Computers and Mathematics with Applications 61 (2011) 250–254

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Computers and Mathematics with Applications

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

Homotopy perturbation method for fractional Fornberg–WhithamequationPraveen Kumar Gupta a,∗, Mithilesh Singh b

a Department of Applied Mathematics, Institute of Technology, Banaras Hindu University, Varanasi, Indiab Department of Mathematics, Dehradun Institute of Technology, Dehradun, India

a r t i c l e i n f o

Article history:Received 4 April 2010Received in revised form 27 October 2010Accepted 27 October 2010

Keywords:Partial differential equationFornberg–Whitham equationCaputo derivativeNonlinear equationHomotopy perturbation method

a b s t r a c t

This article presents the approximate analytical solutions to solve the nonlinearFornberg–Whitham equation with fractional time derivative. By using initial values, theexplicit solutions of the equations are solved by using a reliable algorithm like homotopyperturbation method. The fractional derivatives are taken in the Caputo sense. Numericalresults show that the HPM is easy to implement and accurate when applied to time-fractional PDEs.

© 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Considerable attention has been devoted to the study of the fractional calculus during the past three decades andtheir numerous applications in the area of physics and engineering. The applications of fractional calculus used in manyfields such as electrical networks, control theory of dynamical systems, probability and statistics, electrochemistry ofcorrosion, chemical physics, optics and signal processing can be successfully modelled by linear or nonlinear fractionalorder differential equations. Various definitions of fractional calculus are available in many books [1–3].

The HPM is the new approach for finding the approximate analytical solution of linear and nonlinear problems. Themethodwas first proposed by He [4,5] andwas successfully applied to solve nonlinear wave equation by He [6–8], fractionaldiffusion equation with absorbent term and external force by Das and Gupta [9], fractional convection–diffusion equationwith nonlinear source term by Momani and Yildirim [10], space–time fractional advection–dispersion equation by Yildirimand Kocak [11], fractional Zakharov–Kuznetsov equations by Yildirim and Gulkanat [12], boundary value problems byHe [13], integro-differential equation by El-Shahed [14], non-Newtonian flow by Siddiqui et al. [15], fractional PDEs in fluidmechanics by Yildirim [16], fractional Schrödinger equation [17,18] and nonlinear fractional predator–prey model [19] byHPM, linear PDEs of fractional order by He [20], Monami and Odibat [21], etc. The basic difference of this method from theother perturbation techniques is that it does not require small parameters in the equation which overcomes the limitationsof traditional perturbation techniques. He [4] claimed that the approximations obtained by this method are valid not onlyfor small parameters but for very large parameters. In 2009, Tian and Gao [22] studied the proof of the existence of theattractor for the one-dimensional viscous Fornberg–Whitham equation. Recently, Abidi and Omrani [23] have solved theFornberg–Whitham equation by the homotopy analysis method.

In the present paper, we have to solve the nonlinear time-fractional Fornberg–Whitham equation by the homotopyperturbation method. This equation can be written in operator form as

uαt − uxxt + ux = u uxxx − u ux + 3 uxuxx, t > 0, 0 < α ≤ 1, (1)

∗ Corresponding author. Tel.: +91 09453555044.E-mail address: praveen.rs.apm@itbhu.ac.in (P.K. Gupta).

0898-1221/$ – see front matter© 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.camwa.2010.10.045

P.K. Gupta, M. Singh / Computers and Mathematics with Applications 61 (2011) 250–254 251

subject to the initial condition

u (x, 0) = exp x2

, (2)

where u (x, t) is the fluid velocity, α is constant and lies in the interval (0, 1], t is the time and x is the spatial coordinate.Subscripts denote the partial differentiation unless stated otherwise. Fornberg and Whitham obtained a peaked solution ofthe form u(x, t) = A exp(− 1

2 |x −4t3 |), where A is an arbitrary constant.

2. Preliminaries and notations

In this section, we give some definitions and properties of the fractional calculus [2] which are used further in this paper.

Definition 1. A real function f (t), t > 0, is said to be in the space Cµ, µ ∈ ℜ, if there exists a real number p > µ, such thatf (t) = tp f1(t), where f1(t) ∈ C(0, ∞), and it is said to be in the space Cn

µ if and only if h(n)∈ Cµ, n ∈ N .

Definition 2. The Riemann–Liouville fractional integral operator (Jα) of order α ≥ 0, of a function f ∈ Cµ, µ ≥ −1, isdefined as

Jα f (t) =1

Γ (α)

∫ t

0(t − ξ)α−1f (ξ) dξ, α > 0, t > 0,

J0f (t) = f (t)

where Γ (α) is the well-known gamma function. Some of the properties of the operator Jα , which we will need here, are asfollows:

For f ∈ Cµ, µ ≥ −1, α, β ≥ 0 and γ ≥ −1 :

(1) Jα Jβ f (t) = Jα+β f (t),(2) Jα Jβ f (t) = Jβ Jα f (t),(3) Jαtγ =

Γ (γ+1)Γ (α+γ+1) t

α+γ .

Definition 3. The fractional derivative (Dα) of f (t), in the Caputo sense is defined as

Dα f (t) =1

Γ (n − α)

∫ t

0(t − ξ)n−α−1f (n)(ξ) dξ,

for n − 1 < α < n, n ∈ N, t > 0, f ∈ Cn−1.

The following are two basic properties of the Caputo fractional derivative [20]:

(1) Let f ∈ Cn−1, n ∈ N , then Dα f , 0 ≤ α ≤ n is well defined and Dα f ∈ C−1.

(2) Let n − 1 ≤ α ≤ n, n ∈ N and f ∈ Cnµ, µ ≥ −1. Then

(JαDα) f (t) = f (t) −

n−1−k=0

f (k)(0+)tk

k!.

3. Solution of the problem

We first consider the following time-fractional Fornberg–Whitham equation

Dαt u − Dx x tu + Dxu = u Dx x xu − u Dxu + 3Dxu Dx xu, (3)

with initial condition

u (x, 0) = exp x2

. (4)

According to the HPM [4–8], we construct the following homotopy

Dαt u = p [Dx x tu − Dxu + u Dx x xu − u Dxu + 3Dxu Dx xu], (5)

where the homotopy parameter p is considered as a small parameter (p ∈ [0, 1]). Now applying the classical perturbationtechnique, we can assume that the solution of Eq. (3) can be expressed as a power series in p as given below

u = u0 + p u1 + p2u2 + p3u3 + p4u4 + · · · . (6)

252 P.K. Gupta, M. Singh / Computers and Mathematics with Applications 61 (2011) 250–254

When p → 1, Eq. (5) corresponds to Eq. (3) and Eq. (6) becomes the approximate solution of Eq. (3), that is, of Eq. (1).Substituting Eq. (6) in Eq. (5) and comparing the like powers of p, we obtain the following set of linear differential equations

p0 : Dαt u0 = 0, (7)

p1 : Dαt u1 = Dx x tu0 − Dxu0 + u0 Dx x xu0 − u0 Dxu0 + 3Dxu0 Dx xu0, (8)

p2 : Dαt u2 = Dx x tu1 − Dxu1 + u0 Dx x xu1 + u1 Dx x xu0 − u0 Dxu1 − u1 Dxu0 + 3Dxu0 Dx xu1 + 3Dxu1 Dx xu0, (9)

p3 : Dαt u3 = Dx x tu2 − Dxu2 + u0 Dx x xu2 + u1 Dx x xu1 + u2 Dx x xu0 − u0 Dxu2 − u1 Dxu1

− u2 Dxu0 + 3Dxu0 Dx xu2 + 3Dxu1 Dx xu1 + 3Dxu2 Dx xu0, (10)

and so on.The method is based on applying the operator Jαt (the inverse operator of the Caputo derivative Dα

t ) on both sides of Eqs.(7)–(10), then we obtain

u0(x, t) = exp x2

, (11)

u1(x, t) = −12

exp x2

tα

Γ (α + 1), (12)

u2(x, t) = −18

exp x2

t2α−1

Γ (2α)+

14

exp x2

t2α

Γ (2α + 1), (13)

u3(x, t) = −132

exp x2

t3α−2

Γ (3α − 1)+

18

exp x2

t3α−1

Γ (3α)−

18

exp x2

t3α

Γ (3α + 1). (14)

Proceeding in this manner, the rest of the components un can be obtained and the series solutions are thus entirelydetermined.

Finally, we approximate the analytical solution u (x, t) by the truncated series

u (x, t) = limN→∞

ΦN(x, t), (15)

where ΦN(x, t) =∑N−1

n=0 un(x, t).The above series solutions generally converge very rapidly. A classical approach of convergence of this type of series is

already presented by Abbaoui and Cherruault [24].

4. Numerical results and discussion

In this section numerical results of the displacement u(x, t) for different time-fractional Brownianmotions α = 2/3, 3/4and for the standard motion α = 1 are calculated for various values of t and x. Here the initial condition is taken asu(x, 0) = exp( x

2 ) as a particular case for showing the nature of the displacement. The numerical results u(x, t) for variousvalues of t, x and α are depicted through Fig. 1(a)–(c) and those for different t and α at x = 1 are given in Fig. 2.

It is seen from Fig. 1(a)–(c) that u(x, t) increases with the increase in both x and t for α =23 ,

34 and α = 1. Fig. 1(d)

clearly shows that, when α = 1, the solution is very near to the exact solution. It is also seen from Fig. 2 that as the value ofα increases, the displacement u(x, t) increases but afterwards its nature is opposite.

5. Conclusion

In this work, the homotopy perturbationmethod has been applied for finding the approximate solutions of the nonlinearfractional Fornberg–Whitham equation. The numerical results showed that this method has very good accuracy andreductions in the size of calculations comparable to other perturbation techniques.

In HPM, according to the homotopy technique, a homotopy with an embedding parameter p ∈ [0, 1] is constructed,and the embedding parameter is considered as a ‘‘small parameter’’, which can take full advantages of the traditionalperturbation methods and homotopy techniques. This method contains the homotopy parameter p, which provides uswith a simple way to control the convergence region of solution series for large values of t . It is obvious that the HPM isa very powerful, easy and efficient technique for solving various kinds of nonlinear problems in science and engineeringwithout many assumptions and restrictions. It can also be applied in real problems, where differential equations governingthe process are nonlinear and boundary conditions are complicated.

P.K. Gupta, M. Singh / Computers and Mathematics with Applications 61 (2011) 250–254 253

Fig. 1. The behaviour of the u (x, t) w.r.t. x and t are obtained, when (a) α = 2/3 (b) α = 3/4 (c) α = 1 (d) exact solution.

Fig. 2. Plots of u(x, t) vs. t at x = 1 for different values of α.

Acknowledgements

The authors of this article express their sincere thanks to the reviewers for their valuable suggestions in the improvementof the article. The first author is grateful to the CSIR, NewDelhi, India for the financial support under the SRF (9/13(296)/2010-EMR-I) scheme.

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