International Journal of Analysis and Applications
ISSN 2291-8639
Volume 10, Number 1 (2016), 9-16
http://www.etamaths.com

HOMOTOPY PERTURBATION METHOD FOR SOLVING THE FRACTIONAL

FISHER’S EQUATION

MOUNTASSIR HAMDI CHERIF∗, KACEM BELGHABA AND DJELLOUL ZIANE

Abstract. In this paper, we apply the modified HPM suggested by Momani and al. [23] for solving

the time-fractional Fisher’s equation and we use the classical HPM to derive numerical solutions of the
space-fractional Fisher’s equation. We compared our solution with the exact solution. The results

show that the HPM modified is an appropriate method for solving nonlinear fractional derivative

equations.

1. Introduction

Fractional analysis is a branch of mathematics, was the first debut in 1695 with the question posed
by Leibnitz as follows: what could be the derivative of order (half) of a function x?

From that date to today, the evolution of this branch of mathematics where he became a major
development has many uses, among them, for example:

Fractional derivatives have been widely used in the mathematical model of the visco-elasticity of
the material [1].

The electromagnetic problems can be described using the fractional integro-differential equations
[2].

In biology, it was deduced that the membranes of biological organism cells have the electrical
conductance of fractional order [3], and then is classified into groups of non-integer order models.

In economics, some finance systems can display a dynamic fractional order [4].
In addition to the above, we find that the development of this branch has also led to the emergence

of linear and nonlinear differential equations of fractional order, which became requires researchers to
use conventional methods to solve them. Among these methods there is the homotopy perturbation
method (HPM). This method was established in 1998 by He ([5]-[9]) and applied by many researchers
to solve various linear and nonlinear problems (see [10]- [16]). The method is a powerful and efficient
technique to find the solutions of nonlinear equations. The coupling of the perturbation and homotopy
method is called the homotopy perturbation method. This method can take the advantages of the
conventional perturbation method while eliminating its restrictions [16].

Our concern in this work is to consider the numerical solution of the nonlinear Fisher’s equation
with time and space fractional derivatives of the form

(1) cDαt u =
cDβxu + γu(1 −u), 0 < α 6 1, 1 < β 6 2,

where cDαt u =
∂α

∂tα
, cD

β
t u =

∂β

∂tβ
.

In the case α = 1 and β = 2, this equation become

(2) ut = uxx + γu(1 −u),
which is a Fisher’s partial differential equation.
We will extend the application of the modified HPM in order to derive analytical approximate

solutions to nonlinear time-fractional Fisher’s equation and we use the classical HPM to resolve the

2010 Mathematics Subject Classification. 35R11, 26A33, 47J35.
Key words and phrases. Caputo fractional derivative; homotopy perturbation method; Fisher’s equation; fractional

partial diferential equation.

c©2016 Authors retain the copyrights of
their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

9



10 CHERIF, BELGHABA AND ZIANE

nonlinear space-fractional Fisher’s equation. Precisely, we use the modified homotopy perturbation
method described in [23] for handling an iterative formula easy-to-use for computation. Observing the
numerical results, and comparing with the exact solution, the proposed method reveals to be very close
to the exact solution and consequently, an efficient way to solve the nonlinear time-fractional Fisher’s
equation. This is the raison why we try to use it in this work.

2. Basic definitions

We give some basic definitions and properties of the fractional calculus and the Laplace transform
theory which are used further in this paper. (see [18]-[20]).

Definition 1. Let Ω = [a,b] (−∞ < a < b < +∞) be a finite interval on the real axis R. The
Riemann–Liouville fractional integrals Iαa+f of order α ∈ C (Re(α) > 0) is defined by

(3) (Iαa+f)(t) =
1

Γ (α)

∫ t
a

f(τ)dτ

(t− τ)1−α
, t > 0,Re(α) > 0,

here Γ(α) is the gamma function.

Theorem 2. Let Re(α) > 0 and let n = [Re(α)] + 1. If f(t) ∈ ACn [a,b] , then the Caputo fractional
derivatives (cDαt f)(t) exist almost evrywhere on [a,b] . If α /∈ N, (cDαt f)(t) is represented by:

(4) (cDαt f)(t) =
1

Γ (n−α)

∫ t
a

f(n)(τ)dτ

(t− τ)α−n+1
,

and if α ∈ N, we obtain (cDαt f)(t) = f(n)(t).

Definition 3. Let u ∈ Cn−1, n ∈ N∗. Then the (left sided) Caputo fractional derivative of u is defined
(for t > 0) as

(5) cDαt u(x,t) =
∂αu(x,t)

∂tα
=

{
1

Γ(n−α)
∫ t

0
(t− τ)n−α−1 ∂

nu(x,t)
∂tn

dτ,n− 1 < α < n,n ∈ N∗
∂nu(x,t)
∂tn

,α = n ∈ N.

According to (5), we can obtain:

cDαK = 0, K is a constant, and cDαtβ =

{
Γ(β+1)

Γ(β−α+1)t
β−α,β > α− 1

0,β ≤ α− 1.

3. The Homotopy Perturbation Method
To illustrate the basic ideas of this method, we consider the following nonlinear differential equation

(6) A(u) −f(r) = 0, r ∈ Ω,
with the following boundary conditions

(7) B(u,
∂u

∂n
) = 0, r ∈ Γ,

where A is a general differential operator, f(r) is a known analytic function, B is a boundary operator,
u is the unknown function, and Γ is the boundary of the domain Ω. The operator A can be generally
divided into two operators, L and N, where L is a linear, and N a nonlinear operator. Therefore,
equation (6) can be written as follows:

(8) L(u) + N(u) −f(r) = 0.
Using the homotopy technique, we construct a homotopy v(r,p) : Ω × [0, 1] −→ R, which satisfies

(9) H(v,p) = (1 −p)[L(υ) −L(u0)] + p[A(υ) −f(r)] = 0,
or

(10) H(v,p) = L(v) −L(u0) + pL(u0) + p[N(v) −f(r)] = 0,



FRACTIONAL FISHER’S EQUATION 11

where p ∈ [0, 1] is an embedding parameter, and u0 is the initial approximation of equation (6) which
satisfies the boundary conditions. Clearly, from Eq. (9) and (10) we will have

H(v, 0) = L(v) −L(u0) = 0,(11)
H(v, 1) = A(v) −f(r) = 0.(12)

The changing process of p from zero to unity is just that of v(r,p) changing from u0(r) to u(r).
In topology, this is called deformation and L(v) − L(u0) and A(v) − f(r) are called homotopic. If
the embedding parameter p, (0 6 p 6 1) is considered as a “small parameter”, applying the classical
perturbation technique, we can assume that the solution of equation (9) or (10) can be given as a
power series in p

(13) v = v0 + pv1 + p
2v2 + · · · .

Setting p = 1, results in the approximate solution of Eq. (6)

(14) u = lim
p→1

v = v0 + v1 + v2 + · · · .

The convergence of the series (14) has been proved in ([21], [22]).

3.1. New modification of the HPM . Momani and al. [23] introduce an algorithm to handle in a
realistic and efficient way the nonlinear PDEs of fractional order. They consider the nonlinear partial
differential equations with time fractional derivative of the form

(15)

{
cDαt u(x,t) = f(u,ux,uxx) = L(u,ux,uxx) + N(u,ux,uxx) + h(x,t), t > 0

uk(x, 0) = gk(x), k = 0, 1, 2, ....m− 1,
where L is a linear operator, N is a nonlinear operator which also might include other fractional
derivatives of order less than α. The function h is considered to be a known analytic function and
cDα, m− 1 < α 6 m, is the Caputo fractional derivative of order α.

In view of the homotopy technique, we can construct the following homotopy

(16)
∂um

∂tm
−L(u,ux,uxx) −h(x,t) = p[

∂um

∂tm
+ N(u,ux,uxx) −c Dαt u],

or

(17)
∂um

∂tm
−h(x,t) = p[

∂um

∂tm
+ L(u,ux,uxx) + N(u,ux,uxx) −c Dαt u],

where p ∈ [0, 1]. The homotopy parameter p always changes from zero to unity. In case p = 0, Eq.
(16) becomes the linearized equation

(18)
∂um

∂tm
= L(u,ux,uxx) + h(x,t),

or in the second form, Eq. (17) becomes the linearized equation

(19)
∂um

∂tm
= h(x,t).

When it is one, Eq. (16) or Eq. (17) turns out to be the original fractional differential equation
(15). The basic assumption is that the solution of Eq. (16) or Eq. (17) can be written as a power
series in p

(20) u = u0 + pu1 + p
2u2 + p

3u3 · · · .
Finally, we approximate the solution by



12 CHERIF, BELGHABA AND ZIANE

(21) u(x,t) =

∞∑
n=0

un(x,t).

4. Numerical applications

In this section, we apply the modified homotopy perturbation method for solving Fisher’s equation
with time-fractional derivative and we use the classical HPM to obtain analytical solution for Fisher’s
equation with space-fractional derivative.

4.1. Numerical solutions of time-fractional Fisher’s equation. If β = 2, we obtain the following
form of the time-fractional Fisher’s equation

(22) cDαt u = uxx + γu(1 −u), 0 < α 6 1,
with the initial condition u(x, 0) = f(x).
Application of the New modification of the HPM

In view of Eq. (17), the homotopy of Eq. (22) can be constructed as

(23)
∂u

∂t
= p[

∂u

∂t
+ uxx + γu−γu2 − cDαt u].

Substituting (20) into (23) and equating the terms of the same power p, one obtains the following
set of linear partial differential equations

(24)

p0 : ∂u0
∂t

= 0,

p1 : ∂u1
∂t

= ∂u0
∂t

+ u0xx + γu0 −γu20 − cDαt u0,
p2 : ∂u2

∂t
= ∂u1

∂t
+ u1xx + γu1 −γ(2u0u1) − cDαt u1,

p3 : ∂u3
∂t

= ∂u2
∂t

+ u2xx + γu2 −γ(2u0u2 + u21) − cDαt u2,
...

with the following conditions

(25) u0(x, 0) = f(x),

(26) ui(x, 0) = 0 for i = 1, 2, ....

Case 1: Consider the following form of the time-fractional equation (for γ = 1)

(27) cDαt u = uxx + u(1 −u), 0 < α 6 1,
with the initial condition

(28) u(x, 0) = λ.

Using the initial condition (28) and solving the above Eqs. (24) yields

u0 (x,t) = λ,
u1 (x,t) = λ(1 −λ)t,
u2 (x,t) = λ(1 −λ)t + λ(1 −λ)(1 − 2λ)t

2

2!
−λ(1 −λ) t

2−α

Γ(3−α),

u3 (x,t) = λ(1 −λ)t + 2λ(1 −λ)(1 − 2λ)t
2

2!
+ λ(1 −λ)(1 − 6λ + 6λ2)t

3

3!
− 2λ(1 −λ) t

2−α

Γ(3−α)

+λ(1 −λ) t
3−2α

Γ(4−2α) − 2λ(1 −λ)(1 − 2λ)
t3−α

Γ(4−α),
...

and so on. The first four terms of the decomposition series solution for Eq. (27) is given as



FRACTIONAL FISHER’S EQUATION 13

u(x,t) = λ + 3λ(1 −λ)t + 3λ(1 −λ)(1 − 2λ)
t2

2!
+ λ(1 −λ)(1 − 6λ + 6λ2)

t3

3!
(29)

−3λ(1 −λ)
t2−α

Γ(3 −α)
+ λ(1 −λ)

t3−2α

Γ(4 − 2α)
− 2λ(1 −λ)(1 − 2λ)

t3−α

Γ(4 −α)
.

Substituting α = 1 into (30), we get the same approximate solution of nonlinear partial diferential
Fisher’s equation obtained in[24] as

u(x,t) = λ + λ(1 −λ)t + λ(1 −λ)(1 − 2λ)
t2

2!
+ λ(1 −λ)(1 − 6λ + 6λ2)

t3

3!
+ · · ·(30)

=
λet

1 −λ + λet
.

Case 2 Next we consider the following form of the time-fractional Fisher’s equation (for γ = 6)

(31) cDαt u = uxx + 6u(1 −u), 0 < α 6 1,
subject to the initial condition

(32) u(x, 0) =
1

(1 + ex)2
.

The use of the initial condition (32) and solving the above Eq (24), we obtain

u0 (x,t) =
1

(1+ex)2
,

u1 (x,t) =
10ex

(1+ex)3
t,

u2 (x,t) =
10ex

(1+ex)3
t +

50ex(2ex−1)
(1+ex)4

t2

2!
− 10e

x

(1+ex)3
t2−α

Γ(3−α),

u3 (x,t) =
10ex

(1+ex)3
t +

50ex(2ex−1)
(1+ex)4

t2

2!
+

250ex(4e2x−7ex+1)
(1+ex)4

t3

3!

− 20e
x

(1+ex)3
t2−α

Γ(3−α) +
10ex

(1+ex)3
t3−2α

Γ(4−2α) −
50ex(2ex−1)

(1+ex)4
t3−α

Γ(4−α),
...

and so on. The first three terms of the decomposition series solution for Eq. (31) is given as

u(x,t) =
1

(1 + ex)2
+

30ex

(1 + ex)3
t +

100ex(2ex − 1)
(1 + ex)4

t2

2!
+

250ex(4e2x − 7ex + 1)
(1 + ex)4

t3

3!
(33)

−
30ex

(1 + ex)3
t2−α

Γ(3 −α)
+

10ex

(1 + ex)3
t3−2α

Γ(4 − 2α)
−

50ex(2ex − 1)
(1 + ex)4

t3−α

Γ(4 −α)
.

Substituting α = 1 into (34), we obtain:

u(x,t) =
1

(1 + ex)2
+

10ex

(1 + ex)3
t +

50ex(2ex − 1)
(1 + ex)4

t2

2!
+

250ex(4e2x − 7ex + 1)
(1 + ex)4

t3

3!
+ · · ·(34)

=
1

(1 + ex−5t)2

the same solution as presented in [24].

4.2. Numerical solutions of space-fractional Fisher’s equation. Now we consider the following
form of the space-fractional Fisher’s equation with initial condition

(35) ut =
c D

β
t uxx + γu(1 −u), 1 < β 6 2,

(36) u(x, 0) = x2.



14 CHERIF, BELGHABA AND ZIANE

solution.png

Figure 1. (Left): Exact solution for Eq. (31)-(32); (Right): Approximative solution
of Eq. (31)-(32) with four terms for α = 1.

Figure 2. (Left): Approximative solution of Eq. (31)-(32) for α = 0.5; (Right):
Approximative solution of Eq. (31)-(32) for α = 0.9.

The initial condition has been taken as the above polynomial to avoid heavy calculation of fractional
differentiation.

Application of the HPM
According to the HPM, we construct the following homotopy

(37) ut −v0t + p
[
−cDβx −u(1 −u) + v0t

]
, 1 < β 6 2,

where p ∈ [0; 1], v0 = u(x; 0) = x2 and γ = 1.
In view of the HPM, substituting Eq. (20) into Eq. (37) and equating the coefficients of like powers

of p, we get the following set of differential equations

(38)

p0 : ∂u0
∂t

= v0t,

p1 : ∂u1
∂t

=c Dβxu0 + u0 −u20 −v0t,
p2 : ∂u2

∂t
=c Dβxu1 + u1 − 2u0u1,

p3 : ∂u3
∂t

=c Dβxu2 + u2 − (2u0u2 + u21),
...pn : ∂un

∂t
=c Dβxun−1 + un−1 −

(∑n−1
i=0 uiun−i−1

)
, n > 1,

with the following conditions

(39) u0(x, 0) = x
2,ui(x, 0) = 0 for i = 1, 2, ....

Using the initial conditions (39) and solving the above Eqs. (38) yields



FRACTIONAL FISHER’S EQUATION 15

(40)
u0 (x,t) = x

2,
u1 (x,t) = (a1x

2−β + x2 −x4)t,
u2 (x,t) = (a2x

2−2β + a3x
2−β + a4x

4−β + x2 − 3x4 + 2x6)t
2

2!
,

u3 (x,t) = (a5x
2−3β + a6x

2−2β + a7x
2−β + a8x

4−2β + a9x
4−β + a10x

6−β + x2 − 6x4 + 10x6 − 5x8)t
3

3!
,

...

where
a1 =

2
Γ(3−β), a2 =

2
Γ(3−2β), a3 =

4
Γ(3−β), a4 = −

24
Γ(5−β) −

4
Γ(3−β), a5 =

2
Γ(3−3β), a6 =

6
Γ(3−2β),

a7 =
6

Γ(3−β), a8 = −
24

Γ(5−2β) −
4Γ(5−β)

Γ(3−β)Γ(5−2β) −
4

Γ(3−2β) −
4

Γ2(3−β), a9 = −
96

Γ(5−β) −
16

Γ(3−β),

a10 = 2
6!

Γ(7−β) +
48

Γ(5−β) +
12

Γ(3−β).

Here, setting p = 1, we have the following solution for three iterations

u(x,t) = x2 + (a1x
2−β + x2 −x4)t(41)

+ (a2x
2−2β + a3x

2−β + a4x
4−β + x2 − 3x4 + 2x6)

t2

2!

+ (a5x
2−3β + a6x

2−2β + a7x
2−β + a8x

4−2β + a9x
4−β

+ a10x
6−β + x2 − 6x4 + 10x6 − 5x8)

t3

3!
.

Case 1: substituting β = 2 into (42), we obtain

u(x,t) = x2 + (2 + x2 −x4)t + (4 − 15x2 − 3x4 + 2x6)
t2

2!
(42)

+ (−30 − 63x2 − 90x4 + 10x6 − 5x8)
t3

3!
.

Case 2: substituting β = 3
2

into (42), we get

u(x,t) = x2 + (
4
√
π
x

1
2 + x2 −x4)t + (

8
√
π
x

1
2 + x2 −

104

5
√
π
x

5
2 − 3x4 + 2x6)

t2

2!
(43)

+ (
12
√
π
x

1
2 −

39π + 16

π
x + x2 −

416

5
√
π
x

5
2 − 6x4 +

7328

105
√
π
x

9
2 + 10x6 − 5x8)

t3

3!
.

In the same manner, we can obtain the approximate solution of higher order of Eq. (35) by using
the iteration formulas (38) and Maple.

Figure 3. (Left): Approximative solution of Eq. (35)-(36) with β = 2; (Right):
Approximative solution of Eq. (35)-(36) with β = 3/2.



16 CHERIF, BELGHABA AND ZIANE

5. Conclusion

In this work, the modified homotopy perturbation method was successfully used for solving Fisher’s
equation with time-fractional derivative, and the classical HPM has been used for solving Fisher’s
equation with space-fractional derivative. The final results obtained from modified HPM and com-
pared with the exact solution shown that there is a similarity between the exact and the approximate
solutions. Calculations show that the exact solution can be obtained from the third term. That’s why
we say that modified HPM is an alternative analytical method for solving the nonlinear time-fractional
equations.

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Laboratory of mathematics and its applications (LAMAP), University of Oran1, P.O. Box 1524, Oran,
31000, Algeria

∗Corresponding author: mountassir27@yahoo.fr