Computers and Mathematics with Applications 61 (2011) 2829–2842

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Computers and Mathematics with Applications

journal homepage: www.elsevier.com/locate/camwa

Approximate analytical solutions of fractional Benney–Lin equation byreduced differential transform method and the homotopy perturbationmethodPraveen Kumar Gupta ∗

Department of Applied Mathematics, Institute of Technology, Banaras Hindu University, Varanasi – 221 005, India

a r t i c l e i n f o

Article history:Received 25 December 2010Received in revised form 11 March 2011Accepted 14 March 2011

Keywords:Benney–Lin equationFractional Brownian motionCaputo derivativeReduced differential transform methodHomotopy perturbation methodIVPs

a b s t r a c t

In this paper, the approximate analytical solutions of Benney–Lin equation with fractionaltime derivative are obtained with the help of a general framework of the reduceddifferential transform method (RDTM) and the homotopy perturbation method (HPM).RDTM technique does not require any discretization, linearization or small perturbationsand therefore it reduces significantly the numerical computation. Comparing themethodology (RDTM)with some known technique (HPM) shows that the present approachis effective and powerful. The numerical calculations are carried out when the initialconditions in the form of periodic functions and the results are depicted through graphs.The eight different cases have studied and proved that the method is extremely effectivedue to its simplistic approach and performance.

© 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Historically, comparatively little was known about the extraordinary range of behavior presented by the solutions ofnonlinear partial differential equations. Most of the fundamental phenomena that now drive present time research work,including solitons, chaos, stability, blow-up and singularity formation, asymptotic properties, etc., remained undetectedor at best dimly perceived in the pre-computer era. The past six decades have witnessed a remarkable blossoming inour understanding, due, in large part, to the insight offered by the availability of high performance computers coupledwith great advances in the understanding and development of suitable numerical approximation methods. New analyticalmethods, new mathematical theories, coupled with new computational algorithms have precipitated this gyration in ourunderstanding and study of nonlinear systems, an activity that continues to grow in intensity and breadth. Each leap incomputing power coupled with theoretical advances has led to yet deeper understanding of nonlinear phenomena, whilesimultaneously demonstrating how far we have yet to go. To make sense of this bewildering variety of methods, equations,and results, it is essential to build a firm foundation on, first of all, linear systems theory, and afterward, nonlinear algebraicequations and nonlinear ordinary differential equations.

In present time, fractional differential equations have garnered a lot of attention and appreciation recently due to theirability to provide an exact description of different nonlinear phenomena. The process of development of models based onfractional order differential systems has lately gained popularity in the investigation of dynamical systems. The advantageof fractional order systems is that they allow greater degrees of freedom in the model. The field of chaos has also grabbedthe attention of the researchers and this contributes to a significant amount of the ongoing research these days. The field ofnonlinear sciences provides many hot topics of research due to their practical applications in various fields of engineeringlike electricity, information sciences, communications etc. and also in medical sciences. Recently, Tripathi et al. [1–4] havestudied the fractional models for peristaltic transport in the area of biomechanics.

∗ Tel.: +91 9453555044.E-mail address: praveen.rs.apm@itbhu.ac.in.

0898-1221/$ – see front matter© 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.camwa.2011.03.057

2830 P.K. Gupta / Computers and Mathematics with Applications 61 (2011) 2829–2842

Synchronization of fractional order chaotic systems is one of the potent research areas due to its extensive applicationin communication theory and control processing. In 2008, Xu et al. [5] has studied the chaos synchronization between twodifferent fractional order chaotic systemsbyusing active control. Another important areawhere researchers are now focusedis the role of chaotic attractors in fractional order systems. Recent studies reveal that chaotic order systems can also besynchronized. Sheu et al. [6] has achieved success in finding lowest order system to yield chaos in the study of the dynamicsof a fractional order Newton–Leipnik system [7]. Sheu et al. [8], on analyzing the influence of the system parameters on thesystem dynamics concluded that the appropriate selection of the parameters can restrain or generate chaos.

In 1966, D. J. Benney [9] studied the long waves on the liquid films where he find some attractive results and introducedthe Benney–Lin equation, afterward modified by Lin [10]. This general equation describes the progression of long waves invarious problems in fluid dynamics.

In this article, we consider the initial value problem (IVP) of Benney–Lin equation [11,12] with fractional time derivativeuαt + uux + uxxx + β(uxx + uxxxx) + ηuxxxxx + ux = 0, 0 < α ≤ 1, β > 0, η ∈ R,

u(x, 0) = ϕ(x). (1)

For α = 1, β = 0, i.e., purely dispersive form, Eq. (1) is converted to the standard Kawahara equation (or the fifth-orderKdV equation) that describes the water waves with surface tension [13]. In the purely dissipative form, Eq. (1) reducesto the long wave simplification of the Navier–Stokes equation and has been used to explain different phenomena suchas spatial patterns of the Belousov–Zhabotinsky reaction, surface-tension-driven convection in a liquid film, and unstableflame fronts. Another case for α = 1, η = 0, i.e., the dissipative–dispersive equation, Eq. (1) is converted to the generalizedKuramoto–Sivashinsky equation that describes the waves in the vertical and inclined falling film, in liquid films that aresubjected to interfacial stress from adjacent gas flow, unstable drift waves in plasma, interfacial instability between twoconcurrent viscous fluids and phase evolution for the complex Ginzburg–Landau equation.

After three decades in 1997, Biagioni and Linares [14] derived that IVP connected to the Benney–Lin equation is globallywell posed inHs(R) for s ≥ 0 if β > 0.While if β = 0, Cui et al. [15] have obtained the local well-posedness of the Kawaharaequation for −1 < s < 0 recently.

In 1986, Chinese scientist Zhao [16] first introduced the differential transform method in the engineering domain.Initially, the differential transformmethod was applied to the solution of electric circuit problems. It is a numerical methodbasedon the Taylor series expansionwhich constructs an analytical solution in the formof a polynomial. The establishedhighorder Taylor series method requires only symbolic computation. Another side, the differential transform method obtains apolynomial series solution by means of an iterative procedure. Recently, the application of reduced differential transformmethod is successfully extended to obtain analytical approximate solutions to linear and nonlinear ordinary differentialequations of fractional order [17,18]. A comparison between this method and other approximate analytical method forsolving fractional differential equations is given in [17]. The rapid development in the field of nonlinear sciences during thepast two decades invoked an increasing interest of mathematicians and engineers in these subjects who were stimulated tofind the analytical techniques for solving nonlinear problems. Earlier, the most commonly used methods were perturbationmethods, which suffer from limitations due to the small parameter assumptions that may sometimes have an adverse effecton the solution. Although a considerable amount of research work had already happened before, no such analytical methodwas available for solving these equations.

In this paper,weuse the reduceddifferential transformmethod (RDTM) [19–22] andHe’s homotopyperturbationmethod(HPM) [23–29] to obtain the solutions of the fractional Benney–Lin equation and compare them with each other. We knowthat, the HPMmethod is based on the use of homotopy parameter for classification ofmost favorable values of parameters inbetween [0, 1]. While, RDTM technique does not require any parameter, discretization, linearization or small perturbationsand therefore it reduces significantly the numerical computation. For the standard cases, comparing [12] the methodologywith some known techniques shows that the present approach is effective and powerful.

2. Reduced differential transformmethod

In this section, we introduce the fractional reduced differential transform method used in this paper to obtainapproximate analytical solutions for the fractional Benney–Lin equation (1). This method has been developed in [19] asfollows.

The fractional differentiation in Riemann–Liouville sense is defined by

Dβt0u(x, t) =

1Γ (m − β)

dm

dtm

[∫ t

t0

u(x, ξ)

(t − ξ)1+β−mdξ

], (2)

for m − 1 ≤ β < m,m ∈ Z+, t > t0. Let us expand the analytical and continuous function u(x, t)in terms of a fractionalpower series as follows:

u(x, t) =

∞−k=0

Uk(x)(t − t0)k/α, (3)

where α is the order of fraction and Uk(x) is the fractional differential transform of u(x, t).

P.K. Gupta / Computers and Mathematics with Applications 61 (2011) 2829–2842 2831

In order to avoid fractional initial and boundary conditions, we define the fractional derivative in the Caputo sense.The relation between the Riemann–Liouville operator and Caputo operator is given by

Dβ∗t0u(x, t) = Dβ

t0

u(x, t) −

m−1−i=0

u(i)(x, t0)(t − t0)i

i!

. (4)

By using Eq. (4), from Eq. (2), we obtain fractional derivative in the Caputo sense [10] as follows:

Dβ∗t0u(x, t) =

1Γ (m − β)

dm

dtm

∫ t

t0

u(x, ξ) −

m−1∑i=0

u(i)(x, t0)(ξ−t0)i

i!

(t − ξ)1+β−m

dξ

. (5)

Since the initial conditions are implemented to the integer order derivatives, the transformations of the initial conditionsare defined as follows:

Uk(x) =

If (k/α) ∈ Z+,1

(k/α)!

[dk/α

dtk/αu(x, t)

]t=t0

for k = 0, 1, 2, . . . (αβ − 1)

If (k/α) ∈ Z+ 0,(6)

where,β is the order of fractional differential equation considered. The following theorems that can be deduced fromEqs. (2)and (3) are given below; (for details we can see [20–22]).

Theorem 1. If w(x, t) = u(x, t) ± v(x, t), then Wk(x) = Uk(x) ± Vk(x).

Theorem 2. If w(x, t) = cu(x, t), then Wk(x) = cUk(x), c is a constant.

Theorem 3. If w(x, t) = xmtn, then Wk(x) = xmδ(k − n) where,

δ(k) =

1, k = 00, k = 0.

Theorem 4. If w(x, t) = xmtnu(x, t), then Wk(x) = xmUk−n(x).

Theorem 5. If w(x, t) = u(x, t)v(x, t), then Wk(x) =∑k

r=0 xmUr(x)Vk−r(x) =

∑kr=0 x

mUk−r(x)Vr(x).

Theorem 6. If w(x, t) = u1(x, t)u2(x, t) · · · un−1(x, t)un(x, t), then

Wk(x) =

k−kn−1=0

kn−1−kn−2=0

· · ·

k3−k2=0

k2−k1=0

U1,k1(x)U2,k2−k1(x) · · ·Un−1,kn−1−kn−2(x)Un,k−kn−1(x).

Theorem 7. If w(x, t) =∂m

∂xm u(x, t), then Wk(x) =∂m

∂xm Uk(x).

Theorem 8. If w(x, t) =∂r

∂tr u(x, t), then

Wk(x) = (k + 1)(k + 2) · · · (k + r)Uk+r(x) =(k + r)!

r!Uk+r(x).

Theorem 9. If w(x, t) =∂β

∂tβ u(x, t), then Wk(x) =Γ (β+1+k/α)

Γ (1+k/α)Uk+αβ(x).

Theorem 10. If w(x, t) =∂β1

∂tβ1[u1(x, t)] ∂β1

∂tβ1[u1(x, t)] · · ·

∂βn−1

∂tβn−1[un−1(x, t)] ∂βn

∂tβn [un(x, t)], then

Wk(x) =

k−kn−1=0

kn−1−kn−2=0

· · ·

k3−k2=0

k2−k1=0

Γ (β1 + 1 + k1/α)

Γ (1 + k1/α)

×Γ (β2 + 1 + (k2 − k1)/α)

Γ (1 + (k2 − k1)/α)· · ·

Γ (βn−1 + 1 + (kn−1 − kn−2)/α)

Γ (1 + (kn−1 − kn−2)/α)

×Γ (βn−1 + 1 + (kn−1 − kn−2)/α)

Γ (1 + (kn−1 − kn−2)/α)

Γ (βn + 1 + (kn − kn−1)/α)

Γ (1 + (kn − kn−1)/α)U1,k1+αβ1(x)

×U2,k2−k1+αβ2(x) · · ·Un−1,kn−1−kn−2+αβn−1(x)Un,k−kn−1+αβn(x)

where, αβi ∈ Z+ for i = 1, 2, 3, . . . , n.

2832 P.K. Gupta / Computers and Mathematics with Applications 61 (2011) 2829–2842

3. Solution of the problem by the reduced differential transformmethod

For the illustration of the methodology of the method, we write the Benney–Lin equation in the standard operator form

L(u(x, t)) + R(u(x, t)) + N(u(x, t)) = 0 (7)

with initial condition

u(x, 0) = ϕ(x) (8)

where L(u(x, t)) ≡∂α

∂tα (u(x, t)) is the fractional time derivative operator and R(u(x, t)) ≡∂3

∂x3(u(x, t)) +

β

∂2

∂x2(u(x, t)) +

∂4

∂x4(u(x, t))

+ η ∂5

∂x5(u(x, t)) +

∂∂x (u(x, t)), is linear operator and N(u(x, t)) ≡ u(x, t) ∂

∂x (u(x, t)) isnonlinear operator which have partial derivatives.

According to the RDTM and Theorems 7 and 9, we can construct the following iteration formula for Eq. (7):

Γ (α + 1 + k/q)Γ (1 + k/q)

Uk+αq(x) = −R(Uk(x)) − N(Uk(x)), (9)

where R(Uk(x)) and N(Uk(x)) are the transformations of the functions R(u(x, t)), and N(u(x, t)), respectively.For the convenience of the reader, we can give the first few nonlinear terms:

N(U0(x)) = U0(x)∂

∂xU0(x),

N(U1(x)) = U0(x)∂

∂xU1(x) + U1(x)

∂

∂xU0(x),

N(U2(x)) = U0(x)∂

∂xU2(x) + U1(x)

∂

∂xU1(x) + U2(x)

∂

∂xU0(x).

From initial condition (8), we write

U0(x) = ϕ(x). (10)

Substituting (10) into (9) and by straightforward iterative steps, we get the following Uk(x) (for k = 0, 1, 2, . . . , n) values.

U0(x, t) = ϕ(x), (11)

U1(x, t) = −{ϕ(1)(x) + ϕ(x)ϕ(1)(x) + ϕ(3)(x) + β(ϕ(2)(x) + ϕ(4)(x)) + ηϕ(5)(x)}tα

Γ (α + 1), (12)

U2(x, t) = {2(ϕ(1)(x))2 + 2ϕ(x)(ϕ(1)(x))2 + ϕ(2)(x) + 2ϕ(x)ϕ(2)(x) + (ϕ(x))2ϕ(2)(x) + 3(ϕ(2)(x))2

+ 4βϕ(1)(x)ϕ(2)(x) + 2βϕ(3)(x) + 2βϕ(x)ϕ(3)(x) + 5ϕ(1)(x)ϕ(3)(x) + 10βϕ(2)(x)ϕ(3)(x)+ 10η(ϕ(3)(x))2 + 2ϕ(4)(x) + β2ϕ(4)(x) + 2ϕ(x)ϕ(4)(x) + 6βϕ(1)(x)ϕ(4)(x)+ 15ηϕ(2)(x)ϕ(4)(x) + 4βϕ(5)(x) + 2βϕ(x)ϕ(5)(x) + 7ηϕ(1)(x)ϕ(5)(x) + ϕ(6)(x)+ 2β2ϕ(6)(x) + 2ηϕ(6)(x) + 2ηϕ(x)ϕ(6)(x) + 2βϕ(7)(x) + 2βηϕ(7)(x) + β2ϕ(8)(x)

+ 2ηϕ(8)(x) + 2βηϕ(9)(x) + η2ϕ(10)(x)}t2α

Γ (2α + 1), (13)

and so on.Then the inverse transformation of the set of values {Uk(x)}nk=0 gives approximation solution as,

un(x, t) =

n−k=0

Uk(x)tk

where n is the order of approximate solution.Therefore, the exact solution of problem is given by

u(x, t) = limn→∞

un(x, t). (14)

4. Solution of the problem by the homotopy perturbation method

According to HPM [25–28], we construct the following homotopy of Eq. (1)

Dαt u = −p[uDxu + Dxxxu + β(Dxxu + Dxxxxu) + ηDxxxxxu + Dxu], (15)

where the homotopy parameter p is considered to be small, 0 ≤ p ≤ 1.

P.K. Gupta / Computers and Mathematics with Applications 61 (2011) 2829–2842 2833

100 000

0

–100 000

–200 000

20 000

0

–20 000

–40 000

5000

0

–5000

–10 000

50 000

0

–50 000

–100 000

–0.2

0.0

0.2

–0.2

0.0

0.2

0.2

0.1

0.0

0.2

0.1

0.0

–0.2

0.0

0.2

0.2

0.1

0.0

–0.2

0.0

0.2

0.2

0.1

0.0

a b

c d

Fig. 1. For the Benney–Lin equation with the first initial condition (Eq. (21)) of Eq. (1), reduced differential transform method result for u(x, t) is,respectively (a) α = 1/4, (b) α = 1/2, (c) α = 3/4, (d) α = 1 at β = η = γ = µ = 1.

Now applying the classical perturbation technique, Eq. (15) can be expressed as a power series of p as

u(x, t) = u0 + pu1 + p2u2 + p3u3 + · · · . (16)

When p → 1, Eq. (16) becomes the approximate solution of Eq. (1). Substituting Eq. (16) into Eq. (15) and equating theterms with identical powers of p, we obtain the following set of linear differential equations:

p0 : Dαt u0 = 0, (17)

p1 : Dαt u1 = −[u0Dxu0 + Dxxxu0 + β(Dxxu0 + Dxxxxu0) + ηDxxxxxu0 + Dxu0], (18)

p2 : Dαt u2 = −[u0Dxu1 + u1Dxu0 + Dxxxu1 + β(Dxxu1 + Dxxxxu1) + ηDxxxxxu1 + Dxu1], (19)

and so on.Proceeding as before, we can select u0(x, t) = ϕ(x). Using this selection in (17)–(19) we obtain the successive

approximations and the solution may be obtained.Finally, we approximate the analytical solutions of u(x, t) by the truncated series

u(x, t) = limN→∞

uN(x, t) (20)

where uN(x, t) =∑N−1

n=0 uk(x, t).Here, we take eight different values of ϕ(x) and comparing the results of RDTM and HPM in the form of three- and two-

dimensional figures for each case, we would see that RDTM and HPM solutions are in excellent agreement. Comparisons ofresults are shown in Figs. 1, 3, 5, 7, 9, 11, 13 and 15 for the three-dimensional study, and Figs. 2, 4, 6, 8, 10, 12, 14 and 16 forthe two-dimensional study, for case studies 1–8, respectively.

Case study 1 : ϕ(x) = α − 2µ2 tan(µx) (21)

Case study 2 : ϕ(x) = α − 2µ2 tan2(µx) (22)

Case study 3 : ϕ(x) = α − 2µ2 tanh(µx) (23)

Case study 4 : ϕ(x) = α − 2µ2 tanh2(µx) (24)

2834 P.K. Gupta / Computers and Mathematics with Applications 61 (2011) 2829–2842

x

x

x

x

x

x

x

x

x

x

x

x

x

x

x

–

–– –

– –

–

–

–

– –

––

–

–

– –

x

x

a b

c d

Fig. 2. The comparison of the results of the RDTM and HPM for fractional Benney–Lin equation with the first initial condition (Eq. (21)) for (a) α = 1/4,(b) α = 1/2, (c) α = 3/4, (d) α = 1 at t = 0.8 and β = η = γ = µ = 1.

0

–500 000

0

–200 000

–400 000

–600 000–800 000

0

–50 000

–100 000

0

–100 000

–300 000

–200 000

–1. x 106

–1.5 x 106

–2. x 106

–0.2

0.0

0.2

–0.2

0.0

0.2

–0.2

0.0

0.2

–0.2

0.0

0.20.0

0.1

0.2

0.0

0.1

0.2

0.0

0.1

0.2

0.0

0.1

0.2

a b

c d

Fig. 3. For the Benney–Lin equation with the second initial condition (Eq. (22)) of Eq. (1), reduced differential transform method result for u(x, t) is,respectively (a) α = 1/4, (b) α = 1/2, (c) α = 3/4, (d) α = 1 at β = η = γ = µ = 1.

Case study 5 : ϕ(x) = α − 2µ2 sec(µx) (25)

Case study 6 : ϕ(x) = α − 2µ2 sec2(µx) (26)

P.K. Gupta / Computers and Mathematics with Applications 61 (2011) 2829–2842 2835

– x – x

– x

– x

– x

– x

– x

– x

– x

– x

– x

– x

– x

– x

– x

– x

– x

– x

– x

– x

–

– –

– – – –

– –

x

a b

c d

Fig. 4. The comparison of the results of the RDTM and HPM for fractional Benney–Lin equation with the second initial condition (Eq. (22)) for (a) α = 1/4,(b) α = 1/2, (c) α = 3/4, (d) α = 1 at t = 0.8 and β = η = γ = µ = 1.

–50 000

2000

–2 0000

–10 000

0

10 000

20 000

0

–2000–5000

0

5000

–0.2

0.0

0.2

–0.2

0.0

0.2

–0.2

0.0

0.2

–0.2

0.0

0.20.0

0.1

0.2

0.0

0.1

0.2

0.0

0.1

0.2

0.0

0.1

0.2

0

a b

c d

Fig. 5. For the Benney–Lin equation with the third initial condition (Eq. (23)) of Eq. (1), reduced differential transform method result for u(x, t) is,respectively (a) α = 1/4, (b) α = 1/2, (c) α = 3/4, (d) α = 1 at β = η = γ = µ = 1.

Case study 7 : ϕ(x) = α − 2µ2 sech(µx) (27)

Case study 8 : ϕ(x) = α − 2µ2 sech2(µx). (28)

2836 P.K. Gupta / Computers and Mathematics with Applications 61 (2011) 2829–2842

–

––

–

–

–

–

–

–

– – – –

– –– –

a b

c d

Fig. 6. The comparison of the results of the RDTM and HPM for fractional Benney–Lin equation with the third initial condition (Eq. (23)) for (a) α = 1/4,(b) α = 1/2, (c) α = 3/4, (d) α = 1 at t = 0.8 and β = η = γ = µ = 1.

–200 000

0

200 000

–100 000

10 000

0

–10 000

0

–50 000–20 000

0

–0.2

0.0

0.2 0.0

0.1

0.2

–0.2

0.0

0.2 0.0

0.1

0.2

–0.2

0.0

0.2 0.0

0.1

0.2

–0.2

0.0

0.2 0.0

0.1

0.2

a b

c d

Fig. 7. For the Benney–Lin equation with the fourth initial condition (Eq. (24)) of Eq. (1), reduced differential transform method result for u(x, t) is,respectively (a) α = 1/4, (b) α = 1/2, (c) α = 3/4, (d) α = 1 at β = η = γ = µ = 1.

5. Numerical results and discussion

In this section, the numerical values of the probability density function u(x, t) are calculated for different time-fractionalBrownian motions α =

14 ,

12 ,

34 and also for the standard motion α = 1. It is seen from Figs. 1, 3, 5, 7, 9, 11, 13 and 15

P.K. Gupta / Computers and Mathematics with Applications 61 (2011) 2829–2842 2837

–

–

–

–

–

–

–

–

–– –

– – –

–

–

–

–

a b

c d

Fig. 8. The comparison of the results of the RDTM and HPM for fractional Benney–Lin equation with the fourth initial condition (Eq. (24)) for (a) α = 1/4,(b) α = 1/2, (c) α = 3/4, (d) α = 1 at t = 0.8 and β = η = γ = µ = 1.

0

–100 000

–100 000

0

–10 000

–20 000

–30 000

–40 000

–5000

0

–10 000

–50 000

0

–200 000

–0.2

0.0

0.2 0.0

0.1

0.2

0.0

0.1

0.2

0.0

0.1

0.2

0.0

0.1

0.2

–0.2

0.0

0.2

–0.2

0.0

0.2

–0.2

0.0

0.2

a b

c d

Fig. 9. For the Benney–Lin equation with the fifth initial condition (Eq. (25)) of Eq. (1), reduced differential transform method result for u(x, t) is,respectively (a) α = 1/4, (b) α = 1/2, (c) α = 3/4, (d) α = 1 at β = η = γ = µ = 1.

which graphically describes the three-dimensional variations of u(x, t)w.r.t t and x for standard case and different fractionalvalues of α at β = η = γ = µ = 1. Here, we have taken eight different initial conditions for showing the nature of thedisplacement. The numerical results of u(x, t) for various values of t, x and α are depicted in Figs. 1–16.

2838 P.K. Gupta / Computers and Mathematics with Applications 61 (2011) 2829–2842

– x

– x

– x

– x

– x

– x

– x

–

–

–

–

–

x

– x

–– – – –

– –– –

x

x– x

– x

– x

– x

– x

– x

x

x

x

a b

c d

Fig. 10. The comparison of the results of the RDTM and HPM for fractional Benney–Lin equation with the fifth initial condition (Eq. (25)) for (a) α = 1/4,(b) α = 1/2, (c) α = 3/4, (d) α = 1 at t = 0.8 and β = η = γ = µ = 1.

a b

c d

–0.2

0

–100 000

0

–50 000

–100 000

0

–400 000

–200 000

–800 000

–600 000

–200 000

–300 000

0

–500 000

–1. x 106

–1.5 x 106

–2. x 106

0.0

0.2

–0.2

0.0

0.2

–0.2

0.0

0.2

–0.2

0.0

0.2

0.0

0.1

0.2

0.0

0.1

0.2

0.0

0.1

0.2

0.0

0.1

0.2

Fig. 11. For the Benney–Lin equation with the sixth initial condition (Eq. (26)) of Eq. (1), reduced differential transform method result for u(x, t) is,respectively (a) α = 1/4, (b) α = 1/2, (c) α = 3/4, (d) α = 1 at β = η = γ = µ = 1.

According to numerical solutions, from Figs. 2, 4, 6, 8, 10, 12, 14 and 16, we can see at the time t = 0.8, the values of theapproximate solutions by RDTM and the HPM are quite close. One can also see that their surface graphics and profiles arealmost the same. That is to say the solution obtained by RDTM is efficient and accurate. It is also suggested that RDTM is apowerful method for solving differential equations containing nonlinear terms.

P.K. Gupta / Computers and Mathematics with Applications 61 (2011) 2829–2842 2839

a b

c d

– x

– x

– x

– x

– x

– x

–

– – – –

– ––

– x – x

– x

– x

– x

– x

– x

– x

– x

– x

– x

– x

– x

– x–

x

Fig. 12. The comparison of the results of the RDTM and HPM for fractional Benney–Lin equation with the sixth initial condition (Eq. (26)) for (a) α = 1/4,(b) α = 1/2, (c) α = 3/4, (d) α = 1 at t = 0.8 and β = η = γ = µ = 1.

40 000

20 000

0

–20 000

20 000

10 000

0

–10 000

3000

2000

1000

0

–1000

–0.2

0.0

0.0

0.1

0.2

0.0

0.1

0.2

0.0

0.1

0.2

0.2

–0.2

0.0

0.2

–0.2

5000

0

0.0

0.2

–0.2

0.0

0.2 0.0

0.1

0.2

a b

c d

Fig. 13. For the Benney–Lin equation with the seventh initial condition (Eq. (27)) of Eq. (1), reduced differential transform method result for u(x, t) is,respectively (a) α = 1/4, (b) α = 1/2, (c) α = 3/4, (d) α = 1 at β = η = γ = µ = 1.

6. Conclusion

In this paper, RDTM and He’s HPM are successfully applied to find the solution of the fractional Benney–Linequation of various fractional Brownian motions and standard motion with different initial conditions. Since reduced

2840 P.K. Gupta / Computers and Mathematics with Applications 61 (2011) 2829–2842

–

– –

–

–

–

–

–

– –

––

– –

–

–

a b

c d

Fig. 14. The comparison of the results of the RDTMandHPM for fractional Benney–Lin equationwith the seventh initial condition (Eq. (27)) for (a)α = 1/4,(b) α = 1/2, (c) α = 3/4, (d) α = 1 at t = 0.8 and β = η = γ = µ = 1.

200 000

0

–200 000

–0.2

0.0

0.2

–0.2

0.0

0.2 0.0

0.1

0.2

0.0

0.1

0.2

0.0

0.1

0.2

0.0

0.1

0.2

–0.2

0.0

0.2

–0.2

0.0

0.2

50 000

0

100 000

20 000

10 000

0

–10 000

0

a b

c d

Fig. 15. For the Benney–Lin equation with the eighth initial condition (Eq. (28)) of Eq. (1), reduced differential transform method result for u(x, t) is,respectively (a) α = 1/4, (b) α = 1/2, (c) α = 3/4, (d) α = 1 at β = η = γ = µ = 1.

differential transform technique does not require any discretization, linearization or small perturbations therefore it reducesappreciably the numerical computation. In addition, we compare these two methods and show that the results of the

P.K. Gupta / Computers and Mathematics with Applications 61 (2011) 2829–2842 2841

a b

c d

–

–

–

–

–

––

–– –

– – – –

– –

Fig. 16. The comparison of the results of the RDTM and HPM for fractional Benney–Lin equation with the eighth initial condition (Eq. (28)) for (a) α = 1/4,(b) α = 1/2, (c) α = 3/4, (d) α = 1 at t = 0.8 and β = η = γ = µ = 1.

RDTMmethod are in excellent agreement with results of the HPMmethod and the obtained numerical solutions are showngraphically. We use the mathematical software MATHEMATICA, to calculate the functions obtained from the RDTM andHPM.

Acknowledgments

The author is thankful to the respected reviewers for their valuable suggestions for the improvement of the article. Theauthor is also thankful to the CSIR, NewDelhi, India for the financial support under the SRF (9/13(296)/2010-EMR-I) scheme.

References

[1] D. Tripathi, S.K. Pandey, S. Das, Peristaltic transport of a generalized Burgers’ fluid: application to the movement of chyme in small intestine, ActaAstronaut. (2011) doi:10.1016/j.actaastro.2010.12.010.

[2] D. Tripathi, P.K. Gupta, S. Das, Influence of slip condition on peristaltic transport of a viscoelastic fluid with fractional Burgers’ model, Therm. Sci.(2010) doi:10.2298/TSCI090924043T.

[3] D. Tripathi, Peristaltic transport of a viscoelastic fluid in a channel, Acta Astronaut. 68 (7–8) (2011) 1379–1385.[4] D. Tripathi, Numerical and analytical simulation of peristaltic flows of generalized oldroyd-B fluids, Int. J. Numer. Methods Fluids (2010)

doi:10.1002/fld.2466.[5] C. Xu, G. Wu, J.W. Feng, W. Zhang, Synchronization between two different fractional-order chaotic systems, Int. J. Nonlinear Sci. Numer. Simul. 9

(2008) 89–95.[6] L.J. Sheu, H.K. Chen, J.H. Chen, L.M. Tam,W.C. Chen, K.T. Lin, Y. Kang, Chaos in the Newton–Leipnik systemwith fractional order, Chaos Solitons Fractals

36 (2008) 98–103.[7] R.B. Leipnik, T.A. Newton, Double strange attractors in rigid body motion, Phys. Lett. A 86 (1981) 63–67.[8] L.J. Sheu, L.M. Tam, S.K. Lao, Y. Kang, K.T. Lin, J.H. Chen, H.K. Chen, Parametric analysis and impulsive synchronization of fractional-order

Newton–Leipnik systems, Int. J. Nonlinear Sci. Numer. Simul. 10 (2009) 33–44.[9] D.J. Benney, Long waves on liquid films, J. Math. Phys. 45 (1966) 150–155.

[10] S.P. Lin, Finite amplitude side-band stability of a viscous film, J. Fluid Mech. 63 (1974) 417–429.[11] H.A. Biaginoi, F. Linares, On the Benney–Lin and Kawahara equation, J. Math. Anal. Appl. 211 (1997) 131–152.[12] M. Safari, D.D. Ganji, E.M.M. Sadeghi, Application of He’s homotopy perturbation method and He’s variational iteration methods for solution of

Benney–Lin equation, Int. J. Comput. Math. (2009) doi:10.1080/00207160802524770.[13] N.G. Berloff, L.N. Howard, Solitary and periodic solutions of nonlinear non integrable equations, Stud. Appl. Math. 99 (1997) 1–24.[14] H.A. Biaginoi, F. Linares, On the Benney–Lin and Kawahara equation, J. Math. Anal. Appl. 211 (1997) 131–152.[15] S.B. Cui, D.G. Deng, S.P. Tao, Global existence of solutions for the Cauchy problem of the Kawahara equation with L2 initial data, Acta Math. Sinica 22

(2006) 1457–1466.[16] J.K. Zhou, Differential Transformation and its Applications for Electrical Circuits, Huazhong University Press, Wuhan, China, 1986 (in Chinese).[17] A. Arikoglu, I. Ozkol, Solution of fractional differential equations by using differential transform method, Chaos Solitons Fractals 34 (5) (2007)

1473–1481.[18] Y. Keskin, G. Oturanc, Reduced differential transform method for generalized KdV equations, Math. Comput. Appl. 15 (3) (2010) 382–393.[19] Y. Keskin, Ph.D. Thesis, Selcuk University, 2010.[20] Y. Keskin, G. Oturanc, Reduced differential transform method for partial differential equations, Int. J. Nonlinear Sci. Numer. Simul. 10 (6) (2009)

741–749.

2842 P.K. Gupta / Computers and Mathematics with Applications 61 (2011) 2829–2842

[21] S.S. Chen, C.K. Chen, Application to differential transformation method for solving systems of differential equations, Nonlinear Anal. Real World Appl.10 (2) (2009) 881–888.

[22] Y. Keskin, G. Oturanc, Reduced differential transform method for fractional partial differential equations, Nonlinear Sci. Lett. A 1 (2) (2010) 61–72.[23] J.H. He, Homotopy perturbation technique, Comput. Methods Appl. Mech. Engrg. 178 (1999) 257–262.[24] D. Tripathi, S.K. Pandey, S. Das, Peristaltic flow of viscoelastic fluid with fractional Maxwell model through a model, Appl. Math. Comput. 215 (2010)

3645–3654.[25] J.H. He, A generalized variational principle in micromorphic thermoelasticity, Mech. Res. Comm. 3291 (2005) 93–98.[26] P.K. Gupta, M. Singh, Homotopy perturbation method for fractional Fornberg–Whitham equation, Comput. Math. Appl. 61 (2010) 250–254.[27] S. Das, P.K. Gupta, Rajeev, A fractional predator–prey model and its solution, Int. J. Nonlinear Sci. Numer. Simul. 10 (2009) 873–876.[28] S. Das, P.K. Gupta, An approximate analytical solution of the fractional diffusion equation with absorbent term and external force by homotopy

perturbation method, Z. Naturforsch. 65a (2010) 182–190.[29] Z.M. Odibat, A study on the convergence of variational iteration method, Math. Comput. Modelling 51 (9–10) (2010) 1181–1192.