Accepted Manuscript

The homotopy analysis method for solving the time-fractional Fornberg-Whith‐

am equation and comparison with Adomian’s decomposition method

Mehmet Giyas Sakar, Fevzi Erdogan

PII: S0307-904X(13)00254-0

DOI: http://dx.doi.org/10.1016/j.apm.2013.03.074

Reference: APM 9443

To appear in: Appl. Math. Modelling

Please cite this article as: M.G. Sakar, F. Erdogan, The homotopy analysis method for solving the time-fractional

Fornberg-Whitham equation and comparison with Adomian’s decomposition method, Appl. Math. Modelling

(2013), doi: http://dx.doi.org/10.1016/j.apm.2013.03.074

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers

we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and

review of the resulting proof before it is published in its final form. Please note that during the production process

errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

THE HOMOTOPY ANALYSIS METHOD FOR SOLVING THE

TIME-FRACTIONAL FORNBERG-WHITHAM EQUATION AND

COMPARISON WITH ADOMIAN’S DECOMPOSITION METHOD

M. G. SAKAR, F. ERDOGAN

Abstract. In this paper, we applied relatively new analytical techniques, ho-motopy analysis method (HAM) and Adomian’s decomposition method (ADM)for solving time-fractional Fornberg- Whitham equation. The homotopy anal-

ysis method contains the auxilary parameter, which provides us with a simpleway to adjust and control the convergence region of solution series. The frac-tional derivatives are described in the Caputo sense. A comparison is made

the between HAM and ADM results. The present methods performs extremelywell in terms of efficiency and simplicity. Numerical results for different par-ticular cases of the problem are presented.

1. Introduction

In recent years, there has been a growing interest in the field of fractional cal-culus, i.e. the theory of derivatives and integrals of fractional non-integer order.Considerable interest has been devoted to study of the fractional calculus and theirnumerous applications in the areas of science and engineering. Fractional differ-ential equations are increasingly used to model problems in a number of researchareas including description of fractional random walk, control theory of dynami-cal systems, electrical networks, frequency dependent damping behavior materials,signal processing and system identification, diffusion and reaction processes, creep-ing and relaxation for viscoelastic materials [1,2]. Fractional derivatives providean excellent instrument for the description of memory and hereditary properties ofvarious materials and processes [3].

Most fractional differential equations do not have exact analytic solutions so ap-proximation and numerical methods must be used. A great deal of effort has beenexpended in attempting to find robust and stable numerical and analytical meth-ods for solving fractional differential equations of physical interest. These meth-ods include finite difference method [4-6], Adams-Bashforth-Moulton method [7],Homotopy perturbation method [8-10], Generalized differential transform method[11,12], Variational iteration method [13-15] and Adomian’s decomposition method[16,17]. However, neither perturbation nor non-perturbation method can provideus with a simple way to adjust and control the convergence region and rate of givenseries. Homotopy analysis method (HAM) is proposed 1992 by Liao [18-22]. HAMprovides an effective procedure for explicit, approximate analytical solutions of awide and general class of differential equations. Based on homotopy of topology,

Key words and phrases. Homotopy analysis method, time-fractional Fornberg-Whitham equa-tion, Caputo derivative, Adomian’s decompositon method, Auxiliary parameter.

1

2 M. G. SAKAR, F. ERDOGAN

the validity of the HAM is independent of whether or not there exist small param-eters in the considered equation. Therefore, the HAM can overcome the foregoingrestrictions and limitations of perturbation techniques so that it provides us with apossibility to analyze strongly nonlinear problems [23]. HAM has been successfullyapplied to solve many types of nonlinear problems in science and engineering bymany others [24-28].

We consider linear and nonlinear fractional partial differential equations of theform

Dαt u(x, t) = f(u, ux, ut, uxx, uxxx, uxxt), n − 1 < α ≤ n, t > 0, (1.1)

subject to the initial conditions

u(k)(x, 0) = gk(x), k = 0, 1, 2, ..., n − 1, (1.2)

where n is an integer, f is a linear/nonlinear function, and Dαt (.) = ∂α

∂tα(.) is a

fractional differential operator.The operator form of the nonlinear fractional partial differential equations (1.1)

can be written as follows

Dαt u(x, t) = A(ux, uxxt)+B(u, ux, uxx, uxxx)+C(x, t), n−1 < α ≤ n, t > 0

(1.3)

subject to the initial conditions

u(k)(x, 0) = gk(x), k = 0, 1, 2, ..., n − 1, (1.4)

where A is a linear operator, B is a nonlinear operator and C is a known analyticfunction.

In the present paper, we shall apply HAM and ADM to find the approximate ana-lytical solution of a special case of (1.3)-(1.4) the so-called time-fractional Fornberg-Whitham equation and compare it with the exact solution. This equation can bewritten in operator form as

Dαt u(x, t) = −ux+uxxt+u uxxx−u ux+3uxuxx, t > 0, 0 < α ≤ 1, (1.5)

subject to initial condition

u(x, 0) = e1

2x (1.6)

where u(x, t) is the fluid velocity, α is constant and lies in the interval (0, 1], t isthe time and x is the spatial coordinate.

2. Fractional Calculus

Fractional calculus unifies and generalizes the notions of integer-order differenti-ation and n-fold integration. There exists a vast literature on different definitionsof fractional derivatives. The most popular ones are the Riemann-Liouville and theCaputo derivatives. We have chosen to use the Caputo fractional derivative becauseit allows traditional initial and boundary conditions to be included in the formula-tion of the problem, but for homogeneous initial condition assumption, these twooperators coincide. For more details on the geometric and physical interpretationfor fractional derivatives of both the Riemann-Liouville and Caputo types, see [33].We give some basic definitions and properties of fractional calculus theory whichshall be used in this paper:

3

Definition 2.1. A real function f(x), x > 0, is said to be in the space Cµ,µ ∈ R if there exists a real number p (> µ), such that f(x) = xpf1(x), wheref1(x) ∈ C [0,∞), and it is said to be in the space Cm

µ iff f (m) ∈ Cm, m ∈ N. TheRiemann–Liouville fractional integral operator is defined as follows:

Definition 2.2. The Riemann–Liouville fractional integral operator of orderα ≥ 0, of a function f ∈ Cµ, µ ≥ −1, is defined as

Jαf(x) =1

Γ(α)

x∫

0

(x − t)α−1f(t)dt, α > 0, x > 0,

J0f(x) = f(x).

In this paper only real and positive values of α will be considered.Properties of the operator Jα can be found in [1-3] and we mention only the

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

1. JαJβf(x) = Jα+βf(x),

2. JαJβf(x) = JβJαf(x),

3. Jαxγ = Γ(γ+1)Γ(α+γ+1)x

α+γ .

Lemma 2.1. If m − 1 < α ≤ m, m ∈ N and f ∈ Cmµ , µ ≥ −1, then

DαJαf(x) = f(x),

and,

JαDαf(x) = f(x) −m−1∑

i=0

f (i)(0+)xi

i!, x > 0.

Definition 2.3. The fractional derivative of f(x) in Caputo sense is defined as

Dαf(x) = Jm−αDmf(x) = 1Γ(m)

x∫

0

(x − s)m−α−1f (m)(s)ds,

for m − 1 < α ≤ m, m ∈ N, x > 0, f ∈ Cm−1.

Definition 2.4. For m to be the smallest integer that exceeds α, the Caputotime-fractional derivative operator of order α > 0 is defined as

Dαu(x, t) =

1Γ(m−α)

x∫

0

(t − τ)m−α−1 ∂mu(x,t)∂τm

dτ , m − 1 < α < m

∂mu(x,t)∂tm

, α = m ∈ N

3. Basic ideas of Homotopy Analysis method

We consider the following differential equation

N [u (x, t)] = 0, (3.1)

where, N is a nonlinear differential operator, x and t are independent variables,u (x, t) is an unknown function, respectively, By means of generalizing the tradi-tional homotopy method, Liao [18] constructs the so-called zero-order deformation

4 M. G. SAKAR, F. ERDOGAN

equation

(1 − p) L [φ (x, t; p) − u0 (x, t)] = p~H (x, t) N [φ (x, t; p)] , (3.2)

subject to the following initial conditions:

φ(k)(x, 0; p) = gk(x), k = 0, 1, 2, ...,m − 1, (3.3)

where p ∈ [0, 1] is the embedding parameter, ~ 6= 0 is a non-zero auxiliaryparameter, H (x, t) 6= 0 is non-zero auxiliary function, L = Dα

t (n − 1 < α ≤ n) isan auxiliary linear operator with the following property

L[φ(x, t)] = 0 when φ(x, t) = 0. (3.4)

u0 (x, t) is an initial guess of u (x, t), φ (x, t; p) is an unknown function, respectively.It is important that one has great freedom to choose auxiliary things such as ~ andL in homotopy analysis method. Obviously, when p = 0 and p = 1, it holds

φ (x, t; 0) = u0 (x, t) , φ (x, t; 1) = u (x, t) , (3.5)

respectively. Thus, as p increases from 0 to 1, the solution φ (x, t; p) varies from theinitial guesses u0 (x, t) to the solution u (x, t). Expanding φ (x, t; p) in Taylor serieswith respect to p, we have

φ (x, t; p) = u0 (x, t) +

∞∑

m=1

um (x, t) pm, (3.6)

where

um (x, t) =1

m!

∂mφ (x, t; p)

∂pm

∣

∣

∣

∣

p=0

. (3.7)

If the auxiliary linear operator, the initial guess, the auxiliary operator ~, and theauxiliary function are properly chosen, the series (3.6) converges at p = 1, then wehave

u (x, t) = u0 (x, t) +

∞∑

m=1

um (x, t) , (3.8)

Define the vector

−→un = {u0 (x, t) , u1 (x, t) , .., un (x, t)} .

Differentiating equation (3.2) m-times with respect to the embedding parameter p

and then setting p = 0 and finally dividing them by m!, we obtain the mth-orderdeformation equation, with the assumption H (x, t) = 1

L [um (x, t) − χmum−1 (x, t)] = ~Rm(−→u m−1) (3.9)

subject to the following initial conditions:

u(k)m (x, 0) = 0, k = 0, 1, 2, ...,m − 1, (3.10)

where,

Rm(−→u m−1) =1

(m − 1)!

∂m−1N [φ (x, t; p)]

∂pm−1

∣

∣

∣

∣

p=0

. (3.11)

5

and

χm =

{

0, m ≤ 1,

1, m > 1.(3.12)

Substituting (3.1) into (3.11), and since A is linear operator, Rm(−→u m−1) can begiven by

Rm(−→u m−1) = Dαt um−1 − A(u(m−1)x, u(m−1)xxt)

−1

(m − 1)!

∂m−1B (φ, φx, φxx, φxxx)

∂qm−1

∣

∣

∣

∣

q=0

− (1 − χm) C (x, t) (3.13)

We can apply the operator Jα to both sides of (3.9) to obtain

JαDα [um (x, t) − χmum−1 (x, t)] = ~JαRm(−→u m−1). (3.14)

Using the definition (2.3) and the initial conditions (1.2), we obtain

um (x, t) = χmum−1 (x, t) + ~JαRm(−→u m−1). (3.15)

Finally, we will approximate HAM solution (3.8) by the truncated series

φm(x, t) =

m−1∑

k=0

uk (x, t) . (3.16)

The mth-order deformation equation (3.9) is linear and thus can be easily solved,especially by means of symbolic computation software such as Mathematica andMaple.

4. Convergence Theorem of HAM

Theorem 4.1. As long as the series u (x, t) = u0 (x, t)++∞∑

m=1um (x, t) converges,

where um (x, t) is governed by (3.9) under the definitions (3.11) and (3.12), it mustbe a solution of (1.3)

Proof : See [18].

5. Basic ideas of Adomian’s Decomposition Method

The Adomian’s decompositon method requires that the time-fractional Fornberg-Whitham equation (1.5) be expressed in terms of operatorm form as

Dαt u(x, t) + Lu(x, t) + Nu(x, t) = 0 (5.1)

where the notation Lu(x, t) = −Lxxtu(x, t)+Lxu(x, t) symbolizes the linear differ-ential operator and the notation Nu(x, t) = 1

2Lx

[

u2(x, t) − Lxxu2(x, t)]

symbolizesthe nonlinear operator. Applying the operator Jα

t , the inverse operator Dαt to both

side of Eq. (5.1) yields

u (x, t) =

m−1∑

k=0

∂ku

∂tk

(

x, 0+) tk

k!− Jα

t [Lu(x, t) + Nu(x, t)] . (5.2)

6 M. G. SAKAR, F. ERDOGAN

The ADM suggests the solution u(x, t) be decomposed into the infinite series ofcomponents

u (x, t) =∞∑

n=0

un (x, t) (5.3)

and the nonlinear opeartor Nu (x, t) is decomposed as follows:

Nu (x, t) =∞∑

n=0

An (5.4)

where An are so-called the Adomian’s Polynomials. The Adomian’s PolynomialsAn can be calculated for all form of nonlinearity according to specific algorithmsconstructed by Adomian [29,30]. The general form of formula for An Adomian’sPolynomials is

An =1

n!

[

dn

dλnN

(

∞∑

k=0

λkuk (x, t)

)]

λ=0

, n ≥ 0. (5.5)

Substitution the decomposition series (5.3) and (5.4) into both sides of (5.2)gives

∞∑

n=0

u (x, t) =m−1∑

k=0

∂ku

∂tk

(

x, 0+) tk

k!− Jα

[

L

(

∞∑

n=0

un(x, t)

)

+∞∑

n=0

An

]

(5.6)

from this equation, the iterates are determined by following recursive formula

u0 (x, t) =

m−1∑

k=0

∂ku

∂tk

(

x, 0+) tk

k!(5.7)

u1 (x, t) = −Jα [L (u0(x, t)) + A0] (5.8)

u2 (x, t) = −Jα [L (u1(x, t)) + A1] (5.9)

un+1 (x, t) = −Jα [L (un(x, t)) + An] (5.10)

This formula is easy to compute by using symbolic computation software such asMathematica and Maple or by writing a computer code to get as many polynomialsas we need in the calculation of the numerical as well as explicit solutions. Finally,The approximate solution by the truncated series

φk(x, t) =k−1∑

n=0

un (x, t) and u(x, t) = limk→∞

φk(x, t). (5.11)

where φk(x, t) =k−1∑

n=0un (x, t) denote the k-term approximation to u.

The decomposition series solution generally converge very rapidly, A classical ap-proach of convergence of Adomian’s decomposition method is presented by Abbouiand Cherruault [31]

7

6. Application of HAM and ADM

To solve eq.(1.5)-(1.6) by means of HAM, we choose the initial approximation

u0 (x, t) = u (x, 0) = e1

2x (6.1)

Eq. (1.5) suggests the nonlinear operator as

N [φ (x, t; p)] = Dαt φ (x, t; p)−

∂3φ (x, t; p)

∂x2∂t+

∂φ (x, t; p)

∂x−φ (x, t; p)

∂3φ (x, t; p)

∂x3

+φ (x, t; p)∂φ (x, t; p)

∂x− 3

∂φ (x, t; p)

∂x

∂2φ (x, t; p)

∂x2(6.2)

and the linear operator

L [φ (x, t; p)] = Dαt φ (x, t; p) (6.3)

with the property L (C1) = 0 where C1 is the integration constant. Using abovedefinition, with assumption H (x, t) = 1, we construct the zeroth-order deformationequation

(1 − p) L [φ (x, t; p) − u0 (x, t)] = p~N [φ (x, t; p)] , (6.4)

Obviously, when p = 0 and p = 1,

φ (x, t; 0) = u0 (x, t) , φ (x, t; 1) = u (x, t) ,

Therefore , as the embedding operator p increases from 0 to 1, φ (x, t; p) varies fromthe initial guesses u0 (x, t) to the solution u (x, t). Then, we obtain the mth-orderdeformation equation

L [um (x, t) − χmum−1 (x, t)] = ~Rm(−→u m−1), (6.5)

Where

Rm(−→u m−1) = Dαt um−1 (x, t) −

∂3um−1 (x, t)

∂x2∂t+

∂um−1 (x, t)

∂x

+

m−1∑

k=0

[ −uk(x, t)∂3um−1−k(x, t)

∂x3+ uk(x, t)

∂um−1−k(x, t)

∂x]

+m−1∑

k=0

[−3∂um−1−k(x, t)

∂x

∂2um−1−k(x, t)

∂x2]. (6.6)

The solution of mth-order deformation equation (6.5) for m ≥ 1 becomes

um (x, t) = χmum−1 (x, t) + ~J−1Rm(−→u m−1). (6.7)

from (6.1) and (6.7), we now successively obtain

u0(x, t) = e1

2x (6.8)

u1(x, t) =1

2~e

1

2x tα

Γ (α + 1)(6.9)

8 M. G. SAKAR, F. ERDOGAN

u2(x, t) =

(

4~ + 3~2

8

)

e1

2x t2α−1

Γ (2α)+

(

~2

4

)

e1

2x t2α

Γ (2α + 1)(6.10)

u3(x, t) =

(

16~ + 24~2 + 9~

3

32

)

e1

2x t3α−2

Γ (3α − 1)+

(

8~2 + 6~

3

16

)

e1

2x t3α−1

Γ (3α)

+

(

~3

8

)

e1

2x t3α

Γ (3α + 1)(6.11)

u4 (x, t) =

(

64~ + 144~2 + 108~

3 + 27h4

128

)

e1

2x t4α−3

Γ (4α − 2)

+

(

48~2 + 72~

3 + 27~4

64

)

e1

2x t4α−2

Γ (4α − 1)

+

(

12~3 + 9~

4

32

)

e1

2x t4α−1

Γ (4α)+

(

~4

16

)

e1

2x t4α

Γ (4α + 1)(6.12)

u5 (x, t) =

(

256~ + 768~2 + 864~

3 + 432~4 + 815~

5

512

)

e1

2x t5α−4

Γ (5α − 3)

+

(

64~2 + 144~

3 + 108~4 + 27~

5

64

)

e1

2x t5α−3

Γ (5α − 2)

+

(

48~3 + 72~

4 + 27~5

64

)

e1

2x t5α−2

Γ (5α − 1)

+

(

4~4 + 3~

5

16

)

e1

2x t5α−1

Γ (5α)+

(

~5

32

)

e1

2x t5α

Γ (5α + 1)(6.13)

and so on, Therefore, we use six term in evaluating the approximating solutionu (x, t) by the truncated series

uHAM =

5∑

i=0

ui. (6.14)

We remark that the exact travelling wave solution to the above initial valueproblem is given by [10]

u(x, t) = e1

2x− 2

3t (6.15)

To solve eq.(1.5)-(1.6) by means of ADM, Considering the given initial condition,

we can assume u0(x, t) = e1

2x as an initial approximation. We next use the recursive

relations (5.8)-(5.10) to obtain the rest of components of un(x, t)

u1 (x, t) = −Jα [−Lxxt(u0) + Lx(u0) + A0]

u1(x, t) = −1

2e

1

2x tα

Γ (α + 1)(6.16)

u2 (x, t) = −Jα [−Lxxt(u1) + Lx(u1) + A1]

9

u2(x, t) = −1

8e

1

2x t2α−1

Γ (2α)+

1

4e

1

2x t2α

Γ (2α + 1)(6.17)

u3 (x, t) = −Jα [−Lxxt(u2) + Lx(u2) + A2]

u3(x, t) = −1

32e

1

2x t3α−2

Γ (3α − 1)+

1

8e

1

2x t3α−1

Γ (3α)−

1

8e

1

2x t3α

Γ (3α + 1)(6.18)

u4 (x, t) = −Jα [−Lxxt(u3) + Lx(u3) + A3]

u4 (x, t) = −1

128e

1

2x t4α−3

Γ (4α − 2)+

3

64e

1

2x t4α−2

Γ (4α − 1)

−3

32e

1

2x t4α−1

Γ (4α)+

1

16e

1

2x t4α

Γ (4α + 1)(6.19)

u5 (x, t) = −Jα [−Lxxt(u4) + Lx(u4) + A4]

u5 (x, t) = −1

512e

1

2x t5α−4

Γ (5α − 3)+

1

64e

1

2x t5α−3

Γ (5α − 2)

−3

64e

1

2x t5α−2

Γ (5α − 1)+

1

16e

1

2x t5α−1

Γ (5α)−

1

32e

1

2x t5α

Γ (5α + 1)(6.20)

and so on, Therefore, we use six term in evaluating the decomposition solutionu (x, t) by the truncated series

uADM =

5∑

i=0

ui. (6.21)

.. .. .. ¶t H0, 0L

_ _ _ _ ¶tt H0, 0L

- - - - - ¶ttt H0, 0L

.........¶tttt H0, 0L

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

0

2

4

6

� h

�uHx

,tL

Figure 1. The ~-curve at (0,0) for 6th-order HAM approximatesolution (6.14) when α = 1.

10 M. G. SAKAR, F. ERDOGAN

Table 1. Absolute errors for differences between the exact solu-tion (6.15) and 6th-order HAM approximate given by for ~ = −1(or ADM) and α = 1

xi/ti 0.2 0.4 0.6 0.8 1-4 3.74057E-6 5.04192E-6 1.17973E-5 1.04731E-5 7.66092E-7-2 1.01679E-5 1.37054E-5 3.20684E-5 2.84688E-5 2.08245E-60 2.76393E-5 3.72550E-5 8.71710E-5 7.73862E-5 5.66069E-62 7.51314E-5 1.01270E-4 2.36955E-4 2.10357E-4 1.53873E-54 2.04228E-4 2.75280E-4 6.44112E-4 5.71811E-4 4.18272E-5

Figure 2. For Fornberg-Whitham equation with the initial con-dition of Eq.(1.6), HAM result when ~ = −1 (or ADM) for u(x, t)is, respectively (a) α = 1, (b) α = 0.9, (c) α = 0.8, (d) α = 0.7,(e) exact solution (6.15)

7. Numerical results and discussion

The series (6.14) contain the auxiliary parameter ~. As pointed out by Liao[18], the auxiliary parameter ~ can be employed to adjust the convergence regionof the homotopy analysis solution. This ~-curve, contain a horizontal line segment.

11

aL

0 2 4 6 8 10

0

10

20

30

40

50

60

70

� x

�uH

x,tL

bL

0 2 4 6 8 10

0

10

20

30

40

50

60

70

� x

�uH

x,tL

cL

0 2 4 6 8 10

0

20

40

60

� x

�uH

x,tL

dL

0 2 4 6 8 10

0

20

40

60

80

� x�

uHx,

tL

eL

0 2 4 6 8 10

0

10

20

30

40

50

60

70

� x

�uH

x,tL

Figure 3. For Fornberg-Whitham equation with the initial con-dition of Eq.(1.6), HAM result when ~ = −1 (or ADM) and t = 1for u(x, t) is, respectively (a) α = 1, (b) α = 0.9, (c) α = 0.8, (d)α = 0.7, (e) exact solution (6.15)

Table 2. Absolute errors for differences between the exact solu-tion (6.15) and 6th-order ADM solution (6.21) when t = 6 andα = 1.

xi uexact uADM |uexact-uADM |-4 0.0024787521 0.0056037265 3.12497E-3-2 0.0067379469 0.0152325081 8.49456E-30 0.0183156388 0.0414062500 2.30906E-22 0.0497870683 0.1125538569 6.27667E-24 0.1353352832 0.3059531040 1.70617E-1

This horizontal line segment denotes the valid region of ~ which guaranteed theconvergence of related series. To investigate the influence of ~ on the solution series,we plot the so-called ~-curve of u(0, 0) obtained from the 6th-order HAM solutionas shown in Figure 1. It is observed the valid region for ~ is −1.1 < ~ < −0.9. Weget the same values as in ADM (6.21) or HPM when ~ = -1. Therefore, the HAM

12 M. G. SAKAR, F. ERDOGAN

Table 3. Absolute errors for differences between the exact solu-tion (6.15) and 6th-order HAM solution (6.14) for ~ =-1.04 whent = 6 and α = 1.

xi uexact uHAM |uexact-uHAM |-4 0.0024787521 0.0015123400 9.66413E-4-2 0.0067379469 0.0041109600 2.62698E-30 0.0183156388 0.0111748000 7.14088E-32 0.0497870683 0.0303761000 1.94109E-24 0.1353352832 0.0825709000 5.27644E-2

. . . Exact solution

.......6th order HAM solution ( Ñ= -1.04 )

.. .. 6th order HAM solution ( Ñ= -1.01 )

_ _ _6th order HAM solution ( Ñ= -1)

-5 0 5

0.0

0.2

0.4

0.6

0.8

� x

�uHx

,tL

Figure 4. The results obtained by HAM for various ~ by 6thorder HAM approximate solution (6.14) for α = 1 in comparisonwith the exact solution (6.15) when −8 < x < 8 and t = 6.

is rather general and contains the ADM and HPM. In Table 1, we compute theabsolute errors for differences between the exact solution (6.15) and approximatesolution (6.14) obtained by the HAM (~ = -1). In Table 2, we compute the absoluteerrors for differences between the exact solution (6.15) and ADM (6.21) solutionwhen t = 4 and α = 1. In Figure 2, we study the diagrams of results obtained byHAM (~ = -1) for different particular cases of α. The exact solution (6.14) andHAM solution (6.15) for different particular cases of α are prensented graphicallyin Figure 3, when t = 1. In Figure 4, we study the diagrams of results obtained byHAM for ~ = -1 and ~ = -1.04 in comparison with the exact solution (6.15), onecan see that the best value of ~ in this case is not −1. But then, the HAM solution(6.15) has the same shape as the exact solution for large range of t, i.e., t = 6 asshown in Figure 4 and Table 3, when we take ~ = -1.04. We can remark highterrors for ~ = -1 (see Table 2). Therefore, we can say that HAM is more effectivethan ADM.

13

8. Conclusion

In this paper, the homotopy analysis method has been applied for finding theapproximate analytical solution of the nonlinear time-fractional Fornberg-Whithamequation. The explicit series solution of Fornberg-Whitham equation are obtained,which are the same as those results given by Adomian’s decomposition mehodand Homotopy perturbation method for ~ = -1. This accords with the conclusionthat HAM logically contains the ADM and HPM. The HAM provides us with aconvenient way to the control the convergence region of solution series for largevalues of t, which is a fundamental qualitative difference in analysis between HAMand other methods. The results show that HAM is powerful mathematical tool forsolving nonlinear fractional partial differential equations having wide applicationsin science and engineering.

14 M. G. SAKAR, F. ERDOGAN

References

[1] K. B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.[2] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.[3] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.

[4] V. E. Lynch, B. A. Carreras, D. del-Castillo-Negrete, K. M. Ferriera-Mejias, H. R. Hicks,Numerical methods for the solution of partial differential equations of fractional order, J.Comput. Phys. 192 (2003) 406-421.

[5] M. M. Meerschaert, C. Tadjeran, Finite difference approximations for two-sided space-

fractional partial differential equations, Appl. Numer. Math. 56 (2006) 80-90.[6] C. Tadjeran, M. M. Meerschaert, A second-order accurate numerical method for the two-

dimensional fractional diffusion equation, J. Comput. Phys. 220 (2007) 813-823.[7] K. Diethelm, N. J. Ford, A. D. Freed, Detailed error analysis for a fractional Adams method,

Numer. Algorithms 36 (2004) 31-52.

[8] Q. Wang, Homotopy perturbation method for fractional KdV equation, Appl. Math. Comput.190 (2007) 1795-1802.

[9] A. Yıldırım, An algorithm for solving the fractional nonlinear Schrodinger equation by meansof the homotopy perturbation method, Int. J. Nonlinear Sci. Numer. Simul. 10 (4) (2009) 445-451.

[10] P. K. Gupta, M. Singh, Homotopy perturbation method for fractional Fornberg-Whithamequation, Comput. Math. Appl. 61 (2011) 250-254.

[11] S. Momani, Z. Odibat, A novel method for nonlinear fractional partial differential equations:combination of DTM and generalized Taylor’s formula, J. Comput. Appl. Math. 220 (1-2)

(2008) 85-95.[12] J. Liu, G. Hou, Numerical solutions of the space- and time-fractional coupled Burgers equa-

tions by generalized differential transform method, Appl. Math. Comput. 217 (16) (2011)7001-7008.

[13] R. Yulita Molliq, M. S. M. Noorani, I. Hashim, R. R. Ahmad, Approximate solutions offractional Zakharov-Kuznetsov equations by VIM, J. Comput. Appl. Math. 233 (2) (2009)103-108.

[14] M. G. Sakar, F. Erdogan, A. Yıldırım, Variational iteration method for the time-fractional

Fornberg-Whitham equation, Comput. Math. Appl. 63 (9)(2012) 1382-1388.[15] M. Inc, The approximate and exact solutions of the space- and time-fractional burgers equa-

tions with initial conditions by variational iteration method, J. Math. Anal. Appl. 345 (1)(2008) 476-484.

[16] H. Jafari, V. D. Gejji, Solving a system of nonlinear fractional differential equations using

Adomian decomposition, Appl. Math. Comput. 196 (2006) 644-651.[17] S. Momani, Z. Odibat, Analytical solution of a time-fractional Navier-Stokes equation by

Adomian decomposition method, Appl. Math. Comput. 177 (2006) 488-494.[18] S.J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman

and Hall/CRC Press, Boca Raton, 2003.[19] S.J. Liao, An approximate solution technique which does not depend upon small parameters:

A special example, Internat. J. Non-Linear Mech. 30 (1995) 371-380.[20] S.J. Liao, On the homotopy analysis method for nonlinear problems, Appl. Math. Comput.

147 (2004) 499-513.

[21] S.J. Liao, Notes on the homotopy analysis method: Some definitions and theorems, Commun.Nonlinear Sci. Numer. Simulat. 14 (2009) 983-997.

[22] S.J. Liao, Comparison between the homotopy analysis method and the homotopy perturba-tion method, Appl. Math. Comput. 169 (2005) 1186-1194.

[23] S.J. Liao, An analytic approach to solve multiple solutions of a strongly nonlinear problem,

Appl. Math. Comput. 169 (2005) 854-865.[24] L. Song, H. Zhang, Application of homotopy analysis method to fractional KdV-Burgers-

Kuramoto equation, Phys. Lett. A 367 (1-2) (2007) 88-94.[25] M. Ganjiani, Solution of nonlinear fractional differential equations using homotopy analysis

method, Appl. Math. Model. 34 (6) (2010) 1634-1641.[26] F. Abidi, K. Omrani, The homotopy analysis method for solving the Fornberg-Whitham

equation and comparison with Adomian’s decomposition method, Comput. Math. Appl. 59(2010) 2743-2750.

15

[27] O. Abdulaziz, I. Hashim, A. Saif, Series solutions of time-fractional PDEs by homotopy analy-

sis method, Differential Equations and Nonlinear mechanics (2008), doi:10.1155/2008/686512[28] A. Molabahrami and F. Khani, The homotopy analysis method to solve the Burgers-Huxley

equation, Nonlin. Anal. Real. World Appl. 10 (2009) 589-600.[29] G. Adomian, A review of the decomposition method in applied mathematics, J. Math. Anal.

Appl. 135 (1988) 44-501.[30] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer

Academic Publishers, Boston, 1999.[31] K. Abbaoui, Y. Cherruault, New ideas for proving convergence of decomposition methods,

Comput. Math. Appl. 29 (7) (1995) 103-108.

[32] B. Fornberg, G. B. Whitham, A numerical and theoretical study of certain nonlinear wavephenomena. Phil. Trans. R. Soc. Lond. A. v289 (1978) 373-404.

[33] I. Podlubny, Geometric and Physical Interpretation of Fractional Integration and FractionalDifferentiation, Fractional Calculus and Applied Analysis, 5 (4) (2002) 367-386.

Yuzuncu Yil University, Faculty of Sciences, Department of Mathematics, 65080,

Van, Turkey

E-mail address: giyassakar@hotmail.com(Sakar), ferdogan@yyu.edu.tr(Erdogan)