International Journal of Analysis and Applications
ISSN 2291-8639
Volume 14, Number 1 (2017), 77-87
http://www.etamaths.com

MODIFIED HOMOTOPY ANALYSIS METHOD FOR NONLINEAR

FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS

D. ZIANE∗ AND M. HAMDI CHERIF

Abstract. In this paper, a combined form of natural transform with homotopy analysis method

is proposed to solve nonlinear fractional partial differential equations. This method is called the

fractional homotopy analysis natural transform method (FHANTM). The FHANTM can easily be
applied to many problems and is capable of reducing the size of computational work. The fractional

derivative is described in the Caputo sense. The results show that the FHANTM is an appropriate

method for solving nonlinear fractional partial differentia equation.

1. introduction

The natural transform is a transform defined by an integral like all other transformations defined by
integrals, such as the Laplace transform as well, as the Sumudu transform, where we find only used in
solving of linear differential equations. This transform it was defined by Z. H. Khan and W. A. Khan [1]
in 2008 and it has been used by many researchers in the resolution of linear differential equations
( [2], [3], [4], [5]). But with the presence of some methods, such as the homotopy analysis method
(HAM) ( [6], [7], [8]) that used in the solution of linear and nonlinear differential equations. Then,
with the advent of the compositions of this method with the natural transform, lead to facilitating
the resolution of nonlinear fractional partial differential equations. The objective of this study is to
combine two powerful methods, the first method is ”homotopy analysis method”, the second is called
”the natural transform method”, the fractional derivative is described in the Caputo sense, thus, we
get the modified method ”fractional homotopy analysis natural transform method” (FHANTM), and
we apply this modified method to solve somme exemples of nonlinear fractional partial differential
equations.
The present paper has been organized as follows: In Section 2 some basic definitions and properties
of natural transform. In section 3 we will propose an analysis of the modified method. In section 4
we present three examples explaining how to apply the proposed method (FHANTM). Finally, the
conclusion follows.

2. Basic definitions

In this section, we give some basic definitions and properties of fractional calculus, natral transform
and natural transform of fractional derivatives which are used further in this paper.

2.1. Fractional calculus. There are several definitions of a fractional derivative of order α > 0
(see [9], [10], [11]). The most commonly used definitions are the Riemann–Liouville and Caputo. We
give some basic definitions and properties of the fractional calculus theory which are used further in
this paper.

Definition 2.1. Let Ω = [a,b] (−∞ < a < b < +∞) be a finite interval on the real axis R. The
Riemann–Liouville fractional integral Iα0+f of order α ∈ R (α > 0) is defined by

Received 14th January, 2017; accepted 3rd April, 2017; published 2nd May, 2017.
2010 Mathematics Subject Classification. Caputo fractional derivative; natural transform; homotopy analysis method;

nonlinear fractional equations.
Key words and phrases. Hermite-Hadamard inequality; local fractional integral; fractal space; generalized convex

function.

c©2017 Authors retain the copyrights of
their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

77



78 D. ZIANE AND M. HAMDI CHERIF

(Iα0+f)(t) =
1

Γ (α)

∫ t
0

f(τ)dτ

(t− τ)1−α
, t > 0, α > 0, (2.1)

(I00+f)(t) = f(t).

Here Γ(·) is the gamma function.

Theorem 2.1. Let α > 0 and let n = [α]+1. If f(t) ∈ ACn [a,b] , then the Caputo fractional derivative
(cDα0+f)(t) exist almost evrywhere on [a,b] .

If α /∈ N, (cDα
0+
f)(t) is represented by

(cDα0+f)(t) =
1

Γ (n−α)

∫ t
0

f(n)(τ)dτ

(t− τ)α−n+1
, (2.2)

where D = d
dx

and n = [α] + 1.

Proof. (see [10]). �

Remark 2.1. In this paper, we consider the time-fractional derivative in the Caputo’s sense. When
α ∈ R+, the time-fractional derivative is defined as

(cDαt u)(x,t) =
∂αu(x,t)

∂tα

=

{
1

Γ(m−α)
∫ t

0
(t− τ)m−α−1 ∂

mu(x,τ)
∂τm

, m− 1 < α < m,
∂mu(x,t)
∂tm

, α = m
(2.3)

where m ∈ N∗.

2.2. Definitions of the N-transform. We give some basic definitions and properties of the N-
Transform which are used further in this paper (see [1], [12], [13]). When the real function f(t) > 0
and f(t) = 0 for t < 0 is sectionwise continuous, exponential order and defined in the set

A =

{
f(t) : ∃M, k1,k2 > 0, |f(t)| < Me

|t|
kj , if t ∈ (−1)j × [0, ∞)

}
.

Definition 2.2. [12] The N-transform of the function f(t) > 0 and f(t) = 0 for t < 0 is defined by

N+ [f(t)] = R(s,u) =
∫ ∞

0

e−stf(ut)dt; s > o,u > o. (2.4)

Where s and u are the transform variables. The original function f(t) in (2.4) is called the inverse
transform or inverse of R(s,u) and it is defined by

N−1 {R(s,u)} = f(t) =
1

2πi

∫ c+i∞
c−i∞

e
st
u R(s,u)ds, (2.5)

2.2.1. N-Transform of fractional derivatives.

Proposition 2.1. If R(s,u) is the N-Transform of the function f(t), then the N-Transform of frac-
tional integral of order α is defined by

N+
[
(Iα0+f)(t)

]
=
sα

uα
R(s,u). (2.6)

Proposition 2.2. If R(s,u) is the N-Transform of the function f(t), then the N-Transform of frac-
tional derivative of order α is defined as

N+
[
(cDα0+f)(t)

]
=
sα

uα
R(s,u) −

n−1∑
k=0

sα−(k+1)

uα−k
f(k)(0). (2.7)



MODIFIED HOMOTOPY ANALYSIS METHOD FOR FDE 79

2.2.2. Somme properties of the N-transform. Here are some properties of the N-Transform:
1. If N+ {f(t)} = R(s,u) then, N+ {f(at)} = 1

a
R(s,u).

2. Generalised N-Transform For any value of n the generalised N-Transform of function f(t) > 0
is defined by

N+ {f(t)} = R(s,u) =
∞∑
n=0

n!anu
n

sn+1
. (2.8)

3. N-Transform of derivative If f(n)(t) is the nth derivative of function f(t), then its N-
Transform is given by

N+
{
f(n)(t)

}
= Rn(s,u) =

sn

un
R(s,u) −

n−1∑
k=0

sn−(k+1)

un−k
f(k)(0). (2.9)

For n = 1 and n = 2, (2.9) gives the N-Transform of first and second derivatives of f(t)

N+
{
f

′
(t)
}

= R1(s,u) =
s

u
R(s,u) −

1

u
f(0). (2.10)

N+
{
f

′′

(t)
}

= R2(s,u) =
s2

u2
R(s,u) −

s

u2
f(0) −

1

u
f

′
(0). (2.11)

4. N-Transform of integral

N+ {f(t)} = R(s,u) then N+
{∫ t

0
f(r)dr

}
= u

s
R(s,u).

5. The function f(t) in set A is multiplied with shift function tn then

N+ {tnf(t)} =
sn

un
dn

dun
unR(s,u). (2.12)

And

N+
{

tα

Γ(α + 1)

}
=

uα

sα+1
, α > 0.

3. Fractional Homotopy Analysis N-Transform Method (FHANTM)

To illustrate the basic idea of this method, we consider a general nonlinear nonhomogeneous time-
fractional partial differential equation

cDαt U(x,t) + LU(x,t) + RU(x,t) = g(x,t), (3.1)

where m = 1, 2, . . . , and the initial conditions

∂m−1U(x,t)

∂tm−1

∣∣∣∣
t=0

= fm−1(x), m = 1, 2, . . . , (3.2)

where cDαt U(x,t) is the Caputo fractional derivative of the function U(x,t), L is the linear dif-
ferential operator, R represents the general nonlinear differential operator, and g(x,t) is the source
term.

Applying the N-Transform (denoted in this paper by N+) on both sides of (3.1), we get

N+ [cDαt U(x,t)] + N
+ [LU(x,t) + RU(x,t) −g(x,t)] = 0. (3.3)

Using the property of the N-Transform, we have the following form

N+ [U(x,t)] −
uα

sα

n−1∑
k=0

sα−(k+1)

uα−k
U(k)(x, 0) +

uα

sα
N+ [LU(x,t) + RU(x,t) −g(x,t)] = 0 (3.4)

Or



80 D. ZIANE AND M. HAMDI CHERIF

N+ [U(x,t)] −
n−1∑
k=0

uk

sk+1
U(k)(x, 0) +

uα

sα
N+ [LU(x,t) + RU(x,t) −g(x,t)] = 0. (3.5)

Define the nonlinear operator

R[φ(x,t; p)] = N+ [φ(x,t; p)] −
n−1∑
k=0

uk

sk+1
φ(k)(x, 0,p) (3.6)

+
uα

sα
N+ [Lφ(x,t; p) + Rφ(x,t; p) −g(x,t; p)]

By means of homotopy analysis method [6], we construct the so-called the zero-order deformation
equation

(1 −p)N+[φ(x,t; p) −U0(x,t)] = phH(x,t)R[φ(x,t; p)], (3.7)
where p is an embedding parameter and p ∈ [0, 1], H(x; t) 6= 0 is an auxiliary function, h 6= 0 is an

auxiliary parameter, N+ is an auxiliary linear N-Transform operator. When p = 0 and p = 1, we have{
φ(x,t; 0) = U0(x,t),
φ(x,t; 1) = U(x,t).

(3.8)

When p increases from 0 to 1, the φ(x,t,p) various from U0(x,t) to U(x,t). Expanding φ(x,t; p) in
Taylor series with respect to p, we have

φ(x,t; p) = U0(x,t) +

+∞∑
m=1

Um(x,t)p
m, (3.9)

where

Um(x,t) =
1

m!

∂mφ(x,t; p)

∂pm
|p=0 . (3.10)

When p = 1, the (3.9) becomes

U(x,t) = U0(x,t) +

+∞∑
m=1

Um(x,t). (3.11)

We define the vectors

−→
U n = {U0(x,t),U1(x,t),U2(x,t), . . . ,Un(x,t)}. (3.12)

Differentiating (3.7) m−times with respect to p, then setting p = 0 and finally dividing them by
m!, we obtain the so-called mth-order deformation equation

N+[Um(x,t) −χmUm−1(x,t)] = hpH(x,t)<m(
−→
U m−1(x,t)), (3.13)

where

<m(
−→
U m−1(x,t)) =

1

(m− 1)!
∂m−1R[φ(x,t; p)]

∂pm−1
|p=0 , (3.14)

and

χm =

{
0, m 6 1,
1, m > 1.

Applying the inverse N-Transform on both sides of (3.13), we have

Um(x,t) = χmUm−1(x,t) + N−1
[
hpH(x,t)<m(

−→
U m−1(x,t))

]
. (3.15)

The mth deformation equation (3.15) is a linear which can be easily solved. So, the solution of (3.1)
can be written into the following form



MODIFIED HOMOTOPY ANALYSIS METHOD FOR FDE 81

U(x,t) =

M∑
m=0

Um(x,t), (3.16)

when M →∞, we can obtain an accurate approximation solution of (3.1).

4. Application of the FHANTM

In this section, we apply the homotopy analysis method (HAM) coupled with the N-Transform for
solving somme exemples of nonlinear time-fractional partial differential equations.

Example 4.1. Consider the following nonlinear time-fractional KdV equation

cDαt U − 3(U
2)x + Uxxx = 0, 0 < α 6 1, (4.1)

with the initial condition

U(x, 0) = 6x. (4.2)

Applying the N-Transform on both sides of (4.1), we get

N+ [U] −
1

s
U(x, 0) +

uα

sα
N+
[
−3(U2)x + Uxxx

]
= 0. (4.3)

From (4.3) and the initial condition (4.2), we have

N+ [U] −
1

s
6x−

uα

sα
N+
[
3(U2)x −Uxxx

]
= 0. (4.4)

We take the nonlinear part as

R[φ(x,t,p)] = N+ [φ] −
1

s
6x−

uα

sα
N+
[
3(φ2)x −φxxx

]
. (4.5)

We construct the so-called the zero-order deformation equation with assumption H(x; t) = 1, we
have

(1 −p)N+[φ(x,t; p) −U0(x,t)] = phR[φ(x,t; p)]. (4.6)
When p = 0 and p = 1, we can obtain{

φ(x,t; 0) = U0(x,t),
φ(x,t; 1) = U(x,t).

Therefore, we have the mth order deformation equation

N+[Um(x,t) −χmUm−1(x,t)] = h<m(
−→
U m−1(x,t)). (4.7)

Operating the inverse N-Transform operator on both sides of (4.7), we get

Um(x,t) = χmUm−1(x,t) + N−1[h<m(
−→
U m−1(x,t))]. (4.8)

From (4.8), we have

U1(x,t) = hN−1[ <1(
−→
U 0(x,t))],

U2(x,t) = U1 + hN−1[ <2(
−→
U 1(x,t))], (4.9)

U3(x,t) = U2 + hN−1[ <3(
−→
U 2(x,t))],

...

where

<1(
−→
U 0(x,t)) = N+ [U0] −

1

s
6x−

uα

sα
N+
[
3(U20 )x − (U0)xxx

]
,

<2(
−→
U 1(x,t)) = N+ [U1] −

uα

sα
N+ [3(2U0U1)x − (U1)xxx] , (4.10)



82 D. ZIANE AND M. HAMDI CHERIF

<3(
−→
U 2(x,t)) = N+ [U2] −

uα

sα
N+
[
3(2U0U2 + U

2
1 )x − (U2)xxx

]
,

...

Using the initial condition (4.2), the iteration formulas (4.9) and (4.10), we obtain

U0(x,t) = 6x,

U1(x,t) = (−h)(6x)
36

Γ(α + 1)
tα,

U2(x,t) = (−h)(1 + h)(6x)
36

Γ(α + 1)
tα + 2h2(6x)

(36)2

Γ(2α + 1)
t2α, (4.11)

U3(x,t) = (−h)(1 + h)2(6x)
36

Γ(α + 1)
tα + 4(h2 + h3)(6x)

(36)2

Γ(2α + 1)
t2α

+(−h3)(6x)(36)3
[

4

Γ(2α + 1)
+

1

Γ2(α + 1)

]
Γ(2α + 1)

Γ(3α + 1)
t3α,

...

The other components of the FHANTM can be determined in a similar way. Finally, the approximate
solution of (4.1) in a series form is

U(x,t) = 6x + (−h)(3 + 3h + h2)(6x)
36

Γ(α + 1)
tα + (6h2 + 4h3)(6x)

(36)2

Γ(2α + 1)
t2α

+(−h3)(6x)(36)3
[

4

Γ(2α + 1)
+

1

Γ2(α + 1)

]
Γ(2α + 1)

Γ(3α + 1)
t3α + · · · (4.12)

When h = −1, the approximate solution of (4.1), is given by

U(x,t) = 6x

[
1 +

36

Γ(α + 1)
tα + 2

(36)2

Γ(2α + 1)
t2α

+ (36)3
(

4

Γ(2α + 1)
+

1

Γ2(α + 1)

)
Γ(2α + 1)

Γ(3α + 1)
t3α + · · ·

]
, (4.13)

and when α = 1, we obtain

U(x,t) = 6x
[
1 + 36t + (36t)2 + (36t)3 + · · ·

]
. (4.14)

That gives

U(x,t) =
6x

1 − 36t
, |36t| < 1, (4.15)

which is an exact solution to the nonlinear KdV equation as presented in [14].

Example 4.2. Consider the following nonlinear time-fractional partial differential equation

cDαt U − 2
x2

t
UUx = 0, t > 0, 1 < α 6 2, (4.16)

with the initial conditions

U(x, 0) = 0, Ut(x, 0) = x. (4.17)

Applying the N-Transform on both sides of (4.16), we get

N+ [U] −
1

s
U(x, 0) −

u

s2
Ut(x, 0) +

uα

sα
N+
[
−2

x2

t
UUx

]
= 0. (4.18)

From (4.18) and the initial conditions (4.17), we have



MODIFIED HOMOTOPY ANALYSIS METHOD FOR FDE 83

Figure 1. Shows the exact solution and approximate solutions of (4.1) for different
values of α when x = 1. We note that the graph has changed its position according to
the values of α, if the value of α is closer to 1, we see that the graph corresponding
to this value takes the position closest to the graph of the exact solution.

N+ [U] −
u

s2
x−

uα

sα
N+
[
2
x2

t
UUx

]
= 0. (4.19)

We take the nonlinear part as

R[φ(x,t,p)] = N+ [φ] −
u

s2
x−

uα

sα
N+
[
2
x2

t
φφx

]
. (4.20)

We construct the so-called the zero-order deformation equation with assumption H(x; t) = 1, we
have

(1 −p)N+[φ(x,t; p) −U0(x,t)] = phR[φ(x,t; p)]. (4.21)
When p = 0 and p = 1, we can obtain{

φ(x,t; 0) = U0(x,t),
φ(x,t; 1) = U(x,t).

Therefore, we have the mth order deformation equation

N+[Um(x,t) −χmUm−1(x,t)] = h<m(
−→
U m−1(x,t)). (4.22)

Operating the inverse N-Transform operator on both sides of (4.22), we get

Um(x,t) = χmUm−1(x,t) + N−1[h<m(
−→
U m−1(x,t))]. (4.23)

From the formula (4.23), we have

U1(x,t) = hN−1[<1(
−→
U 0(x,t))],

U2(x,t) = U1 + hN−1[<2(
−→
U 1(x,t))], (4.24)

U3(x,t) = U2 + hN−1[ <3(
−→
U 2(x,t))],

...

where

<1(
−→
U 0(x,t)) = N+ [U0] −x

u

s2
−
uα

sα
N+
[
2
x2

t
U0U0x

]
,

<2(
−→
U 1(x,t)) = N+ [U1] −

uα

sα
N+
[
2
x2

t
(U0U1x + U1U0x)

]
, (4.25)



84 D. ZIANE AND M. HAMDI CHERIF

<3(
−→
U 2(x,t)) = N+ [U2] −

uα

sα
N+
[
2
x2

t
(U0U2x + U1U1x + U2U0x)

]
,

...

and so on. Consequently, while taking the initial conditions (4.17), and according to (4.24) and
(4.25), the first few components of the Homotopy Analysis N-Transform Method of (4.16) are derived
as follows

U0(x,t) = xt,

U1(x,t) = (−h)
2x3

Γ(α + 2)
tα+1,

U2(x,t) = (−h)(1 + h)
2x3

Γ(α + 2)
tα+1 + h2

16x5

Γ(2α + 2)
t2α+1, (4.26)

U3(x,t) = (−h)(1 + h)2
2x3

Γ(α + 2)
tα+1 + h2(1 + h)

32x5

Γ(2α + 2)
t2α+1

+(−h3)
[

32 × 6
Γ(2α + 2)

+
24

Γ2(α + 2)

]
Γ(2α + 2)

Γ(3α + 2)
x7t3α+1,

...

The other components of the (FHANTM) can be determined in a similar way. Finally, the approx-
imate solution of (4.16) in a series form is

U(x,t) = xt + (−h)
[
2 + h + (1 + h)2

] 2x3
Γ(α + 2)

tα+1 + h2(3 + 2h)
16x5

Γ(2α + 2)
t2α+1 (4.27)

+(−h3)
[

32 × 6
Γ(2α + 2)

+
24

Γ2(α + 2)

]
Γ(2α + 2)

Γ(3α + 2)
x7t3α+1 + · · ·

When h = −1, the approximate solution of (4.16) is given by

U(x,t) = xt +
2x3

Γ(α + 2)
tα+1 +

16x5

Γ(2α + 2)
t2α+1 (4.28)

+

[
32 × 6

Γ(2α + 2)
+

24

Γ2(α + 2)

]
Γ(2α + 2)

Γ(3α + 2)
x7t3α+1 + · · · .

When α = 2, the (4.28) becomes

U(x,t) = xt +
1

3
(xt)3 +

2

15
(xt)5 +

17

315
(xt)7 + · · · . (4.29)

On the other hand, we see that the development of the function U(x,t) = tan(xt) according to the
Taylor series in the vicinity of t = 0 is given by

U(x,t) = xt +
1

3
(xt)3 +

2

15
(xt)5 +

17

315
(xt)7 + 0(

t

x
)8. (4.30)

Therefore, we conclude that

U(x,t) = tan(xt), (4.31)

which is the exact solution of (4.16) in the case α = 2.



MODIFIED HOMOTOPY ANALYSIS METHOD FOR FDE 85

Figure 2. Shows the exact solution and approximate solutions of (4.16) for different
values of α when x = 1/2. We note that the graph has changed its position according
to the values of α, if the value of α is closer to 2, we see that the graph corresponding
to this value takes the position closest to the graph of the exact solution.

Example 4.3. Finaly, we consider the nonlinear time-fractional partial differential equation

cDαt U −
3

8

[
(Uxx)

2
]
x

=
3

2
t, 2 < α 6 3, (4.32)

with the initial conditions

U(x, 0) = −
1

2
x2, Ut(x, 0) =

1

3
x3, Utt(x, 0) = 0. (4.33)

Applying the N-Transform on both sides of (4.32), we get

N+ [U] −
1

s
U(x, 0) −

u

s2
Ut(x, 0) +

uα

sα
N+
[
−

3

8

[
(Uxx)

2
]
x

]
=

3

2

uα+1

sα+2
. (4.34)

From (4.34) and the initial conditions (4.33), we have

N+ [U] +
1

s

x2

2
−
u

s2
x3

3
−
uα

sα
N+
[

3

8

[
(Uxx)

2
]
x

]
=

3

2

uα+1

sα+2
. (4.35)

We take the nonlinear part as

R[φ(x,t,p)] = N+ [φ] +
1

s

x2

2
−
u

s2
x3

3
−

3

2

uα+1

sα+2
−
uα

sα
N+
[

3

8

[
(φxx)

2
]
x

]
. (4.36)

We construct the so-called the zero-order deformation equation with assumption H(x; t) = 1, we
have

(1 −p)N+[φ(x,t; p) −U0(x,t)] = phR[φ(x,t; p)]. (4.37)
When p = 0 and p = 1, we can obtain{

φ(x,t; 0) = U0(x,t),
φ(x,t; 1) = U(x,t).

Therefore, we have the mth order deformation equation

N+[Um(x,t) −χmUm−1(x,t)] = h<m(
−→
U m−1(x,t)). (4.38)

Operating the inverse N-Transform operator on both sides of (4.38), we get

Um(x,t) = χmUm−1(x,t) + N−1[h<m(
−→
U m−1(x,t))]. (4.39)

From the formula (4.23), we have



86 D. ZIANE AND M. HAMDI CHERIF

U1(x,t) = hN−1[<1(
−→
U 0(x,t))],

U2(x,t) = U1 + hN−1[<2(
−→
U 1(x,t))], (4.40)

U3(x,t) = U2 + hN−1[ <3(
−→
U 2(x,t))],

...

where

<1(
−→
U 0(x,t)) = N+ [U0] +

1

s

x2

2
−
u

s2
x3

3
−

3

2

uα+1

sα+2
−
uα

sα
N+
[

3

8

[
(U0xx)

2
]
x

]
,

<2(
−→
U 1(x,t)) = N+ [U1] −

uα

sα
N+
[

3

8
(U0U1xx + U1U0xx)x

]
, (4.41)

<3(
−→
U 2(x,t)) = N+ [U2] −

uα

sα
N+
[

3

8
(U0U2xx + U1U1xx + U2U0xx)x

]
,

...

Consequently, while taking (4.33), and according to (4.40) and (4.41), the first few components of
the Homotopy Analysis N-Transform Method of (4.32), are derived as follows

U0(x,t) = −
1

2
x2 +

1

3
x3t,

U1(x,t) = (−h)6x
tα+2

Γ(α + 3)
,

U2(x,t) = (−h)(1 + h)6x
tα+2

Γ(α + 3)
, (4.42)

U3(x,t) = (−h)(1 + h)26x
tα+2

Γ(α + 3)
,

...

Un(x,t) = (−h)(1 + h)n−16x
tα+2

Γ(α + 3)

The other components of the (FNTHAM) can be determined in a similar way. Finally, the approx-
imate solution of (4.32) in a series form is given by

U(x,t) = −
1

2
x2 +

1

3
x3t + (−h)6x

tα+2

Γ(α + 3)
+ (−h)(1 + h)6x

tα+2

Γ(α + 3)
(4.43)

+(−h)(1 + h)26x
tα+2

Γ(α + 3)
+ · · · + (−h)(1 + h)n−16x

tα+2

Γ(α + 3)
.

When h = −1, we obtain the exact solution of (4.32)

U(x,t) = −
1

2
x2 +

1

3
x3t + 6x

tα+2

Γ(α + 3)
. (4.44)

We note that from the term U2, all terms are equal to zero.
When α = 3, the (4.44) becomes

U(x,t) = −
1

2
x2 +

1

3
x3t +

1

20
xt5, (4.45)

which is the exact solution of (4.32) in the case α = 3.



MODIFIED HOMOTOPY ANALYSIS METHOD FOR FDE 87

Figure 3. Shows the exact solution and approximate solutions of (4.32) for different
values of α when x = 1/2. We note that the graph has changed its position according
to the values of α, if the value of α is closer to 3, we see that the graph corresponding
to this value takes the position closest to the graph of the exact solution.

5. Conclusion

The coupling of homotopy analysis method (HAM) and the natural transform method proved very
effective to solve nonlinear fractional partial differential equations. The proposed algorithm provides
the solution in a series form that converges rapidly to the exact solution if it exists. The advantage
of this method is its ability to combine two powerful methods for obtaining exact or approximate
solutions for nonlinear fractional partial differential equations. From the obtained results, it is clear
that the FHANTM yields very accurate approximate solutions using only a few iterates.

References

[1] Z. H. Khan and W. A. Khan, N-transform properties and applications, NUST J. Eng. Sci, 1 (2008), 127-133.

[2] R. Silambarasn and F. B. M. Belgacem, Applications of the Natural Transform to Maxwell’s Equations, Prog. Elect.
Res. Symp. Proc. Suzhou. China, (2011), 899-902.

[3] S. K. Q. Al-Omari, On the Application of Natural Transforms, Int. J. Pure Appl. Math. 85 (2013), 729-744.

[4] M. Junaid, Application Of Natural Transform To Newtonian Fluid Problems, Int. J. Sci. Technol. 5 (2016), 138-147.
[5] D. Loonker and P. K. Banerji, Solution of Fractional Ordinary Differential Equations by Natural Transform, Int. J.

Math. Eng. Sci. 2 (2013), 1-7.

[6] S.J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, Ph.D. Thesis, Shanghai
Jiao Tong University, 1992.

[7] S. J. Liao, Beyond Perturbation: Introduction to Homotopy Analysis Method, Chapman and Hall/CRC Press, Boca
Raton, 2003.

[8] S. J. Liao, On the homotopy analysis method for nonlinear problems, Appl. Math. Comput. 147 (2004), 499-513.

[9] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
[10] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations,

Elsevier, Amsterdam, 2006.

[11] K. Diethelm, The Analysis Fractional Differential Equations, Springer-Verlag Berlin Heidelberg 2010.
[12] F. B. M. Belgacem and R. Silambarasn, Theory of natural transform, Math. Eng. Sci. Aerospace, 3 (2012), 105-135.

[13] D. Loonker and P.K. Banerji, Solution of Fractional Ordinary Differential Equations by Natural Transform, Int. J.
Math. Eng. Sci. 2 (2013), 1-7.

[14] A. Wazwaz, The variational iteration method for rational solutions for KdV, K(2,2), Burgers, and cubic Boussinesq

equations, J. Comput. Appl. Math, 207 (2007), 18-23.

Laboratory of mathematics and its applications (LAMAP), University of Oran1 Ahmed Ben Bella, Oran,
31000, Algeria

∗Corresponding author: djeloulz@yahoo.com


	1. introduction
	2. Basic definitions
	2.1. Fractional calculus
	2.2. Definitions of the N-transform

	3. Fractional Homotopy Analysis N-Transform Method (FHANTM)
	4. Application of the FHANTM
	5. Conclusion
	References