Reduction Formula for Linear Fuzzy Equations

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Reduction Formula for Linear Fuzzy Equations

N.A. Rajab

1

, A.M. Ahmed

2

, O.M. Al-Faour

3

1,3

Applied Sciences Department, University of Technology, Baghdad- Iraq.

1

[email protected], [email protected], [email protected] 2

Mathematical Department -College of Education, Ibn Al-Haitham, University of Baghdad Baghdad University, Baghdad-Iraq

Abstra ct-- Re cently fuzzy initial value problems or fuzzy diffe rential equ ations, an d fuzzy integral e quations h ave recei ved consi derable amount of attentions. In this paper, a redu ction formula to re du ce fuzzy linear differential equations to fuzzy linear integro-differential e quations of Volte rra type is produ ce d. Also the fuzzy linear integral equations of Volterra type to fuzzy linear i ntegro-differential equations of Volterra type is re duce d with s ome examples.

Fuzzy reduction theorem helps to reduce every fuzzy linear integro-differential equation of Volterra type of order , to fuzzy linear integro-differential equation of Volterra type of first order is introduced. Fuzzy function, which is use d to define the functions in each fuzzy equ ation, is a fuzzy bun ch function.

Ind ex Te rm-- Fuzzy bun ch fun ction , Fuzzy deferential equations , Fuzzy Volte rra integral e quations, Fuzzy integro -differential equations, Fuzzy reduction.

I. INTRODUCTION

The top ics o f fu zzy integ ra l equ at ions are g ro wing inte rest for so me t ime; espec ia lly its re lat ionsh ip to fu zzy contro l has be en rap id ly deve lop ed in recent yea rs [1]. When the syste m mod e led unde r th e d iffe rent ia l sense , it is fina lly g ives a fu zzy d ifferent ia l equat ion , a fu zzy int egra l equat ion o r a fu zzy iteg ro -d ifferent ia l equat ion and h ence , the so lut ion o f integ ro -d ifferent ia l equat ions have a ma jo r ro le in the fie lds o f sc ien ce and eng ine ering [2]. We kno w that so lv ing fu zzy int egra l eq uat ions requ ires approp riat e and app licab le de fin it ions of fu zzy funct ion and fu zzy integ ra l of fu zzy funct ion [3]. The fu zzy mapp ing was int roduced by Ch ang and Zadeh [4], lat er, Dubo is and Prade [5] p resented a n e le menta ry fu zzy ca lcu lus based on the e xtension p rin c ip le . Se ikka la [6] de fined the fu zzy derivat iv e wh ich is th e g ene ra lizat ion o f the Hu kuh ara derivat iv e, the fu zzy integ ra l wh ich is the sa me as Dubo is and Prade [5]; and by mean o f th e e xtens ion prin c ip le o f Zadeh , showed th at the fu zzy in it ia l v a lue p rob le m of th e form

̃́( ) ( ( ) ) ̃( ) ̃

has a un ique fu zzy solut ion wh en f sat isfies the genera lized Lipsch it z cond it ion wh ich gua rante es a unique so lut ion o f th e dete rmin istic in it ia l va lue p rob le m. Ka leva [7] stud ied the Cau chy p rob le m o f fu zzy diffe rent ia l equa t ions; a lso Pa rk and Han in [8] studied fu zzy d iffe rent ia l equ at ions, in [ 9] they a re d iscus sed the e xistenc e and un iqueness fo r th e solut ions o f fu zzy diffe rent ia l equ at ions, we re as in [10] d iscussed th e

e xistence and uniqueness theorem for a solution of fuzzy Volterra integral equations of the form,

̃( ) ̃( ) ∫ ̃( ) ̃( )

Pa rk and Jeong [11] introdu ced the e xist ence and uniquen ess of fu zzy Vo lt erra -Fredho lm integ ra l equat ions of the form,

̃( ) ̃( ) ∫ ̃( ) ̃( ) ∫ ̃( ) ̃( )

The e xist ence and un iqueness o f so lut ions of non-lin ea r fu zzy Vo lt erra -Fredho lm integ ra l equat ions was d iscussed by Balachandran and Prakash [12] such as.

̃( ) ̃( ) ∫ ̃( ) ̃( ) ∫ ̃( ̃( )) . A lso , Pa rk and Jeong [13] int roduc ed the e xist ence and uniquen ess of th e so lut ions of fu zzy integ ro -d iffe rent ia l equations of the form,

̃( )

̃( ) ∫ ̃( ) ̃( ) ∫ ̃( ̃( )) .

Many resea rche rs aft er that d iscussed the methods o f solut ions o f these kinds o f equat ions, due to d ifferent understand to fu zzy nu mb er space , th ere a re d ifferent resea rch methods to d iscuss fu zzy equat ions [1,2,3,14,15,16,17,18,19,20,21,22].

In th is pape r we use fu zzy bunch funct ions to de fine eve ry equat ion, and pay at tent ion by find ing a gen era l fo rmu la o f reduct ion to reduce fu zzy d iffe rent ia l equat ions , and fu zzy Vo lte rra lin ea r integ ra l equat ions to fu zzy Vo lte rra lin ea r integ ro -d iffe rent ia l equ at ions. A lso fuzzified the ordinary reduction theorem in [23], helps to reduce every fu zzy linear integro-diffe rential equation of Volterra type of ordern1, to fuzzy linear integro-differentia l equation of Vo lterra type of first order.

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II. PRLIM ENATRIES

Let ( ( ) ) denote the fa mily of all none mpty compact convex subset of , the addition and scalar mu ltip licat ion in

( ) are defined as usual. Let ̃ and ̃ be two nonempty subset of . The distance between ̃ and ̃ is defined by Hausdorff metric

( ̃ ̃) 2 ‖ ‖ ‖ ‖

3

Where ‖ ‖ denote the usual Euc lidean norm in . It is c lear that ( ( ) ) becomes a metric space[8,21,22].

Let * ̃ , -| ̃ ( ) ( ) +, ̃ is

called fuzzy number, where

1) ̃ is norma l, that is there e xist an such that

̃( )

2) ̃ is fu zzy convex set (that is ̃( ( ) )

* ̃( ) ̃( )+ ) 3) ̃ is upper semicontinuous on

4) , ̃- * + compact set.

For , denote , ̃- * +, then the -level set is defined as , - ( ), for all Note that for every , the Singleton fuzzy number

̃ ̃( ) { ,

If is a function, then according to Zadeh’s e xtension principle we can e xpand to by the equation ̃( ̃ ̃)( ) _{( )} * ̃( ) ̃( )+

It is well known that

, ( ̃ ̃)- (, ̃- , ̃- ), for all ̃ ̃ , and continuous function .

Especially for addition and scaler multiplication, we have

, ̃ ̃- , ̃- , ̃- , ̃- , ̃- ,

where ̃ ̃ [8,21,22].

1) Definition [19,20]

Let ̃ be a fuzzy bunch function such that, if ̃ {( )} then ̃ is fin ite, and it is a lso written in the

form, ̃( ) *( ( ) )+ , for all

2) Definition [8,21,22]

Let ̃ . The integral of ̃ over denoted by

∫ ̃( ) is defined level-wise by the equation

0∫ ̃( ) 1 ∫ ̃ ( ) 2∫ ( ) | 3, fo r

all

1) Proposition [8, 21, 22]

Let ̃ ̃ be integrable and , then

1) ∫ ( ̃( ) ̃( )) ∫ ̃( ) ∫ ̃( )

2) ∫ ̃( ) ∫ ̃( )

3) Definition [ 8, 21, 22]

A mapping ̃ is differentiable at if there exist ̃́( ) such that the limits

̃( ) ̃( ) and ̃( ) ̃( ),

exist and equal ̃́( )

2) Proposition [8,21, 22]

If ̃ ̃ are differentiable and , then

( ̃ ̃)́( ) ̃́( ) ̃́( ) ( ̃)́( ) ̃́( )

3) Proposition [8,21,22]

Let ̃ be diffe rentiable and assume that the derivative ̃́( ) is integrable over T, then for each we have ̃( ) ̃( ) ∫ ̃́( )

4) Definition [20]

Let ̃ , and be the ordinary derivative operator. To define the fuzzy derivative operator ( ̃), when

we have, ( ̃)( ) * ( )| +,

In general ( ̃)( ) * ( )| +, If ̃ *( )| + then ( ̃)( )

*( )| +

And( ̃)( ) *( )| +.

III. REDUCTION FORM ULA

This section will obtained how the fuzzy linear d ifferentia l equations and fuzzy linear integral equations of Volterra type reduce to fuzzy linear integro-diffe rential equations of Volterra type.

A. Reduction of Fuzzy Linear Differential Equations

In this section fuzzy linear initia l value d ifferentia l equations of order is reduced to fuzzy linear integro-diffe rential equations of Volterra type of order

, this is obtained in the ne xt e xa mple . Without loss of generality let T=[a,b].

Example1

Let ̃( ) ̃( ̃( ) ) for al , - (3-1)

with in itia l conditions ̃( ) ̃ ̃́(0)= ̃ and ̃́́(0)= ̃ ; where ̃ is linear fuzzy function in ̃, and ̃ is fuzzy differentiable function. ̃ ̃ , and ̃ are fuzzy numbers.

Reducing equation (3-1) to linear integro-differentia l equation of Volterra type is as follow:

( ( ) ) ( ( ) ) ( ( ) )

for all , - , - ( ).

( ( ) ) ( ( ) ( ) ) for

all , - ( ).

which is equivalent to ( ( ) ( ) ( ) ) for all , - ( ) ( 3-2) whenever the initial conditions becomes:

1) For ̃( ) ̃

since ̃( ) ( ( ) ) for all ( ), and ̃ ( ) for all ( ), then ( ) ( ) for all ( ),

( ( ) ) for all ( ) (3-3)

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̃́( ) ̃( ) ( ( ) ) for a ll (

),

̃́( ) ( ( ) ) for all ( ), and ̃ ( ) for all ( ). then ( ( ) ) ( ) for all ( ),

( ( ) ) for a ll ( ;

(3-4)

3) For ̃́́( ) ̃

As in the second condition we calculate ( ( )

) for all ( ), (3-5) Solv ing equation (3-2) with in itia l condition, given in

equation (3-3), (3-4), and (3-5), we have ( ) ( ) ( ) (3-6)

with initial conditions ( ) , ́ ( ) , and ́́ ( )

, which is an ordinary differential equation for all

, - ( )..

By integrating both sides of equation (3-6) we obtain

( ) ∫ ( ) ∫ ( ) (3-7)

with the initial conditions ́ ( ) ( ) , for all

, -, and all ( ). Then equation (3-7) linear integro-differential equation of second order of Volterra type,

and ( ( ) ∫ ( ) ∫ ( ) )for all

, -, with initial conditions ( ́ ( ) ) ( ( ) ) for all ( ),

it can be written in the form

( ( ) ) (∫ ( ) ∫ ( ) )

For all , - ( ), Such that

( ( ) ) (∫ ( ) ) (∫ ( ) )

( )

for all , - ( ). by definition (4) and proposition (2) we have

( ( ) ) ∫ ( ( ) ) ∫ ( ( ) )

( )

for all , - ( ), this equation is equivalent to

̃( ) ̃( ) ∫ ̃( ) ; ̃( ) ̃ ̃́(0)

̃

for all , -, which is fuzzy linear integro-differential equations of Volterra type of order 2.

Integrate equation (3-7), yields

( ) ∫ ( )( ( ) ( ))

( ) , (3-8) for all , - ( ); equation (3-8) written in the form

( ( ) ∫ ( )( ( ) ( )) );

( ( ) ), (3-9) for all , - ( ); equation (3-9) equivalent to ( ( ) ) (∫ ( )( ( )

( )) ) ( ) ( )

( ( ) ) ( ), for all , -

( );

by definition (2) and definition (4), the above equation

written in the form

( ( ) ) ∫ ( )( ( ) ( ) ) ( )

( )

with ( ( ) ) ( ), for all , -

( );

( ( ) ) ∫ (( )( ( ) ) ( ))

( ) ( ) with ( ( ) ) ( ),

for all , - ( ); then

̃( ) ∫ ( ) . ̃( ) ̃( ) / ̃ ̃

̃( ) ̃ , for all , -. (3-10) Equation (3-9) is fuzzy linear integro-differential equation of first order of Volterra type.

Example2 Suppose that

, ̃ ( ) ̃ ( ) ̃( )-[ ̌( )]=[ ̌( )]

with ̃( ) ̃ ̃́( ) ̃ and ́́( ) ̃ , is fuzzy initial value differential equation, for all , - where ̌ and

, ̃ -, are fuzzy level-wise continuous function on [0,b] from , and ̃ is a fuzzy differentiable function from ̃ ̃ , and ̃ are fuzzy numbers. We have

, ( ( ) ) ( ( ) )

( ( ) )-,( ( ) )- ,( ( )

)-for all , - ( );

,( ( ) )- ( ( ) ) ,( ( ) )- ( ( ) )

,( ( ) )- ( ( ) ), ( ) - , ( )

-by applying definition (4) the above equation become

( ( ) ) ( ( ) ( ) ) ( ( ) ( )

( ) ( ) ) ( ( ) )

( ( ) ( ) ( ) ( ) ( )

( ) ( ) ) ( ( ) )

( ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) )

With initial condition which is found in the same way of the initial condition in example (1) in equations (3.3),(3.4), and (3.5).

Such that we have the initial conditions

( ( ) ) ( ( ) ) and

( ( ) ) for all ( ) (3-11) We have an ordinary initial differential equation

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (3-12)

with the initial conditions

( ( ) ) ( ( ) ) and ( ( ) )

for all , - ( ); Now, set ̃ ( ) ̃( ) (3-13)

Where ̃( ) is fuzzy integrable function from . Let ̃( )has the form of fuzzy function according to definition (1) we have

̃( ) ( ( ) ),

And ̃( ) ( ( ) ) for all , -

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such that ( ( ) ) ( ( ) ) and ( ( ) ( ) ) (3-14) by integrating

(

_x

)

_D

2

_u

(

_x

)

i





to obtain

∫ ( ) ( )

so ( ) ∫ ( ) (3-15) for all , - ( ); By integrating equation (3-15) we obtains ( ) ∫ ( ) ( ) such that

( ) ∫ ( ) ( ) (3-16) for all , - ( ).

Now, to reduce the initial differential equation (3.12) to linear integro-differential equation of Volterra type, put equations (3.13),(3.14), and (3.15) in equation (3.12) to obtain

( ) ( ) ( ) ( ),∫ ( ) -

,∫ ( ) ( )

( )

for all , - ( );

Such that linear integro-differential equation of Volterra type of first order will be

( ( ) ( ) ( )

( ) ( ) ( )

( ) ∫ , ( ) ( ) ( )- ( ) ) (3-17) with initial condition ( ( ) , ) for all

, - ( );

For all ( ); equation (3-17) written in the form

( ( ) ( ) ( ) ) ( ( ) ( ) (

) ( ) ∫ , ( ) ( ) ( )- ( ) ) (3.18) with initial condition ( ( ) ) ( ),

for all ( ); ( ( ) )

( ( ) ( ) ) ( ( ) ) ( ( ) )

(( ) ( ) )

(∫ , ( ) ( ) ( )- ( ) )

by definitions (2) and definitions (4) the above equation become ( ( ) ) ( ( ) )( ( ) )

( ( ) ) ( )( ( ) ) ,( ) ( ) -( ( ) ) ∫ ,( ( ) ) (

)( ( ) )-( ( ) ) ̃ ( ) ̃( ) ̃( )

̃( ) ̃ ̃( ) [ ̃ ̃ ̃ ] ∫ , ̃ ̃(

)- ̃( )

with initial condition ̃ ( ) ̃ ; be a fuzzy linear integro-differential equation of Volterra type of first order.

B. Reduction of Fuzzy Linear Integral Equations

In this section we discuss the possibility of reducing fuzzy linear Vo lterra integral equations of second kind to fuzzy linear Volterra integro-differential equations.

Consider fuzzy linear Volterra integral equations of second kind

̃( ) ̃( ) ∫ ̃( ) ̃( ) for all , - (3-19)

Assuming that ̃ ̃are levelwise continuous, differentiable functions from , K~is levelwise continuous fuzzy function from , where , - , -, when all functions in equation (3-18) are defined according to the

definition (1) this equation is written as follows:

( ( ) ) ( ( ) ) ∫ ( ( ) )( ( ) )

For all ( );

( ( ) ) ( ( ) ) ∫ ( ( ) ( ) )

( ( ) ) ( ( ) ) (∫ ( ) ( ) )

( ( ) ) ( ( ) ∫ ( ) ( ) )

( ( ) ( ) ∫ ( ) ( ) )

then ( ) ( ) ∫ ( ) ( ) is an ordinary integral equation of Vo lterra type of second kind, for a ll

( ).

Assuming that and are smooth functions, diffe rentiating the Volterra integral equation of second kind and using the generalized formu la of fundamental theorem of integral

calculus (Le ibnitz generalized formu la), to obtain

( ) ( ) ( ) ( ) ∫ ( )

( )

With ( ) ( )

Which is reduced to an initial value differential equation of first order if does not depend on X, otherwise it is linear integro-differential equation.

Such that we have for all ( );

. ( ) ( ) ( ) ( ) ∫ ( )

( ) /

with( ( ) ( ) ) ( ( ) ) . ( )

( ) ( ) ∫ ( )

( ) /

with( ( ) ) ( ( ) )

( ( ) )

( ( ) ) ( ( ) ( ) ) .∫ ( )

( ) /

with( ( ) ) ( ( ) )

by definitions (2), and definitions (4) we have

( ( ) ) ( ( ) ) ( ( ) )( ( ) )

∫ .

( ( ) )/( ( ) )

with( ( ) ) ( ( ) ) ̃( ) ̃( )

̃( ) ̃( ) ∫ (

( ̃( )) ̃( ) with ̃( ) ̃( )

It is fuzzy linear integro-differential of Volterra type of first order if ̃ does not depend on

x

.

IV. FUZZY REDUCTION FORM ULA

This section will be fu zzified the ordinary reduction theorem [23] that is helps us to reduce every fuzzy linear integro-diffe rential equation of Volterra type of order , to fuzzy linear integro-diffe rential equation of Vo lterra type of first order.

Theorem 4-1

Let

, ∑ ̃

( ) - ̃( ) ̃( ) ∫ ̃( ) ̃( ) (4-1)

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̃( ) ̃ ̃́( ) ̃ ̃( )( ) ̃ for all , -

Where ̃, and ̃ ( ) are fuzzy level-wise continuous, integrable functions from , ̃ is a fuzzy level-wise continuous integrable function from , -

, - , and ̃ ̃ ̃ are fuzzy numbers.

Then equation (4-1) reduced to fuzzy linear integro-differential of Volterra type of first order of the form

, ̃- ̃( ) ̃( ) ∫ ̃( ) ̃( )

With ̃( ) ̃ (4-2) where

̃ ( )

∑ ,( )_{( )} )

, ̃

∑ ( )

∑ ( ) ̃

( ) _{( ) ̃}

-- _{( )}

∫ ( ) _{̃( )}_; And

̃( ) ∑ ( )

( )

∑ ( ) ( ) ̃ ( )_{( )}

( )

( ) ∑ ( ) ̃

( )

( ) _{( )}∫ ( ) ̃( )

With

∑ for all

.

Proof

Since every function in equation (4-1) defined as in definition (1), equation (4-1) becomes as follow:

, ∑ ( ( ) ) -,( ( ) )- ( ( ) )

∫ ( ( ) )( ( ) )

for all , and ( )

(4-3)

with initial conditions

{

( ( ) )

( ́ ( ) )

( ( )_{( )}

)

For equation (4-3) we have

, ,( ( ) )-

∑ ( ( ) ), ,( ( ) )-

( ( ) )

∫ ( ( ) )(( ( ) )

by definition (2), and definition (4), the above equation becomes ( ( ) ) ∑ ( ( ) )( ( ) )

( ( ) ) ∫ ( ( ) ( ) )

by definition (4) the above equation becomes

( ( ) ) ∑ ( ( ) ( ) ) ( ( ) )

(∫ ( ) ( ) )

which is equivalent to

( ( ) ∑ ( )

( ) ) ( ( )

∫ ( ( ) ( ) ) ( ( )

∑ ( ( )

( ) ( ) ∫ ( ( ) ( ) )

we have

( ) ∑ ( ( ) ( )

( ) ∫ ( ( ) ( ) (4-4)

which is an ordinary linear integro-differential equation of first order of Volterra type for all ( ) Now equation (4-4) is reduced to first order integro-differential equation.

For all ( ) we have

( ) ( ) ( ) ( ) (∑ ( )

( )

,

∑ _{( )}

∑ ( ) ( ) ( ) ( ) ( ) -)

∫ ( )

( ) ( )

∫ ∑ ( )

( )

∑ ( )( ( ) ( ) ( ) ( ))

∫ ( )

( ) ∑ ( ) ( )( ) ( )

∫ ∫ ( )

( ) ( ) ( ) (4.5) for all ( ) we have

( ( ) _{( )}( ) ( ) ∑ ( )_{( )}

,

∑ ( )

∑ ( ) ( ) ( ) ( ) ( )

∫ ( )

( ) ( )

∫ ∑ ( )

( )

∑ ( )( ( ) ( ) ( ) ( ))

∫ ( )

( ) ∑ ( )

( )_{( )} _{( )}

∫ ∫ ( )

( ) ( ) ( ) ) (4-6) which is equivalent to

( ( ) ( ) ( ) ( ) ) (∑ ( )

( )

,

∑ ( )

∑ ( ) ( ) ( ) ( ) ( ) -

∫ ( )

( ) ( )

∫ ∑ ( )

( )

∑ ( )( ( ) ( ) ( ) ( ))

∫ ( )

( ) ∑ ( ) ( )( ) ( )

∫ ∫ ( )

( ) ( ) ( ) ) (4-7)

( ( ) ) ( _{( )}( ) ( ) ) .∑ ( )_{( )}

0

∑ ( )

∑ ( ) ( ) ( ) ( ) ( ) 1 /

.∫ ( )

( ) ( ) /

.∫ ∑ ( )

( )

∑ ( )( ( ) ( ) ( ) ( ) /

(∫ ( )_{( )} ∑ ( )

( )( ) ( ) )

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By definitions (2), and definition (4) the above equation becomes

( ( ) ) ( ( ) ( ) ) ( ( ) )

.∑ ( )

( )

0( )

∑ ( )

∑ ( )( ( ) ( ) ( ) )( ( ) )-/

∫ ( )_{( )} ( ( ) /

∫ ,∑ ( )_{( )}

∑ ( )( ( ) ( ) ( ) )( ( ) )-

∫ ( )_{( )} ∑ ( ) (

( )_{( )}₎₍_{( )}₎

∫ ∫ ( )

( ) ( ( ) )( ( ) ) (4-9) such that the above equation becomes ̃

̃( )( ) ̃( )

(∑ ( )_{( )}

, ̃

∑ ( )

∑ ( ) ̃( ) ( ) ( ) ̃ -)

∫ ( )

( ) ̃( )

∫ ∑ ( )

( )

∑ ( ) ̃

( )_{( )}

̃( )

∫ ( )

( ) ̃

( )_{( )}_{̃( ) ∫ ∫} ( )

( ) ̃( ) ̃( ) (4-10)

It is linear integro-differential equation of Volterra type of first order.

Example3

Consider the fuzzy linear integro-differentia l of Vo lterra type of order three

, ̃ ( ) ̃( ) ̃ ( )- ̃( ) ̃( )

∫ ̃( ) ̃( ) (4-11)

with initial conditions ̃( ) ̃ ̃́( ) ̃ ̃́( ) ̃ , and; we apply theorem (4-1) to reduce equation (4-8) to first order linear integro-differential equation of Volterra type. Since each functions in the equation (4-11) has the form of the function in definition (1) we have:

̃( ) *( ( ) ) ( ( ) )+ ̃( ) *( ) ( )+

̃( ) *( ( ) ) ( ( ) )+ ̃( ) *( ) ( ₎₊

̃( ) *( ( ) ) ( ( ) )+

̃( ) *( ) ( ₎₊

̃( ) *( ( ) ) ( ( ) )+

̃( ) *( ( ) ) ( ( ) ₎₊

̃( ) *( ( ) ) ( ( ) )+ ̃( ) *( _{) (} ₎₊ with initial conditions

̃( ) *( ( ) ) ( ( ) )+

̃́( ) *( ́ ( ) ) ( ́ ( ) )+

̃́́( ) {( ́́ ( ) ) ( ́́ ( ) )}

̃́( ) {( ́́ ( ) ) ( ́́ ( ) )}

according to equation (4.2) in theorem (4-1) we have

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ∫ ( ) ( )

( ) ( ) ( ) ( ) ( )

∫ _{( )}

( ) ́ ( ) ́́ ( ) , (4-12) and

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ∫ ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) _∫ _{( )}

( ) ́ ( ) ́́ ( ) (4-13) According to equation (4-5) equations (4-12) and (4-13) becomes

( ) ( ) ( ) ( ( ) ( ) ́ ( ) )

∫ ( ) ( ) ∫ ́( ) ( ) ∫ ( )( ( )

́( ) ́́( )) ( ) ∫ ∫ ( ) ( ) ( )

such that

, - ( ) ( ) ∫ ( ) (

₎ _{( )} _{( )}

and , - ( )

∫ _{( )}

( ) Where

̃( ) * +

̃( ) * ( ) ₊

̃( ) *( ) ( ₊ and ̃ * +.

V. CONCLUSIONS

Fuzzy integro-differential equation of first order obtained from different forms of fuzzy equations, in this paper, we found it from fuzzy differential equations, fuzzy integral equations, and fuzzy integro differential equations of high order, special type of integral equations, and integro -differential equations are used, i.e. equations of Volterra type. For further work we suggest use Fredholm integral equations, and Fredholm integro-differential equations of high order to reduce it to Fredholm integro-differential equation of first order.

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