L25052056

H. Saeedi, F. Samimi / International Journal of Engineering Research and Applications
(IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 5, September- October 2012, pp.052-056

He’s Homotopy Perturbation Method for Nonlinear Ferdholm
Integro-Differential Equations Of Fractional Order
H. Saeedi *, F. Samimi **
*(Department of Mathematics, Shahid Bahonar University of Kerman, Iran
** (Department of Mathematics, Islamic Azad University of Zahedan, Iran

ABSTRACT
In this paper, homotopy perturbation method is devoted to the search for better and more efficient
applied to solve non-linear Fredholm integro- solution methods for determining a solution,
differential equations of fractional order. approximate or exact, analytical or numerical, to
Examples are presented to illustrate the ability of nonlinear models. So the aim of this work is to
the method. The results reveal that the proposed present a numerical method (Homotopy-perturbation
methods very effective and simple and show that method) for approximating the solution of a
this method can be applied to the non-fractional nonlinear fractional integro-differential equation of
cases. the second kind:
1
𝐷𝛼 𝑓 𝑥 − 𝜆 𝑘 𝑥, 𝑡 𝐹 𝑓 𝑡 𝑑𝑡
Keywords - Fractional calculus, Fredholm 0
integro-differential equations, Homotopy- = 𝑔(𝑥) (1)
perturbation method Where 𝐹(𝑓 𝑡 ) = 𝑓 𝑡 𝑞 𝑞>1
With these supplementary conditions:

I. INTRODUCTION 𝑓 (𝑖) 0 = 𝛿 𝑖 , 𝑖 = 0,1, … , 𝑟 − 1,
The generalization of the concept of derivative 𝑟−1 < 𝛼 ≤ 𝑟 , 𝑟
𝐷 𝛼 𝑓(𝑥) to non-integer values of a goes to the ∈ 𝑁 (2)
beginning of the theory of differential calculus. In Where,𝑔 ∈ 𝐿2 ([0, 1)), 𝑘 ∈ 𝐿2 ([0,1)2 ) are known
fact, Leibniz, in his correspondence with Bernoulli, functions, f(x) is the unknown function, 𝐷 𝛼 is the
L’Hopital and Wallis (1695), had several notes about Caputo fractional differentiation operator and q is a
1
the calculation of 𝐷 2 𝑓(𝑥). Nevertheless, the positive integer. There are several techniques for
development of the theory of fractional derivatives solving such equations like Adomian decomposition
and integrals is due to Euler, Liouville and Abel method [7, 9], collocation method [8], CAS wavelet
(1823). However, during the last 10 years fractional method [2] and differential transform method [1].
calculus starts to attract much more attention of Most of the methods have been utilized in linear
physicists and mathematicians. It was found that problems and a few numbers of works have
various; especially interdisciplinary applications can considered nonlinear problems. In this paper,
be elegantly modeled with the help of the fractional homotopy perturbation method is applied to solve
derivatives. For example, the nonlinear oscillation of non-linear Fredholm integro-differential equations of
earthquake can be modeled with fractional fractional order.
derivatives [3], and the fluid-dynamic traffic model
with fractional derivatives caneliminate the II. FRACTIONAL CALCULUS
deficiency arising from the assumption of continuum There are several definitions of a fractional derivative
traffic flow [4]. In the fields of physics and of order 𝛼> 0. The two most commonly used
chemistry, fractional derivatives and integrals are definitions are the Riemann–Liouville and Caputo.
presently associated with the application of fractals in Definition1.2.The Riemann-Liouville fractional
the modeling of electro-chemical reactions, integral operator of order α ≥ 0 is defined as
irreversibility and electro magnetism [10], heat
conduction in materials with memory and radiation 𝐽𝛼 𝑓 𝑥
𝑥
problems. Many mathematical formulations of 1
mentioned phenomena contain nonlinear integro- = (𝑥 − 𝑡) 𝛼 −1 𝑓 𝑡 𝑑𝑡, 𝛼 > 0, 𝑥
Γ(𝛼) 0
differential equations with fractional order. Nonlinear > 0, 3
phenomena are also of fundamental importance in 𝐽0 𝑓 𝑥 = 𝑓(𝑥)
various fields of science and engineering. The
nonlinear models of real-life problems are still IIt has the following properties:
difficult to be solved either numerically or
theoretically. There has recently been much attention

52 | P a g e

Γ 𝛾+1 The solution, usually an approximation to the
𝐽𝛼 𝑥𝛾 = 𝑥 𝛼+𝛾 , solution, will be obtained by taking the limit as p
Γ 𝛼+ 𝛾+1
𝛾 > −1, (4) tends to 1,
Definition2.2.The Caputo definition of fractal 𝑈 = lim 𝑉 = 𝑉0 + 𝑉1 + 𝑉2
𝑝→1
derivative operator is given by +⋯ (9)
To illustrate the homotopy perturbation method, for
𝐷 𝛼 𝑓 𝑥 = 𝐽 𝑚−𝛼 𝐷 𝑚 𝑓 𝑥 nonlinear Ferdholm integro-differential equations of
𝑥
1 fractional order, we consider
= 𝑥
Γ(𝑚 − 𝛼) 0
− 𝑡 𝑚 −𝛼−1 𝑓 𝑚 (𝑡)𝑑𝑡, (5) 1− 𝑝 𝐷 𝛼 𝑓𝑖 𝑥
1
Where, m − 1 ≤ α ≤ m, m∈N, x >0. It has the + 𝑝 𝐷 𝛼 𝑓𝑖 𝑥 − 𝑔 𝑖 𝑥 − 𝜆 𝑘 𝑥, 𝑡 [𝑓𝑖 𝑡 ] 𝑞 𝑑𝑡
0
following two basic properties:
=0 10
Or
𝐷𝛼 𝐽𝛼 𝑓 𝑥 = 𝑓 𝑥 ,
𝐽𝛼 𝐷𝛼 𝑓 𝑥 𝐷 𝛼 𝑓𝑖 𝑥 = 𝑝 𝑔𝑖 𝑥
= 𝑓 𝑥 1
𝑚 −1
𝑘 +
𝑥𝑘 + 𝜆 𝑘 𝑥, 𝑡 [𝑓𝑖 𝑡 ] 𝑞 𝑑𝑡 (11)
− 𝑓 (0 ) , 0
𝑘!
𝑘 =0 P is an embedding parameter which changes from
𝑥>0 (6) zero to unity. Using the parameter p, we expand the
solution of the Eq. (1) in the following form:
𝑓𝑖 𝑡 = 𝑓0 + 𝑝𝑓1 + 𝑝2 𝑓2 + ⋯. (12)
III. HOMOTOPY PERTURBATION METHOD Substituting (12) into (11) and collecting the terms
FOR SOLVING EQ. (1) with the same powers of p, we obtain a series of
To illustrate the basic concept of homotopy equations of the form:
perturbation method, consider the following non- 𝑝0 : 𝐷 𝛼 𝑓0 = 0 (13)
linear functional equation: 1
𝑝1 : 𝐷𝛼 𝑓 = 𝑔 𝑥 + 𝜆
1 𝑘 𝑥, 𝑡 𝑓0 𝑡 𝑞
𝑑𝑡 (14)
0
𝐴 𝑈 = 𝑓 𝑟 (7) 1
With the following boundary conditions: 𝑝2 : 𝐷 𝛼 𝑓2 = 𝜆 𝑘 𝑥, 𝑡 𝑓 𝑡
1
𝑞
𝑑𝑡 (15)
0
𝜕𝑈 .
𝑈, = 0, 𝑟∈Γ .
𝜕𝑛
Where A is a general functional operator, B is a It is obvious that these equations can be easily solved
boundary operator, f(r) is a known analytic function, by applying the operator 𝐽 𝛼 , the inverse of the
and Γ is the boundary of the domain Ω. Generally operator𝐷 𝛼 , which is defined by (3). Hence, the
speaking the operator A can be decomposed into two components 𝑓𝑖 (i = 0, 1, 2 , ...) of the HPM solution
parts L and N, where L is a linear and N is a non- can be determined. That is, by setting p = 1. In (11)
linear operator. Eq. (7), therefore, can be rewritten as we can entirely determine the HPM series
the following: solutions, 𝑓 𝑥 = ∞ 𝑓𝑖 𝑥 .
𝑘=0

𝐿 𝑈 + 𝑁 𝑈 − 𝑓 𝑟 =0 IV. APPLICATIONS
In this section, in order to illustrate the method, we
We construct a homotopy 𝑉 𝑟, 𝑝 : Ω × 0,1 → 𝑅, solve two examples and then we will compare the
which satisfies obtained results with the exact solutions.
𝐻 𝑉, 𝑝 = 1 − 𝑝 𝐿 𝑉 − 𝐿 𝑈0 Example 4.1. Consider the following fractional
+ 𝑝 𝐴 𝑉 − 𝑓 𝑟 = 0, nonlinear integro-differential equation:
1
𝑝 ∈ 0,1 , 𝑟∈Ω (8) 32 2
𝐷𝛼 𝑓 𝑥 = 2− 𝑥 + 𝑥 2 𝑡 2 𝑓 𝑡 2 𝑑𝑡,
Or 15 0
𝐻 𝑉, 𝑝 = 𝐿 𝑉 − 𝐿 𝑈0 + 𝑝𝐿 𝑈0 0 ≤ 𝑥 < 1,
+ 𝑝 𝑁 𝑉 − 𝑓 𝑟 =0 0< 𝛼≤1 (16)
Where, 𝑢0 is an initial approximation for the solution With this supplementary condition 𝑓(0) = 1.
of Eq. (7).In this method, we use the homotopy For 𝛼 = 1, we can get the exact solution
parameter p to expand 𝑓 𝑥 = 1 + 2𝑥.
1
For 𝛼 = , According to (11) we construct the
𝑉 = 𝑉0 + 𝑝𝑉1 + 𝑝2 𝑉2 + ⋯ 4
following homotopy

53 | P a g e

1
1
𝐷 4 𝑓 𝑥 = 𝑝(𝑔 𝑥 + 𝑥2 𝑡 2 𝑓 𝑡 2
𝑑𝑡) (17)
0
Substituting (12) into (17)
1
𝑝0 ∶ 𝐷 4 𝑓0 𝑥 = 0
1
1
𝑝1 ∶ 𝐷 4 𝑓 𝑥 = 𝑔 𝑥 +
1 𝑥 2 𝑡 2 [𝑓 𝑡 ]2 𝑑𝑡
0
0
1
1
𝑝2 : 𝐷 4 𝑓 𝑥 =
2 𝑥 2 𝑡 2 [2𝑓0 𝑡 𝑓1 𝑡 ]𝑑𝑡
0
1
3 1
𝑝 ∶ 𝐷 4 𝑓3 𝑥 = 𝑥 2 𝑡 2 [2𝑓0 𝑡 𝑓2 𝑡 + 𝑓12 𝑡 ]𝑑𝑡
0
1
1
𝑝4 ∶ 𝐷 4 𝑓 𝑥 =
4 𝑥 2 𝑡 2 [2𝑓0 𝑡 𝑓3 𝑡
0
+ 2𝑓2 𝑡 𝑓 𝑡 ]𝑑𝑡
1
.
.
.
Consequently, by applying the operators 𝐽 𝛼 to the
above sets Example 4.2. Consider the following nonlinear
Fredholm integro-differential equation, of order
1
𝑓0 𝑥 = 1 𝛼=2
1
1 2
1154 1 764 9 𝐷 2 𝑓 𝑥 = 𝑔 𝑥 + 𝑥𝑡 𝑓 𝑡 𝑑𝑡
𝑓 =
1 𝑥 4− 𝑥 4 0
523 541 0≤ 𝑥<1 (18)
893 9
𝑓2 (𝑥) = 1386 𝑥 4 1 17
Where, 𝑔 𝑥 = 3 𝑥 − 12 𝑥 with these
718 9 Γ( )
2
𝑓3 𝑥 = 𝑥 4 supplementary conditions𝑓 0 = 1, with the exact
1169
317 9 solution, 𝑓 𝑥 = 1 + 𝑥
𝑓4 𝑥 = 1250 𝑥 4
According to (11) we construct the following
. homotopy:
. 1
1 2
. 𝐷 2 𝑓 𝑥 = 𝑝(𝑔 𝑥 + 𝑥𝑡 𝑓 𝑡 𝑑𝑡) (19)
Therefore the approximations to the solutions of 0
1 Substituting (12) into (18)
Example 4.1. For 𝛼 = 4 will be determined as 1
𝑝0 ∶ 𝐷 2 𝑓0 𝑥 = 0
1
∞ 1
𝑓 𝑥 = 𝑓𝑖 𝑥 = 𝑓0 + 𝑓 + 𝑓2 + 𝑓3 + 𝑓4 + ⋯
1
𝑝1 ∶ 𝐷 2 𝑓 𝑥 = 𝑔 𝑥 +
1 𝑥𝑡[𝑓0 𝑡 ]2 𝑑𝑡
𝑖=0 0
1
1154 4 1
𝑥 = 1+ 𝑝2 ∶ 𝐷 2 𝑓2 𝑥 = 𝑥𝑡[2𝑓0 𝑡 𝑓 𝑡 ]𝑑𝑡
1
523 0
1 1
The approximations to the solutions for 𝛼 = is 1
2821 2
2 𝑝3 ∶ 𝐷 2 𝑓3 𝑥 = 𝑥𝑡[2𝑓0 𝑡 𝑓2 𝑡 + 𝑓12 𝑡 ]𝑑𝑡
𝑓 𝑥 = 1+ 𝑥. 0
1250 1
3 1
Hence for 𝛼 = 4 , 𝑝4 ∶ 𝐷 2 𝑓4 𝑥 = 𝑥𝑡[2𝑓0 𝑡 𝑓3 𝑡
346 4 0
𝑓 𝑥 = 1+ 𝑥3. + 2𝑓2 𝑡 𝑓 𝑡 ]𝑑𝑡
159 1
1 1 3
Fig. 1 shows the numerical results for 𝛼 = . . . .
4 2 4
.
The comparisons show that as   1 , the .
approximate solutions tend to 𝑓(𝑥) = 1 + 2𝑥, Consequently, by applying the operators 𝐽 𝛼 to the
which is the exact solution of the equation in the case above sets
of 𝛼 = 1. 𝑓0 𝑥 = 1

1379 3
𝑓 = 𝑥−
1 𝑥 2
2000
41 3
𝑓2 (𝑥) = 𝑥 2
200

54 | P a g e

617 3 232 7
𝑓3 𝑥 = 𝑥 2 𝑓3 𝑥 = 𝑥 4
5269 921
471 3 779 7
𝑓4 𝑥 = 𝑥 2 𝑓4 𝑥 = 5000 𝑥 4
6173
. .
. .
. .
Therefore the approximations to the solutions of
Example 4.2. Will be determined as Therefore the approximations to the solutions of
∞
Example 4.3. Will be determined as
𝑓 𝑥 = 𝑓𝑖 𝑥 = 𝑓0 + 𝑓 + 𝑓2 + 𝑓3 + 𝑓4 + ⋯
1 ∞
𝑖=0 𝑓 𝑥 = 𝑓𝑖 𝑥 = 𝑓0 + 𝑓1 + 𝑓2 + 𝑓3 + 𝑓4 + ⋯
=1+ 𝑥 𝑖=0
And hence,𝑓 𝑥 = 1 + 𝑥, which are the exact = 1 + 𝑥2
solution. And hence 𝑓 𝑥 = 1 + 𝑥 2 , which are the exact
Example 4.3. Consider the following nonlinear solution.
Fredholm integro-differential equation, of order
3
𝛼=4 V. CONCLUSION
This paper presents the use of the He’s homotopy
3 8 5 15 perturbation method, for non-linear ferdholm
𝐷 4 𝑓 𝑥 = 5
𝑥 4− 𝑥 integro-differential equations of fractional order. We
5Γ 8
4 usually derive very good approximations to the
1
3
solutions. It can be concluded that the He’s homotopy
+ 𝑥𝑡 𝑓 𝑡 𝑑𝑡 perturbation method is a powerful and efficient
0
0≤ 𝑥<1 (20) technique infinding very good solutions for this kind
of equations.
With these supplementary Condition 𝑓 0 = 1, with
the exact solution, 𝑓 𝑥 = 1 + 𝑥 2
According to (11) we construct the following ACKNOWLEDGEMENTS
homotopy: The authors wish to thank the reviewers for their
3 useful Comments on this paper.
𝐷 4𝑓 𝑥
1
3
= 𝑝(𝑔 𝑥 + 𝑥𝑡 𝑓 𝑡 𝑑𝑡) (21)
0
Substituting (12) into (21) REFERENCES
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1
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𝑝1 ∶ 𝐷 4 𝑓 𝑥 = 𝑔 𝑥 +
1 𝑥𝑡[𝑓0 𝑡 ]3 𝑑𝑡
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1
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+ 3𝑓0 𝑡 𝑓 𝑡 2 ]𝑑𝑡
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𝑓0 𝑥 = 1 fractional differential equations and their
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1 𝑥 4
[5] J.H. He, Compute.Methods Appl. Mech.
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𝑓2 (𝑥) = 479 𝑥 4

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[6] J.H. He, Int. J. Nonlinear Mech. (2000),
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[7] Momani S, Noor M. Numerical methods for
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