Computational and Applied Mathematics Journal 2015; 1(3): 67-71

Published online April 20, 2015 (http://www.aascit.org/journal/camj)

Keywords Fuzzy Number,

Parametric Form of a Fuzzy

Number,

Fuzzy Integral Equations,

Homotopy Analysis Method,

Approximate Solution,

Simple Algorithm

Received: March 2, 2015

Revised: March 16, 2015

Accepted: March 17, 2015

Solving a System of Fuzzy Integral Equations by an Analytic Method

Maryam Mosleh, Mahmood Otadi*

Department of Mathematics, Firoozkooh Branch, Islamic Azad University, Firoozkooh, Iran

Email address Maryammosleh79@yahoo.com (M. Mosleh), mahmoodotadi@yahoo.com (M. Otadi),

otadi@iaufb.ac.ir (M. Otadi)

Citation Maryam Mosleh, Mahmood Otadi. Solving a System of Fuzzy Integral Equations by an Analytic

Method. Computational and Applied Mathematics Journal. Vol. 1, No. 3, 2015, pp. 67-71.

Abstract In this paper, we use new parametic formof fuzzy numbers and convert a system of fuzzy

integral equations to two system of integral equations in crisp. Then we solve a system of

fuzzy integral equations by means of an analytic technique, namely the homotopy analysis

method (HAM). Using the HAM, it is possible to find the exact solution or an approximate

solution of the problem. The results reveal that the proposed method is very effective and

simple. Numerical examples are presented to illustrate the proposed model.

1. Introduction

The fuzzy mapping function was introduced by Chang and Zadeh [6]. Later, Dubois and

Prade [7] presented an elementary fuzzy calculus based on the extension principle also the

concept of integration of fuzzy functions was first introduced by Dubois and Prade [7].

The topics of fuzzy integral equations (FIE) and fuzzy differential equations which

growing interest for some time, in particular in relation to fuzzy control, have been rapidly

developed in recent years. A few of these equations can be solved explicitly, it is often

necessary to resort to numerical techniques which are appropriate combinations of

numerical integration and interpolation [5, 22, 21]. There are several numerical methods

for solving linear Fredholm fuzzy integral equations of the second kind [3, 4].

In 1992, Liao [8] employed the basic ideas of the homotopy in topology to propose a

general analytic method for nonlinear problems, namely homotopy analysis method

(HAM), [9, 10]. Abbasbandy [1, 2] applied homotopy perturbation method (HPM), which

is a special case of HAM, to solve Riccati differential equation. Also he used HAM for

solving quadratic Riccati differential equation [24] and mixed Volterra-Fredholm integral

equations [14]. After this, the HAM has been applied to obtain solution to a linear

Fredholm fuzzy integral equation of the second [20]. The purpose of this paper is to extend

the application of the HAM for solving a nonlinear system of fuzzy integral equations. We

shall apply HAM to find the approximate analytical solutions of a system of fuzzy

Fredholm integral equations of the second kind.

2. Preliminaries

In this section the basic notations used in fuzzy calculus are introduced. We start by

defining the fuzzy number.

Definition 1 ([18]). A fuzzy number is a fuzzy set [0,1]=: 1 Iu →R such that

i. u is upper semi-continuous;

ii. 0=)(xu outside some interval ],[ da ;

iii. There are real numbers b and c , ,dcba ≤≤≤ for which

68 Maryam Mosleh and Mahmood Otadi: Solving a System of Fuzzy Integral Equations by an Analytic Method

1. )(xu is monotonically increasing on ],[ ba ,

2. )(xu is monotonically decreasing on ],[ dc ,

3. cxbxu ≤≤1,=)( .

The set of all the fuzzy numbers (as given in Definition 1) is

denoted by E .

An alternative definition which yields the same E is given

by Kaleva [17, 19].

Definition 2.A fuzzy number u is a pair ),( uu of

functions )(ru and )(ru , 10 ≤≤ r , which satisfy the

following requirements:

i. )(ru is a bounded monotonically increasing, left

continuous function on (0,1] and right continuous at 0 ;

ii. )(ru is a bounded monotonically decreasing, left

continuous function on (0,1] and right continuous at 0 ;

iii. 1),0()( ≤≤≤ rruru .

A crisp number α is simply represented by

( ) = ( ) = ,0 1.u r u r rα ≤ ≤ This fuzzy number space as shown

in [25], can be embedded into the Banach space

= [0,1] [0,1]B C C× .

For arbitrary ))(),((=)),(),((= rvrvvruruu and R∈k

we define addition and multiplication by k as

( )( ) = ( ( ) ( )),

( )( ) = ( ( ) ( )),

( ) = ( ), ( ) = ( ), 0,

( ) = ( ), ( ) = ( ), < 0.

u v r u r v r

u v r u r v r

ku r ku r ku r ku r ifk

ku r ku r ku r ku r if k

+ +

+ +

≥

Definition 3 ([15]). For arbitrary fuzzy numbers ,,vu we

use the distance

0 1( , ) = {| ( ) ( ) |,| ( ) ( ) |}rD u v sup max u r v r u r v r≤ ≤ − −

and it is shown that ),( DE is a complete metric space [23].

Definition 4 ([13, 15]). Let 1],[: Ebaf → , for each

partition },,,{= 10 ntttP … of ],[ ba and for arbitrary

nitt iii ≤≤∈ − ],1,[ 1ξ suppose

1

=1

1

= ( )( ),

:= {| |, = 1,2, , }.

n

p i i i

i

i i

R f t t

max t t i n

ξ

∆

−

−

−

−

∑

…

The definite integral of )(tf over ],[ ba is

0( ) =b

pa

f t dt lim R∆→∫

provided that this limit exists in the metric D .

If the fuzzy function )(tf is continuous in the metric D ,

its definite integral exists [15] and also,

( ( ; ) ) = ( ; ) ,

( ( ; ) ) = ( ; ) .

b b

a a

b b

a a

f t r dt f t r dt

f t r dt f t r dt

∫ ∫

∫ ∫

3. System of Fuzzy Integral Equations

In this section, we consider the system of nonlinear

Fredholm integral equation type is given by [16]

( ) = ( ) ( , , ( )) , [ , ],b

ax s y s k s t x t dt s a b+ ∈∫ (1)

and

1

1

1

( ) = ( ( ), , ( )) ,

( ) = ( ( ), , ( )) ,

( , , ( )) = ( ( , , ( )), , ( , , ( ))) ,

T

n

T

n

T

n

x s x s x s

y s y s y s

k s t x t k s t x t k s t x t

…

…

…

(2)

k is an arbitrary given kernel function and )(sy is a given

function of ],[ bas ∈ . For the linear case, it is assumed that

njitxtsktxtsk ji ,1,2,=,),()],([=))(,,( , … . If )(sy is a

crisp function then the solution of above Eq. (1) is crisp as

well. However, if )(sy is a fuzzy function this equation may

only possess fuzzy solution.

For solving system of fuzzy nonlinear Fredholm integral

equation (1), we may replace Eq. (1) by the equivalent system

( ) = ( ) ( , , ( )) = ( ) ( , , , )) ,

( ) = ( ) ( , , ( )) = ( ) ( , , , ) ,

b b

a a

b b

a a

x s y s k s t x t dt y s F s t x x dt

x s y s k s t x t dt y s G s t x x dt

+ +

+ +

∫ ∫

∫ ∫ (3)

which possesses a unique solution Bxx ∈),( which is a

fuzzy function, i.e. for each s , the pair ));(),;(( rsxrsx is

a fuzzy number.

The parametric form of Eqs. (3) is given by

( ; ) = ( ; ) ( , , ( ; ), ( ; )) ,

( ; ) = ( ; ) ( , , ( ; ), ( ; ))

b

a

b

a

x s r y s r F s t x t r x t r dt

x s r y s r G s t x t r x t r dt

+

+

∫

∫ (4)

for [0,1].∈r For the linear case, it is assumed that

njitxtsktxtsk ji ,1,2,=,),()],([=))(,,( , … and we have

( , ) ( ; ), ( , ) 0,( , , ( ; ), ( ; )) =

( , ) ( ; ), ( , ) < 0,

k s t x t r k s tF s t x t r x t r

k s t x t r k s t

≥

and

( , ) ( ; ), ( , ) 0,( , , ( ; ), ( ; )) =

( , ) ( ; ), ( , ) < 0.

k s t x t r k s tG s t x t r x t r

k s t x t r k s t

≥

Computational and Applied Mathematics Journal 2015; 1(3): 67-71 69

In the next section, we define the HAM as an analytical

algorithm for approximating the solution of this system of

integral equation. Then we find the approximate solutions for

);( rsx and );( rsx for each 10 ≤≤ r and ].,[ bas ∈

4. Homotopy Analysis Method

Figure 1. The h-curves of 8th-order of approximation solution given by HAM.

Figure 2. Left: the exact solution and right: the HAM solution for 1x .

Figure 3. Left: the exact solution and right: the HAM solution for 2x .

Figure 4. Comparison between the exact solutions and the HAM solutions: left: for 1(0.5; ; 0.4)x r − and right:for 2 (0.5; ; 0.4)x r − .

70 Maryam Mosleh and Mahmood Otadi: Solving a System of Fuzzy Integral Equations by an Analytic Method

In this section the ideas of HAM are introduced [11]. Let φ

be a function of the homotopy parameter q , then

=0

1( ) = | ,

!

m

m qmD

m q

ϕϕ ∂∂

is called the mth-order homotopy derivative of φ , where

0≥m is an integer [12].

Consider the system of Fredholm integral equations in the

crisp case

( ) = ( ) ( , , ( )) , [ , ],b

aX s Y s K s t X t dt s a b+ ∈∫ (5)

and

1

1

1

( ) = ( ( ), , ( )) ,

( ) = ( ( ), , ( )) ,

( , , ( )) = ( ( , , ( )), , ( , , ( ))) .

T

n

T

n

T

n

X s X s X s

Y s Y s Y s

K s t x t K s t x t K s t x t

…

…

…

(6)

Furthermore, Eq. (5) suggests to define the nonlinear

operator N as follows:

[ ( ; )] = ( ; ) ( ) ( , , ( )) ,b

aN s q s q Y s K s t t dtϕ ϕ ϕ− − ∫ (7)

and we choose the auxiliary linear operator L as follows:

[ ( ; )] = ( ; ).L s q s qϕ ϕ (8)

suppose that [0,1]∈q denotes an embedding parameter,

0≠h an auxiliary parameter, and 0)( ≠sH an auxiliary

function. By using the above definitions, we construct the

zero-order deformation equation as follows:

0(1 ) [ ( ; ) ( )] = ( ) [ ( ; )],q L s q s qhH s N s qϕ ψ ϕ− − (9)

where )(0 sψ is an approximate guess of the exact solution

of system of fuzzy integral equations and );( qsφ is an

unknown function which depends also on

convergence-parameters and auxiliary functions. Expanding

);( qsφ with respect to the embedding parameter ,q in

Taylor series, we obtain following equation:

0 =1( ; ) = ( ) ( ) ,m

m ms q s s qϕ ψ Σ ψ+∞+

where ( ) = [ ]m m

s Dψ ϕ .

Operating on Eq. (9) with ,mD we have the so-called

mth-order deformation equation as follows:

1 1[ ( ) ( )] = ( ) ( ),

m m m m mL s s hH s Rψ χ ψ ψ− −− (10)

where

0, 1,=

1, 2m

m

mχ

≤ ≥

and

1 1( , ) = ( [ ( ; )]),m m mR s D N s qψ ϕ− − (11)

also, 0

= { ( ), , ( )}m mpsi s psi sψ … and

1 1( , ) = ( ) ( , , ( )) (1 ) ( ).b

m m m ma

R s s K s t X t dt Y sψ ψ χ− − − − −∫

Therefore, the above relation to find the series pattern

solution of Eq. (5) for ,1,2,= …m by choosing

),(=)(0 sYsψ is as follows:

1 1

0

( ) ( ) ( , ),( ) =

( ) = ( ).

m m m m

m

s hH s R ss

s Y s

χ ψ ψψ

ψ− −+

(12)

Similarly, by applying the homotopy perturbation method

(HPM) and substituting 1= −h and ,=)( IsH in (12) the

recurrent relation to find the series pattern solution of Eq. (5),

for ,1,2,= …m is as follows

1

0

( ( , , ( ; )))( ) =

( ) = ( ).

b

ma

m

D K s t t q dts

s Y s

ϕψψ

−

∫ (13)

For illustrate the method proposed in this work, we consider

the following example.

Example 1. Consider the system of fuzzy integral equations

1 12

1 1 2 10 0

1 1

2 1 2 20 0

( ) ( ) 2 ( ) = ( ),

( ) 4 ( ) 2 ( ) = ( ),

x s sx t dt s x t dt y s

x s stx t dt sx t dt y s

+ + + +

∫ ∫

∫ ∫

where 2 2 2

1 ( ) = ( 2 2,7 2 ) ( 1, 4 ),3

sy s s r r r r r r+ + − + + + −

2

2 ( ) = ( 3 3,10 3 ),y s s r r r+ + − 0 , 1s t≤ ≤ for 0 1.r≤ ≤

To find the valid region of h by the 8th-order of

approximation solution which are shown in figure 1.

The exact solution in this case is given by

),1,4(=);( 22

1 rrrsrsx −++

),1,3(=);(2 rrsrsx −+

and are shown in figures 2-3. As a special case of the HAM

when 1,= −h the HPM cannot guarantee the convergence

of solution series where this is the case where Liao introduced

the convergence-control parameter to improved the traditional

HAM. The exact and obtained solution by means of HAM of

fuzzy integral equation in this example at 0.5=s are shown

in figure 4.

Computational and Applied Mathematics Journal 2015; 1(3): 67-71 71

5. Summary and Conclusions

In this work, we proposed an efficient and simple method to

solve system of fuzzy integral equations. The main advantage

of HAM is compute the series pattern solution of fuzzy

integral equations. The results show that the proposed method

is a promising tool for this type of fuzzy integral equations.

Also we extend solving the linear Fredholm fuzzy integral

equations of the second kind [20]. The HAM is more better

than another analytic methods, because this method provided

us with a convenient way to control the convergence of an

approximating series. Also for 1,= −h the HAM reducing

to the HPM.

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