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International Journal of Computer Applications (0975 – 8887)
Volume 121 – No.24, July 2015
9
System of Linear Fractional Integro-Differential
Equations by using Adomian Decomposition Method
M. H.Saleh
Mathematics Department
Faculty of Science Zagazig University
Zagazig, Egypt
D.Sh.Mohamed Mathematics Department
Faculty of Science Zagazig University
Zagazig, Egypt
M.H.Ahmed Mathematics Department
Faculty of Science Zagazig University
Zagazig, Egypt
M.K. Marjan Mathematics Department
Faculty of Science Zagazig University
Zagazig, Egypt
ABSTRACT
In this paper, Adomian decomposition method is applied to
solve system linear fractional integro-differential equations.
The fractional derivative is considered in the Caputo sense.
Special attentions are given to study the convergence of the
proposed method. Finally, some numerical examples are
provided to show that this method is computationally
efficient.
Keywords
System linear fractional integro-differential equations,
Adomian decomposition method, Caputo fractional
derivative, Riemann-Liouville
1. INTRODUCTION The Adomian decomposition method was first proposed by
Adomian and used to solve a wide class of linear and integral
differential equations. The method is very powerful in finding
the solutions for various physical problems. Some important
problems in science and engineering can usually be reduced to
a system of integral and fractional integro-differential
equations. Integro-differential equations has attracted much
attention and solving this equation has been one of the
interesting tasks for mathematicians. In this paper we try to
introduce a solution of system of linear fractional integro-
differential equations in the following form:
with initial conditions:
Adomian decomposition offers certain advantages over
routine numerical methods. Numerical methods use
discretization which gives rise to rounding off errors causing
loss of accuracy, and requires large computer power and time.
Adomian decomposition method is better since it does not
involve discretization of the variables hence is free from
rounding off errors and does not require large computer
memory or time. There are only a few techniques for the
solution of fractional integro-differential equations, since it is
relatively a new subject in mathematics. These methods are:
Adomian decomposition method ([2], [6], [7], [9], [10]),
Iterative Decomposition Method [8], the collocation method
[3] and fractional differential trams form method ([1],[11]). In
this study presented, fractional differentiations and integration
are understood in Remann - Liouville sense.
The paper has been organized as follows. Section 2 gives
notations and basic definitions. Section 3 consists of main
results of the paper, in which Adomian decomposition of the
system of fractional integro-differential equations has been
developed. Some illustrative examples are given in Section 4
followed by the discussion and conclusions presented in
Section 5.
2. BASIC DEFINITIONS In this section we introduce some basic definitions and
properties of fractional calculus.
Definition 2.1 A real function , 0,f x x is said to
be in the space , ,C R if there exists a real number
,p such that 1 ,pf x x f x where
1 0, .f Cx
International Journal of Computer Applications (0975 – 8887)
Volume 121 – No.24, July 2015
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Definition 2.2 A real function , 0,f x x is said to
be in the space , ,kC k N if .kf C
Definition 2.3 I denotes the fractional integral operator of
order α in the sense of Riemann-Liouville, defined by
0 1
1, 0,
Γ
, 0,
x t
x t
fdt
I
x
f
f
x
(2.1)
Definition 2.4 Let 1, m .mf C N Then the
Caputo fractional derivative of ,f x defined by
0 1
1, 0 1 ,
Γ
, ,
xm
m
m
m
fdt m m
mD f
d fm
t
x t
x
d
x
Nx
(2.2)
Or
, 1, 2, .m
m
m
dD f x I f x m
dx
(2.3)
Hence, we have the following properties:
1
0
.
Γ 1(2)
Γ 1
(3) .
(4) 0 , 0 1 .
1)
!
(
.
kmk
k
I I f x I f x I I f x
I x x
D I f x f x
xI D f x f x f m m N
k
(2.4)
3. ADOMIAN DECOMPOSITION
METHOD FOR SOLVING THE SYSTEM Consider equations (1.1) with initial conditions (1.2) where
is the operator defined as (2.4). Operating with on
both sides of the equation (1.1) as follows:
In which
by substituting from (3.2) into (3.1), we get
International Journal of Computer Applications (0975 – 8887)
Volume 121 – No.24, July 2015
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Adomian decomposition method decomposed the solution of
ui(x) as the following:
by substituting from (3.4) into (3.3), we get
We set
And
And also we take
or generally we have recursive relations as follows:
International Journal of Computer Applications (0975 – 8887)
Volume 121 – No.24, July 2015
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4. NUMERICAL RESULTS In this section, we have applied Adomian decomposition
method for solving system of linear fractional Integro-
differential equations with known exact solution. All the
results are calculated by using the symbolic computation
software Maple 16.
Example 4.1
Consider the following system of fractional integro-
differential equation:
The adomian decomposition method suggests that by applying
the inverse operator = which is the inverse of to
both sides of (4.1), (4.2) and by using equation (2.4) we have
By assuming
And with starting of the initial approximation
International Journal of Computer Applications (0975 – 8887)
Volume 121 – No.24, July 2015
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Table 1, figure 1 and figure 2 show the comparison between
approximate and exact for Adomian decomposition method.
International Journal of Computer Applications (0975 – 8887)
Volume 121 – No.24, July 2015
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Example 4.2
Consider the following system of fractional integro-
differential equation:
The adomian decomposition method suggests that by applying
the inverse operator = which is the inverse of to
both sides of (4.7), (4.8) and by using equation (2.4) we have
Substituting from equation (3.4) into equations (4.9) and
(4.10) we get:
By assuming
and with starting of the initial approximation
International Journal of Computer Applications (0975 – 8887)
Volume 121 – No.24, July 2015
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Table 2, figure 3 and figure 4 show the comparison between
approximate and exact for adomain decomposition method.
International Journal of Computer Applications (0975 – 8887)
Volume 121 – No.24, July 2015
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Example 4.3
Consider the following system of fractional integro-
differential equation:
The adomian decomposition method suggests that by applying
the inverse operator = which is the inverse of to
both sides of (4.13), (4.14) and by using equation (2.4) we
have
International Journal of Computer Applications (0975 – 8887)
Volume 121 – No.24, July 2015
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Substituting from equation (4.2) into equations (4.15) and
(4.16) we get:
By assuming
and with starting of the initial approximation
International Journal of Computer Applications (0975 – 8887)
Volume 121 – No.24, July 2015
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Table 3, figure 5 and figure 6 show the comparison between
approximate and exact for Adomian decomposition method.
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Volume 121 – No.24, July 2015
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5. CONCLUSION In this article, Adomian decomposition method has been
successfully applied to find the solution of a system of linear
Fredholm fractional integro-differential equations are
presented in Table 1, 2 and 3, for differential result of x to
show the stability of the method. The approximate solution
obtained by Adomian decomposition method is compared
with exact solution.
6. REFERENCES [1] A. Arikoglu, and I. Ozkol , Solution of fractional
differential equations by using differential transform
method, Chaos Soliton Fractals. 34(2007), 1473-1481.
[2] A. M. Wazwaz, The combined Laplace transform-
Adomian decomposition method for handling nonlinear
Volterra integro- differential equations, Appl. Math.
Comput. 216, 2010, pp. 1304-1309.
[3] E. A. Rawashdeh, Numerical of fractional integro-
differential equations by collocation method, Appl. Math.
Comput. 176 (2006) 1-6.
[4] I. Podlubny, Fractional Differential Equations, Academic
press, New York, 1999.
[5] I. Podlubny, 1999. Fractional differential equations: an
introduction to fractional derivatives, fractional
differential equations, to methods of their solution and
some of their applications. New York: Academic press.
[6] Hashim I (2006). Adomian decomposition method for
solving BVPs for fourth-order integro-differential
equations. Journal of Computational and Applied
Mathematics 193, pp. 658^a_A _S664.
[7] R.C. Mittal, R. Nigam, Solution of fractional integro-
differential equations by Adomian decomposition
method, Int. J. of Appl. Math. and Mech, 4(2) (2008) 87-
94.
[8] O. A, Taiwo, and Odetunde, O. S., Approximation of
Multi-Order Fractional Differential Equations by an
Iterative Decomposition Method, Amer. J. Engr. Sci.
Tech. Res., 1 (2):1-9, (2013).
[9] Momani, S. and A. Qaralleh, 2006. An efficient method
for solving systems of fractional integro-differential
equations, Applied Mathematics and Computation, 52:
459-470.
[10] Momani, S. and M.A. Noor, 2006. Numerical methods
for fourth order fractional integro-differential equations,
Applied Mathematics and Computation, 182: 754-760.
[11] Rawashdeh, E.A., 2006. Numerical solution of fractional
integro-differential equations by collocation method,
Applied Mathematics and Computation, 176: 1-6.
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