–600

Fig. 1. The h¯ -curves of u 00approx (0, 0) and u 000 approx (0, 0)at the 5th order of approximation.

  u 2 (x, t) = h¯ (1 + h¯ ) βγ f (x) − β(γ + η) f 3 (x) + βη f 5 (x) + α f 2 (x) f 0 (x) − λ f 00 (x) t h¯ 2  2 2 + β γ f (x) − 4β 2 γ (γ + η) f 3 (x) + 3β 2 (γ 2 + 4γ η + η2 ) f 5 (x) 2 − 8β 2 η(γ + η) f 7 (x) + 5β 2 η2 f 9 (x) + 4αβγ f 2 (x) f 0 (x) − 8αβ(γ + η) f 4 (x) f 0 (x)    
2 + 12αβη f 6 (x) f 0 (x) + 2 3βλ (γ + η) + 2 α 2 − 5βηλ f 2 (x) f (x) f 0 (x)    
3 − 2αλ f 0 (x) − 2βγ λ f 00 (x) + 6βλ (γ + η) + α 2 − 10βηλ f 2 (x) f 2 (x) f 00 (x)  − 8αλ f (x) f 0 (x) f 00 (x) − 2αλ f 2 (x) f 000 (x) + λ2 f (4) (x) t 2 , .. .. 4. Result analysis In this section, five examples are presented P to illustrate the effectiveness of the HAM. We use six and five terms in evaluating approximate solution u approx = nm=0 u m for Examples 1 and 2 respectively. In the first two examples, we first plot the so-called h¯ -curves of u 00approx (0, 0) and u 000 ¯ , which corresponds approx (0, 0) to discover the valid region of h to the line segment nearly parallel to the horizontal axis. Example 1. Consider the Burgers–Huxley equation  2  2 2 2  u t + u u x − u x x = u(1 − u )(u − 1), t > 0, 0 ≤ x ≤ 1, 3   1  2 1 1 1   u(x, 0) = + tanh x , 0 ≤ x ≤ 1, 2 2 3

(22)

with the exact solution    1 2 1 1 1 u(x, t) = + tanh . (3x + t) 2 2 9 Recall the results in Section 3.2, from the h¯ -curves (Fig. 1), it is found that, when −1.3 ≤ h¯ < 0, the solution series (12) converges to the exact solution (22). Fig. 2 shows the comparison between the exact solution of (22) and u approx . Example 2. Consider the Burgers–Fisher equation  2 2  u t + u u x − u x x = u(1 − u ), t > 0, 0 ≤ x ≤ 1,    1 2 1 1 1  − tanh x , 0 ≤ x ≤ 1, u(x, 0) = 2 2 3

(23)

A. Molabahrami, F. Khani / Nonlinear Analysis: Real World Applications 10 (2009) 589–600

597

Fig. 2. (a) The exact solution and (b) the fifth order of approximation at h¯ = −0.003.

Fig. 3. The h¯ -curves of u 00approx (0, 0) and u 000 approx (0, 0)at the 4th order of approximation.

with the exact solution    1 2 1 1 1 10 u(x, t) = − tanh x− t . 2 2 3 9 Recall the results in Section 3.2, from the h¯ -curves (Fig. 3), it is found that, when −1.3 ≤ h¯ < −0.6, the solution series (12) converges to the exact solution (23). Fig. 4 shows the comparison between the exact solution of (23) and u approx .

598

A. Molabahrami, F. Khani / Nonlinear Analysis: Real World Applications 10 (2009) 589–600

Fig. 4. (a) The exact solution and (b) the fourth order of approximation at h¯ = −0.64.

Example 3. Consider the one dimensional heat equation  u t − λu x x = 0, −∞ < x < +∞, t > 0, u(x, 0) = f (x), −∞ < x < +∞.

(24)

Recall the results in Section 3.2, at h¯ = −1, we have u 1 (x, t) = λ f 00 (x)t, 1 u 2 (x, t) = λ2 f (4) (x)t 2 , 2 1 u 3 (x, t) = λ3 f (6) (x)t 3 , 6 .. .. Thus u(x, t) = f (x) +

+∞ X (λt)n n=1

n!

f (2n) (x).

Example 4. Consider the linear Schr¨odinger equation  u t + iu x x = 0, i2 = −1, −∞ < x < +∞, t > 0, u(x, 0) = f (x), −∞ < x < +∞.

(25)

A. Molabahrami, F. Khani / Nonlinear Analysis: Real World Applications 10 (2009) 589–600

599

According to (25), we have u(x, t) = f (x) +

+∞ X (−it)n n=1

n!

f (2n) (x).

Example 5. Consider the one dimensional heat equation  u t − λu x x = 0, −∞ < x < +∞, t > 0, u(x, 0) = b + ca x + c1 sinh(c2 x) + c3 cosh(c4 x),

−∞ < x < +∞.

According to (25), we have u(x, t) = b + ca x + ca x

+∞ X (λtLn2 a)n n=1

n!

+ c3 cosh(c4 x) + c3 cosh(c4 x)

+ c1 sinh(c2 x) + c1 sinh(c2 x)

= b + ca e

2

n=1

n!

+∞ X (λtc2 )n 4

n=1 x λtLn2 a

+∞ X (λtc2 )n

n!

+ c1 sinh(c2 x)eλtc2 + c3 cosh(c4 x)eλtc4 . 2

2

5. Conclusion In this paper, the homotopy analysis method (HAM) is applied to obtain the solution of Burgers–Huxley equation. Five examples are employed to illustrate its validity, effectiveness and flexibility. The results of the examples show that the HAM is very effective and convenient. It shows that HAM is a promising tool for linear and nonlinear PDEs. In many cases, from physical characteristics and boundary/initial conditions, it is not very difficult to determine the type of base functions convenient to represent solution of a given nonlinear problem. In fact, a solution of a given nonlinear problem may be expressed by different sets of base functions. This paper introduces two theorems which provide us with a convenient way to apply the HAM for nonlinear problems. Also, in this paper, a new suitable and simple way for applying the HAM to the Burgers–Huxley equation is introduced which has extension ability about the base functions. Thus, such an approach can be used for the same problems. In addition, this paper provides us with a convenient way to choose the auxiliary linear operator, auxiliary function, initial guess and the set of base functions suitable to represent the solution of a given nonlinear problem. The HAM provides us with a convenient way to control the convergence of approximation series which is a fundamental qualitative difference in analysis between HAM and other methods. Thus the auxiliary parameter h¯ plays an important role within the frame of the HAM which can be determined by the so-called h¯ -curves. Acknowledgements The authors would like to express their great thankful to Professor Shijun Liao for comments and discussion. Many thanks are due to financial support from the Ilam University, Maragheh University and Bakhtar Institute of Higher Education of Iran. References [1] [2] [3] [4] [5] [6] [7]

W. Malfliet, Solitary wave solutions of nonlinear wave equations, Amer. J. Phys. 60 (7) (1992) 650–654. W. Malfliet, The tanh method: I. Exact solutions of nonlinear evolution and wave equations, Phys. Scripta 54 (1996) 563–568. W. Malfliet, The tanh method. II: Perturbation technique for conservative systems, Phys. Scripta 54 (1996) 569–575. J.D. Cole, Perturbation Methods in Applied Mathematics, Blaisdell, Waltham, MA, 1968. A.H. Nayfeh, Perturbation Methods, Wiley, New York, 2000. A.M. Lyapunov, General Problem on Stability of Motion, Taylor & Francis, London, 1992, (English translation). A.V. Karmishin, A.I. Zhukov, V.G. Kolosov, Methods of Dynamics Calculation and Testing for Thin-Walled Structures, Mashinostroyenie, Moscow, 1990. [8] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic, Dordrecht, 1994.

600 [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]

A. Molabahrami, F. Khani / Nonlinear Analysis: Real World Applications 10 (2009) 589–600 J.H. He, Internat. J. Modern Phys. B 20 (2006) 1141. J.H. He, Phys. Lett. A 350 (2006) 87. A. Molabahrami, F. Khani, Numerical solutions for highly oscillatory integrals, Appl. Math. Comput. (2007), doi:10.1016/j.amc.2007.09.007. A. Molabahrami, F. Khani, S. Hamedi-Nezhad, Phys. Lett. A 370 (2007) 433–436. S.J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, Ph.D. Thesis, Shanghai Jiao Tong University, 1992. S.J. Liao, Internat J. Non-Linear Mech. 34 (1999) 759. S.J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman & Hall/CRC Press, Boca Raton, 2003. S.J. Liao, J. Fluid Mech. 488 (2003) 189. S.J. Liao, Appl. Math. Comput. 147 (2004) 499. S.J. Liao, Int. J. Heat Mass Transfer 48 (2005) 2529. M. Inc, On numerical solution of Burgers’ equation by homotopy analysis method, Phys. Lett. A (2007), doi:10.1016/j.physleta.2007.07.057. Y. Liu, Z. Li, The homotopy analysis method for approximating the solution of the modified Korteweg-de Vries equation, Chaos Solitons Fractals (2007), doi:10.1016/j.chaos.2007.01.148. T. Hayat, Z. Abbas, M. Sajid, Phys. Lett. A 358 (2006) 396. S. Abbasbandy, Int. Commun. Heat Mass Transfer 34 (2007) 380–387. Lina Song, Hongqing Zhang, Phys. Lett. A 367 (2007) 88. L.T. Watson, Globally convergent homotopy methods: A tutorial, Appl. Math. Comput. 13BK (1989) 369–396. Layne T. Watson, Melvin R. Scott, Solving spline-collocation approximations to nonlinear two-point boundary-value problems by a homotopy method, Appl. Math. Comput. 24 (1987) 3X-357. Layne T. Watson, Engineering applications of the Chow–Yorke algorithm, Appl. Math. Comput. 9 (1981) 111–133. Layne T. Watson, Raphael T. Haftka, Modern homotopy methods in optimization, Comput. Math. Appl. Mech. Eng. 74 (1989) 289–305. Y. Wang, D.S. Bernstein, L.T. Watson, Probability-one homotopy algorithms for solving the coupled Lyapunov equations arising in reducedorder H 2 /H ∞ modeling estimation, and control, Appl. Math. Comput. 123 (2001) 155–185. S.J. Liao, Appl. Math. Comput. 169 (2005) 1186. T. Hayat, M. Sajid, Phys. Lett. A 361 (2007) 316. S. Abbasbandy, Phys. Lett. A 360 (2006) 109. J.H. He, Appl. Math. Comput. 156 (2004) 527.