System of Linear Fractional Integro-Differential Equations ... · System linear fractional integro -differential equations, Adomian decomposition method, Caputo fractional derivative,

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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


Zagazig, Egypt

M.H.Ahmed Mathematics Department


Zagazig, Egypt

M.K. Marjan Mathematics Department


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



10

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



11

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:



12

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



13

Table 1, figure 1 and figure 2 show the comparison between

approximate and exact for Adomian decomposition method.



14

Example 4.2





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



15


approximate and exact for adomain decomposition method.



16

Example 4.3





both sides of (4.13), (4.14) and by using equation (2.4) we

have



17

Substituting from equation (4.2) into equations (4.15) and

(4.16) we get:

By assuming

and with starting of the initial approximation



18


approximate and exact for Adomian decomposition method.



19

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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System of Linear Fractional Integro-Differential Equations ... · System linear fractional integro -differential equations, Adomian decomposition method, Caputo fractional derivative,