On The Approximate Solution of Linear Fuzzy Volterra-Integro Differential Equations of the Second Kind

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1 AUSTRALIAN JOURNAL OF BASIC AND APPLIED SCIENCES ISSN: EISSN: 39-8 Journal ome page: On Te Approimate Solution of Linear Fuzzy Volterra-Integro Differential Equation of te Second Kind Saymaa uain ali and Dr. Abdul Kaleq O. Al-Jubory Univerity Tecnology, College of Science Applied, Department of Matematic, Bagdad-Iraq. Univerity Al-Mutaniriya, College of Science, Department of Matematic, Bagdad-Iraq. Addre For Correpondence: Saymaa uain ali, Univerity Tecnology, College of Science Applied, Department of Matematic, Bagdad-Iraq. A R T I C L E I N F O Article itory: Received 9 September 6 Accepted December 6 Publied 3 December 6 Keyword: A B S T R A C T In ti paper, te variation iteration metod (VIM) i adopt to finding te approimate olution of te linear fuzzy Volterra-integro differential equation of econd kind (LFVIDE).Te (VIM) i to contruct correction functional uing general Lagrange multiplier( λ) identified optimally via te variational teory. We proving teorm tudy te convergence approimate olution to te eact olution. Finally, two eample are given and teir reult are own in figure to illutrate te efficiency and accuracy of ti metod. INTRODUCTION Te topic of fuzzy differential equation (FDE) and fuzzy integral equation (FIE) in bot teoretical and numerical point of view ave been developed in recent year. Prior to dicuing fuzzy integro-differential and teir numerical treatment (ZEINALI, M., S. et al., 3; Oama, H., et al., 3.; Zade, L.A. and S.S.L. Cang, 97; Mizumoto, M. and K. Tanaka, 979; Kaleva, O., 987; Goetcel, R., W. Vaman, 986; Puri, M.L. and D. Ralecu, 983; Merkanoon, S., et al., 9; He, J.H., 997), it i neceary to preent and brief introduction to preliminary topic uc a fuzzy number and fuzzy calculu (Oama, H., et al., 3). Te concept of fuzzy et wic wa originally introduced by Zada (97) led to definition of te fuzzy number and it implementation in of te fuzzy control and approimate reaoning problem. Te baic aritmetic tructure for fuzzy number wa later developed by Mizumoto and Tanaka (979). Nomia and Raleu (987) all of wic oberved te fuzzy number a a collection of r level, < r. In ti paper, we ued variational iteration metod (VIM) for olving of linear fuzzy Volterra integro-differential equation of te econd kind (LFVIDE): y () = f () + λ k(, t)y (t)dt (). Baic concept of Fuzzy Set Teory: In ti ection te baic notation ued in fuzzy calculu are introduced. We tart by defining te fuzzy number. Definition (.) (Goetcel, R., W. Vaman, 986): A fuzzy number i a map u: R I = [,] wic atifie? i. u i upper emi-continuou, ii. u()= outide ome interval [c,d] iii. Tere eit real number a,b uc tat c a b d, were Open Acce Journal Publied BY AENSI Publication 6 AENSI Publier All rigt reerved Ti work i licened under te Creative Common Attribution International Licene (CC BY). ttp://creativecommon.org/licene/by/./ To Cite Ti Article: Saymaa uain ali and Dr. Abdul Kaleq O. Al-Jubory., On Te Approimate Solution of Linear Fuzzy Volterra- Integro Differential Equation of te Second Kind. Aut. J. Baic & Appl. Sci., (8): -5, 6

2 5 Saymaa uain ali and Dr. Abdul Kaleq O. Al-Jubory, 6. u() i monotonic increaing on [c,a]. u()i monotonic decreaing on [b,d] 3. u()=,a b. Te et of all te fuzzy number (a given in definition (.)) i denoted by E an alternative definition wic yield te ame E i given by[ 5]. Definition (.):A fuzzy number u i a pair ( u, u) of function u(r), u(r); r Wic atifying te following requirement:. u(r) i a bounded monotic increaing Left continuou function. u(r) i a bounded monotic decreaing Left continuou function 3. u(r) u(r), r. For arbitrary fuzzy number u=(u(r), u(r)), v = (v(r), v (r)) and real contant we define addition u+v and calar multiplication ku a (u + v)( r) = u(r) + v(r) (u )( + v r) = u (r) + v(r) () (ku)(r) = ku(r), (ku)(r) = ku (r) if k (ku)(r) = ku (r), (ku)(r) = ku(r) if k > (3) Te collection of all uc fuzzy number wit addition and multiplication a define by equation () and (3) i denoted by E and i conve cone. Net, we will define te fuzzy function notation and a metric D in E. Definition (.3) (Goetcel, R., W. Vaman, 986): For arbitrary fuzzy number u= (u(r), u(r)), v = (v(r), v r), te quantity D(u. v) = ma{ up α u(r) v(r), up u(r) v α r } I te ditance between u and v. Ti metric i equivalent to te one ued by (Goetcel, R., W. Vaman, 986; Puri, M.L. and D. Ralecu, 983). It i own (Puri, M.L. and D. Ralecu, 983) tat (E, D) i a complete metric pace. Definition (.): A fuzzy function f: R E i aid to be continuou if for arbitrary fied R and > tere eit δ > uc tat < δ, ten D(f(), f( )) < () Definition (.5) (Merkanoon, S., et al., 9): Te Seikkala derivative f () of a fuzzy function f i defined by [f ()] r =[f (;r), f (; r)],r [,], were prime ymbol denoted te derivative wit repect to. Definition(.6): let f : I E be a fuzzy function and I R, ten f i differentiable at, if (I) tere eit an element f ( ) E, uc tat for all > ufficiently mall, tere are f( + ) f( ) ; f( ) f( + ) and ( Lim + ) f( +) f( ) = ( Lim + ) f( ) f( +) = f ( ) (5) Or (II) tere eit an elementf ( ) E, uc tat for all < ufficiently mall, tere are f( + ) f( ; f( ) f( + ) and ( Lim ) f( +) f( ) = ( Lim ) f( ) f( +) = f ( ) (6) were te relation (I) i te claical definition of te fuzzy H-derivative. 3- Fuzzy Integro- Differential Equation: In ti ection, we conider te firt order linear fuzzy Volterra integro-differential equation (LFVIDE). y () = f () + λ k(, t)y (t)dt (7) Were te initial y = y () = (y(, r), y(, r)) = (,), Were y () = d d y (), f : [, b] y(r) i continuou fuzzy number, k i arbitrary continuou function over te region Ω = {(, t) b} and Δ = {(, t, y ()) b,y () y(r)} and y i not to be determined. Eq (7) can be written in term of r Level et a: [y (; r), y (; r)] = [f(; r), f(; r)] + λ k(, t)[y For r (t;r), y(; r)]dt (8)

3 6 Saymaa uain ali and Dr. Abdul Kaleq O. Al-Jubory, 6 Tu, Eq. (7) can be tranformed into te following ytem y (; r) = f(; r) + λ k(, t) y (t;r) dt y (; r) = f(; r) + λ k(, t)y (t; r)dt (9) Wit initial condition y(; r) = y(; r) = -Variational iteration Metod: Te variational iteration metod (VIM) i propoed by He (He, J.H., 997) a a modification of a general Lagrange multiplier metod.ti metod a been own to olve effectively,eaily and accurately a large cla of nonlinear problem wit approimation converging rapidly to a accurate olution. To illutrate it baic idea of te tecnique, we conider following general nonlinear ytem: L[u()] + N[u()] = g() () Were L i linear operator, N i a nonlinear operator, and g() i given continuou function. Te baic caracter of metod i to a correction functional for ytem Eq.() wic u n+ () = u n () + λ(τ){lu n (τ) + Nũ n (τ) g(τ)} () Were λ(τ) i a general Lagrangian multiplier (Kaleva, O., 987) wic can be identified optimally via variational teory, te ubcript n denote te nt-order approimation and ũ n i conider a retricted variation, i.e. δũ n = were L = d, we can contruct te following correction functional dt u n+ () = u n () + λ(τ){u n(τ) + Nũ n (τ) g(τ)} () 5- Variational Iteration Metod Solving Te Linear Fuzzy Volterra-integro Differential Equation (lfvide): Now, we conider te linear fuzzy Volterra-integro differential equation of econd kind: y () = f () + λ k(, t)y (t)dt Ten we ave te following iteration equence u n+ () = u n () + λ(τ){ u n(τ) f (τ) k(, t)ũ n ()d} (3) To find optimal λ, we proceed a following: δu n+ () = δu n () + δ λ(τ){u n(τ) f (τ) k(, t)ũ n ()d} () And upon uing te metod of integration by part, ten Eq.(3) will be reduced to δu n+ () = δu n () + δ λ(τ){u n(τ)} Ten te following tationary condition are obtained: λ = and λ + = Te general Lagrange multiplier terefore, can be readily identified: λ = and by ubtitute in Eq(3), te following iteration formula n i obtained u n+ () = u n () {u n(τ) f (τ) k(, τ)ũ n ()d} (5) Terefore, we can write te following iteration formula y n+ (, r) = y n (, r) {y (τ, r) f(τ, r) k(, τ)f n (, r)d} n y n+ (, r) = y n (, r) { y n (τ, r) f(τ, r) k(, τ)f n (, r)d } y (, r) = and y (, r) =. wit te initial approimation Teorem: Let ũ (C [a, b],. ) be te eact olution of te linear fuzzy Volterra-integro differential equation of (7) and ũ n () C [a, b] be te obtained olution of te equence defined by eq.(5). If Ẽ n () = ũ n () ũ() and k < c, <c< ten te equence of approimate olution {ũ n ()}, n,,... converge to te eact olution ũ(). Proof:- conider linear fuzzy Volterra integro-differential equation of te econd kind:- y () = f () + λ k(, t)y (t)dt Were te approimate olution uing te VIM i given by u n+ () = u n () [u n(τ) f (τ) k(, τ)ũ n ()d] (6) and ince u i eact olution of te linear fuzzy Volterra-integro differential equation of te econd kind u () = u () [u n(τ) f (τ) k(, τ)ũ n ()d] (7)

4 7 Saymaa uain ali and Dr. Abdul Kaleq O. Al-Jubory, 6 Now, ubtracting Eq(6) from Eq.(7) Ẽ n+ () = Ẽ n () [Ẽ n (τ) f (τ) + f (τ) + k(, τ)ũ n ()d] Ẽ n+ () = Ẽ n () Ẽ n (τ) + k(, τ) Ẽ n (τ) d Ẽ n+ () = Ẽ n () Ẽ n () Ẽ n () k(, τ) Ẽ n (τ) d And ince E n() = ũ n () u () wic ave te initial condition of te FVIDE, te E n() =. Hence Ẽ n+ () = k(, τ) Ẽ n (τ) d (8) Now, taking te maimum-norm on bot ide of Eq.(7) Ẽ n+ () = k(, τ) Ẽ n (τ) d Ẽ n+ () k Ẽ n (τ) d Since K i function bounded by c, c (,), ten Ẽ n+ () c Ẽ n (τ) Ẽ n+ () = c Ẽ n (τ) Terefore Ẽ n+ () c Ẽ n (τ) n =,,.. (9) Now, if n=, ten inequality (8) yield to Ẽ () c Ẽ (τ) τ Ẽ () c Ma Ẽ (τ) Ẽ () c ( ) Ma (τ) Ẽ () Alo, if n=, ten form inequality (9) and () we ave E () c E (τ) Subtituting (), in ti inequality we get E () c c Ma E E () = c Ma E () Similarly, for n= and from inequality (8) and (), we ave E 3() c E (τ) Subtituting (3), in ti inequality we get E 3() c c Ma E E 3() = c Ma E 3! And o, in general and uing matematical induction we get: E n() = c n n n Ma E () n! And ince c (,) and a n, ten we will ave te rigt and ide of inequality Eq.() tend to zero, i.e, Ẽ n () a n Ti implie to ũ n () ũ() a n, i.e., te equence of olution obtained from te VIM converge of te eact olution ũ(). 6- Numerical Eample: In te ection, we apply variational iteration metod for olving linear fuzzy volterra- integro differential equation of te econd kind. Eample (): Conider te fuzzy volterra integro differential equation of te econd kind y (; r) = f (; r) + k(, t)y (t; r)dt Were k(, t) = t f (; r) = r ( 3 ) f (; r) = r 3 ( r) Wit te initial approimation y (; r) = and y (; r) = By uing iteration formula, we obtain y (; r) = r( 5) 5 y = (r ) 3 (r+)

5 8 Saymaa uain ali and Dr. Abdul Kaleq O. Al-Jubory, 6 y (; r) = r(8 95) 95 y (; r) = (r )( ) And o on.terefor, it i obviou tat ti olution i converage to eact olution y(; r) = r y(; r) = ( r) Following figure () repreenting te upper and Lower olution of eample () uing different value of r. lower numerical lower eact upper numerical upper eact r = r = r =.75.5 r = Fig. : Upper and lower olution of Eample () for different value of r Eample (): Conider te linear fuzzy volterra integro differential equation of te econd kind y (; r) = f (; r) + k(, t)y (t; r)dt were k(, t) = f (; r) = (r + )( + ) f (; r) = ( + )( r) Wit te initial approimation y (; r) = and y (; r) = By uing iteration formula, we obtain (r + )( + ) y (; r) = y (; r) = ( + )(r ) y (; r) = (r + )( )

6 9 Saymaa uain ali and Dr. Abdul Kaleq O. Al-Jubory, 6 y (; r) = (r )( ) 3 And o on.terefor, it i obviou tat ti olution i converge to eact olution y(; r) = (r + )(e ) y(; r) = ( r)(e ) Following figure () repreenting te upper and Lower olution of eample () uing different value of r. - +low er numerical low er eact * upper numerical upper eact r= r= r= r= Fig. : Upper and lower olution of Eample () for different value of r Concluion: In ti paper te variational iteration metod (VIM) i ued to olve te linear fuzzy volterra- integro differential equation of econd kind. Te reult owed tat te convergence and accuracy of variational iteration metod for numerically olution for (LFVIDE) were in good agreement wit analytical olution. Te computation aociated wit eample and graping in ti paper performed uing matlab (V.7). REFERENCES Goetcel, R., W. Vaman, 986. "Elementary fuzzy calculu, Fuzzy Set Syt.", pp: 3-3. He, J.H., 997. " Variational iteration metod for delay differential equation," Comm. Nonlinear Sci. Numer. Simul., (): Kaleva, O., 987. " Fuzzy differential equation, Fuzzy Set Sytem", (): ttp://d.doi.org/.6/65-(87)99-7 Merkanoon, S., M. Suleiman and Z.A. Majid, 9. "Block Metod for Numerical Solution of Fuzzy Differential Equation",. Int. Mat. Forum, (6): 69-8.

7 5 Saymaa uain ali and Dr. Abdul Kaleq O. Al-Jubory, 6 Mizumoto, M. and K. Tanaka, 979. "Some Propertie of Fuzzy Number", in M. M. Gupta, Rajade R. K. and Yager R. R. (ed.), Advance in Fuzzy Set Teory and Application, Nort Holland, Amterdam, pp: Oama, H., Moammed and Salam A. Amed, 3. " Solving Fuzzy Fractional Boundary Value Problem Uing Fractional Differential Tranform Metod", 6(): 5-3. Puri, M.L. and D. Ralecu, 983. " Differential for fuzzy function" J. Mat. Anal. Appl., 9(): ttp://d.doi.org/.6/-7x(83)969-5 Zade, L.A. and S.S.L. Cang, 97. " On fuzzy mapping and control, IEEE Tran. Sytem Man Cybernetic", : 3-3. ZEINALI, M., S. SHAHMORAD and K. MIRNIA, 3. " Fuzzy integro-differfntial equation: dicrete olution and error etimation", Iranian Journal of Fuzzy Sytem, (): 7-.