Reduction Formula for Linear Fuzzy Equations

Transcription

1 International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:13 No:01 75 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 nuha.abd10@gmail.com, 3 oumera85@yahoo.com, 2 adel.ahmed196@yahoo.com 2 Mathematical Department -College of Education, Ibn Al-Haitham, University of Baghdad Baghdad University, Baghdad-Iraq Abstract-- Re cently fuzzy initial value problems or fuzzy diffe rential equations, and fuzzy integral e quations have recei ved consi derable amount of attentions. In this paper, a reduction formula to re duce fuzzy linear differential equations to fuzzy linear integro-differential e quations of Volte rra type is produce d. Also the fuzzy linear integral equations of Volterra type to fuzzy linear integro-differential equations of Volterra type is re duce d with some 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 equation, is a fuzzy bunch function. Index Term-- Fuzzy bunch function, Fuzzy deferential equations, Fuzzy Volte rra integral e quations, Fuzzy integrodifferential equations, Fuzzy reduction. I. INTRODUCTION The topics of fuzzy integral equations are growing interest for some time; especially its relationship to fuzzy control has been rapidly developed in recent years [1]. When the system modeled under the differential sense, it is finally gives a fuzzy differential equation, a fuzzy integral equation or a fuzzy itegro -differential equation and hence, the solution of integro -differential equations have a major role in the fields of science and engineering [2]. We know that solving fuzzy integral equations requires appropriate and applicable definitions of fuzzy function and fuzzy integral of fuzzy function [3]. The fuzzy mapp ing was introduced by Chang and Zadeh [4], later, Dubois and Prade [5] presented an elementary fuzzy calculus based on the extension principle. Seikkala[6] defined the fuzzy derivative which is the generalization of the Hukuhara derivative, the fuzzy integral which is the same as Dubois and Prade[5]; and by mean of the extens ion principle of Zadeh, showed that the fuzzy initial value problem of the form ( ) has a unique fuzzy solution when f satisfies the generalized Lipschitz condition which guarantees a unique solution of the deterministic initial value problem. Kaleva[7] studied the Cauchy problem of fuzzy differential equations; also Park and Han in [8] studied fuzzy differential equations, in [ 9] they are discussed the existence and uniqueness for the solutions of fuzzy differential equations, were as in [10] discussed the existence and uniqueness theorem for a solution of fuzzy Volterra integral equations of the form, Park and Jeong [11] introduced the existence and uniqueness of fuzzy Volterra -Fredholm integral equations of the form, The existence and uniqueness of solutions of non-linear fuzzy Volterra-Fredholm integral equations was discussed by Balachandran and Prakash [12] such as. ( ). Also, Park and Jeong [13] introduced the existence and uniqueness of the solutions of fuzzy integro-differential equations of the form, ( ). Many researchers after that discussed the methods of solutions of these kinds of equations, due to different understand to fuzzy number space, there are different research methods to discuss fuzzy equations [1,2,3,14,15,16,17,18,19,20,21,22]. In this paper we use fuzzy bunch functions to define every equation, and pay attention by finding a general formula of reduction to reduce fuzzy differential equations, and fuzzy Volterra linear integral equations to fuzzy Volterra linear integro-differential equations. Also fuzzified the ordinary reduction theorem in [23], helps to reduce every fuzzy linear integro-differential equation of Volterra type of order n 1, to fuzzy linear integro-differential equation of Volterra type of first order. The reset of this paper is organized as follows: some basic concepts and properties of fuzzy numbers, fuzzy functions, and there continuity, differentiability, and integration are recalled in section 2. In section 3, we introduced the idea of reduction formula to reduce the fuzzy differential equations and fuzzy linear volterra integ ral equat ions to fu zzy linear Vo lterra integ ro - differential equations. In section 4, we introduce the theorem of fuzzy reduction formula which is reduce every fuzzy linear integro-differential equation of Volterra type of order n 1, to fuzzy linear integro-differential equation of Volterra type of first order. Finally a conclusion is drown in section 5.

2 International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:13 No:01 76 II. PRLIMENATRIES Let ( ) denote the family of all nonempty compact convex subset of, the addition and scalar multiplication 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 Euclidean norm in. It is clear that ( ) becomes a metric space[8,21,22]. Let *, - +, is called fuzzy number, where 1) is normal, that is there exist an such that 2) is fuzzy 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 1) Definition [19,20] Let be a fuzzy bunch function such that, if {} then is finite, and it is also 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 , for 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 differentiable 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 FORMULA This section will obtained how the fuzzy linear differential If is a function, then according to Zadeh s equations and fuzzy linear integral equations of Volterra type extension principle we can expand to by the reduce to fuzzy linear integro-differential equations of equation ( ) * + Volterra type. It is well known that, ( )- (, -, A. Reduction of Fuzzy Linear Differential Equations - ), for all, and In this section fuzzy linear initial value differential continuous function. Especially for addition and scaler multiplication, we have equations of order is reduced to fuzzy linear integrodifferential equations of Volterra type of order, -, -, -, -, -,, this is obtained in the next example. Without loss of generality let T=[a,b]. where [8,21,22]. Example1 Let ( ) for al, - (3-1) (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-differential equation of Volterra type is as follow: ( ) ( ) ( ) for all, -, -. ( ) ( ) for all, -. which is equivalent to ( all, - ( 3-2) ) for whenever the initial conditions becomes: 1) For since ( ) for all, and for all, then for all, ( ) for all (3-3) 2) For,

3 International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:13 No:01 77 ( ) for all ( ), ( ) for all, and for all. then ( ) for all, ( ) for all ( ; (3-4) 3) For As in the second condition we calculate ( ) for all, (3-5) Solving equation (3-2) with initial condition, given in equation (3-3), (3-4), and (3-5), we have (3-6),, 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, -, ( ) ( ) 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, - ;

4 International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:13 No:01 78 such that ( ) ( ) and ( ) (3-14) by integrating 2 i x D ui ( x) 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 integrodifferential 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 Volterra 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 Volterra type of second kind, for all. Assuming that and are smooth functions, differentiating the Volterra integral equation of second kind and using the generalized formula of fundamental theorem of integral calculus (Leibnitz generalized formula), 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 FORMULA This section will be fuzzified the ordinary reduction theorem [23] that is helps us 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. Theorem 4-1 Let, - (4-1) be fuzzy linear integro-differential equation of initial conditions order, with

5 International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:13 No:01 79 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 integrodifferential of Volterra type of first order of the form, - With (4-2) where, ), ) ( -- ; And With Proof. for all Since every function in equation (4-1) defined as in definition (1), equation (4-1) becomes as follow:, ( ) -,( )- ( ) ( )( ) for all, and (4-3) ( ) ( ) {( ) 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 integrodifferential equation. For all we have ( -) for all ( ( (4.5) ) we have, ) (, ( ) ( ) ) (4-6) which is equivalent to ( ) ( (, - ) ) (4-7) ( ) ( )... ( ( / ( ) 1 / 0 / ) (4-8)

6 International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:13 No:01 80 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-differential of Volterra type of order three, - (4-11), 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: *( ) ( )+ * + *( ) ( )+ * + *( ) ( )+ * + *( ) ( )+ *( ) ( )+ *( ) ( )+ * + *( ) ( )+ *( ) ( )+ *( ) ( )+ *( ) ( )+ {( {( ) ( )} ) ( )} 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 integrodifferential 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. REFERENCES [1] M. M. Otadi, Numerical solution of fuzzy integral equations using Berntein polynomials, Australian J. of Basic and App.Science,5(7): (2011) [2] N. Mikaeilvand, and S. Khakrangin, Solving fuzzy Volterra integrodifferential equation by differential transford method, Atlantis Press,(2011) [3] M. Ghanbari, Approximate analytical solutions of fuzzy linear Fredholm integral equations by HAM, IJIM. 4(1) (2012) [4] S.S.L. Chang, L.A. Zadeh, On fuzzy mapping and control, IEEE TRANS, System Man Cybernet. 2(1972) [5] D. Dubois, H. Prade, Towords fuzzy differential calculus, fuzzy set and system, 8 (1982)1-7.

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