Research Article Solving Fuzzy Fractional Differential Equations Using Zadeh s Extension Principle

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1 The Scienific World Journal Volume 3, Aricle ID 5969, pages hp://dx.doi.org/.55/3/5969 Research Aricle Solving Fuzzy Fracional Differenial Equaions Using Zadeh s Exension Principle M. Z. Ahmad, M. K. Hasan, and S. Abbasbandy 3 Insiue of Engineering Mahemaics, Universii Malaysia Perlis, Kampus Teap Pauh Pura, 6 Arau, Perlis, Malaysia School of Informaion Technology, Faculy of Informaion Science and Technology, Universii Kebangsaan Malaysia, 36Bangi,Selangor,Malaysia 3 Deparmen of Mahemaics, Imam Khomeini Inernaional Universiy, Ghazvin , Iran Correspondence should be addressed o M. Z. Ahmad; mzaini@unimap.edu.my Received 5 June 3; Acceped July 3 Academic Ediors: M. Bruzón and S. Momani Copyrigh 3 M. Z. Ahmad e al. This is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. We sudy a fuzzy fracional differenial equaion (FFDE) and presen is soluion using Zadeh s exension principle. The proposed sudy exs he case of fuzzy differenial equaions of ineger order. We also propose a numerical mehod o approximae he soluion of FFDEs. To solve nonlinear problems, he proposed numerical mehod is hen incorporaed ino an unconsrained opimisaion echnique. Several numerical examples are provided.. Inroducion Fracional calculus is an imporan branch in mahemaical analysis. I is a generalisaion of ordinary calculus ha allows nonineger order. A he beginning, i was slowly esablished. However, afer Leibniz and Newon invened differenial calculus, i has been a subjec of ineres among mahemaicians, physiciss, and engineers. Consequenly, he heory of fracional calculus has been exensively developed and influenced in many areas of discipline. The fracional inegral and fracional derivaive of Riemann-Liouville, for example, have been applied o solve many mahemaical problems [ 6]. One of he paricular ineress is he case of solvingfracionaldifferenialequaions [7 9]. The fracional derivaive of Riemann-Liouville, however, has a common characerisic. I requires a quaniy of fracional derivaive of unknown soluion a iniial poin. In pracice, we do no clearly know wha he meaning of he fracional derivaive a ha poin is. In oher words, he required quaniy canno be measured and perhaps may no be available [, ]. The well-known and popularly used mehod in solving fracional differenial equaions is he Capuo fracional derivaive. I allows o specify a quaniy of ineger order derivaives a he iniial poin. This quaniy ypically is available and can be measured. I is herefore no surprising ha here is a vas lieraure dealing wih fracional differenial equaions involving he Capuo fracional derivaive [ 6]. The heory and applicaion of fracional differenial equaions under boh ypes of fracional derivaives have been discussed by many auhors [, 7 5]. Some poenial applicaions have been sudied in [6 8]. In he conex of mahemaical modelling, developing an accurae fracional differenial equaion is no a simple ask. I requires an undersanding of real physical phenomena involved. The real physical phenomena, however, are always pervadedwihuncerainy.thisisobviouswhealing wih living maerials such as soil, waer, and microbial populaions [9]. When a real physical phenomenon is modelled by a fracional differenial equaion, namely, D β ax () =f(, x ()), x( )=x, <β, >a, wecannousuallybesurehahemodelisperfec.for example, he iniial value in () may no be known precisely. Imayakeanyvalueinheformof lesshanx, abou x, or more han x. Classical mahemaics, however, fail o cope wih his siuaion. Therefore, i is necessary o have oher heories in order o handle his issue. Various heories ()

2 The Scienific World Journal exis for describing his siuaion and he mos popular one is he fuzzy se heory [3]. In order o obain a more realisic model han (), Agarwal e al. [3] have aken an iniiaive o inroduce he concep of soluion for fuzzy fracional differenial equaions. This conribuion has moivaed several auhors o esablish some resuls on he exisence and uniqueness of soluion (see [3]). In [33], he auhors derived he explici soluion of fuzzy fracional differenial equaions using he Riemann-Liouville H-derivaive. Recenly, Salahshour e al. [3] applied fuzzy Laplace ransforms [35] o solve fuzzy fracional differenial equaions. Basically, he proposed ideas are a generalisaion of he heory and soluion of fuzzy differenial equaions [36 ]. However, he auhors considered fuzzy fracional differenial equaions under he Riemann-Liouville H-derivaive. Again, i requires a quaniy of fracional H-derivaive of an unknown soluion a he fuzzy iniial poin. In his paper, we propose a new inerpreaion of fuzzy fracional differenial equaions and presen heir soluions analyically andnumerically.theproposedideaisageneralisaionof he inerpreaion given in [ 9], where he auhors used Zadeh s exension principle o inerpre fuzzy differenial equaions. According o Mizukoshi e al. [3], his inerpreaion requires neiher he concep of derivaives of a fuzzy funcion nor he use of selecion heory o obain a soluion o he fuzzy differenial equaion. This paper is organised in he following sequence. In Secion, werecallsomebasicdefiniionsandheoreical backgrounds needed in his paper. In Secion 3, wepresen he soluion of fuzzy fracional differenial equaions analyically and numerically. In Secion, some numerical examples are given. Finally in Secion 5,wegiveconclusions.. Basic Conceps In his secion, we briefly elaborae some definiions and imporan conceps of fracional calculus and fuzzy se... The Fracional Inegral and Fracional Derivaive. The following definiions of fracional inegral and fracional derivaive are adoped from []. Definiion. Arealfuncionx(), >, issaidobeinhe space C μ,μ R, if here exiss a real number p>μ,such ha x() = p x (),wherex () C(, ) and i is said o be in he space C n μ if and only if xn C μ, n N. Definiion. The Riemann-Liouville fracional inegral of x of order β>wih a is defined as I β a x () = Γ(β) a ( s) β x (s) ds, > a, () and for β=, he Riemann-Liouville fracional inegral of x is defined as I ax () =x(). (3) Here, Γ(β) is he well-known gamma funcion defined as Γ(β)= β e d. () Definiion 3. The Capuo fracional derivaive of x of order β>wih a is defined as c D β a x () = Γ(n β) ( s) n β x (n) (s) ds (5) a for n <β n, n N, a, x C n. The following are wo basic properies of he Capuo fracional derivaive [5]. () Le x C n, n N. Then c D β ax, β n,iswell defined and c D β a x C. () Le n <β n, n N,andx C n μ, μ.then I β a ( c D β n a )x() =x() x (k) (a) ( a)k. (6) k! k= The Laplace ransform of he Capuo fracional derivaive is given by [5] L { c D β x ()} =s β n X (s) k= s β k x (k) () ; n <β n... Fuzzy Se Theory. According o Zadeh [3], a fuzzy se is a generalisaion of a classical se ha allows a membership funcion o ake any value in he uni inerval [, ]. Definiion (see [3]). Le U beauniversalse.afuzzyse A in U is defined by a membership funcion A(x) ha maps every elemen in U o he uni inerval [, ]. Definiion 5 (see [38]). Le A be a fuzzy se defined in R. A is called a fuzzy number if (i) A is normal: here exiss x R such ha A(x )=; (ii) A is convex: for all x, y R and λ,iholds ha A(λx ( λ) y) min (A (x), A(y)) ; (8) (iii) A is upper semiconinuous: for any x R, iholds ha A(x ) lim A (x) ; x x (9) ± (iv) [A] = {x R A(x)>}is a compac subse of R. Definiion 6 (see [38]). Le A be a fuzzy number defined in R. The -cu of A is he crisp se [A] ha conains all elemens in R such ha he membership value of A is greaer han or equal o ;hais, (7) [A] = {x U A(x) }, (, ]. ()

3 The Scienific World Journal 3 For a fuzzy number A,is-cus are closed inervals in R andwedenoehemby [A] = [a,a ]. () Definiion 7 (see [5]). A fuzzy number A is called a riangular fuzzy number if is membership funcion has he following form:, if x<a, x a { b a, if a x<b, A (x) = c x { c b, if b x c, {, if x>c, () and is -cus are simply [A] =[a(b a),c (c b)], (, ]. In his paper, we denoe A= (a, b, c) as he riangular fuzzy number and F(R) as he se of all riangular fuzzy numbers. Any crisp funcion can be exed o ake fuzzy se as argumens by applying Zadeh s exension principle [3]. Le f be a funcion from X o Y. Given a fuzzy se A in X, we wan o find a fuzzy se B=f(A)in Y ha is induced by f.if f is a sricly monoone, hen we can ex f o fuzzy se as follows: f (A) (y) = { A(f (y)), if y range (f),, if y range (f). (3) I is clear ha (3) can be easily calculaed by deermining he membership a he poins of he -cus of A.However, in general, he process of finding he fuzzy se B = f(a ) is more complicaed and canno be gahered easily. For example, if f is nonmonoone, hen he problem can arise when wo or more disinc poins in X aremappedohe same poin in Y. If his is he case, hen he above equaion may ake wo or more differen values. This requires a new exension of (3)asshown below: where f (A) (y) = { sup A (x), if y range (f), x f { (y) {, if y range (f), () f (y)={x X f(x) =y}. (5) Some compuaional mehods o compue () can be found in [53, 5]. Theorem 8 (see [55]). If f:r R is coninuous, hen f: F(R) F(R) is well defined and [f(a)] =f([a] ), [, ], A F (R), (6) where f(a)={f(u) u [A] }. For A,B F(R) and λ R, hesumaband he produc λa are defined as follows, respecively: for each [,]. [AB] = [A] [B], (7) [λb] =λ[a] Definiion 9 (see [56]). If A and B are wo fuzzy numbers, hen he disance D beween A and B is defined as D (A,B) = sup max { a b, a b }. (8) [,] In [57], he auhors have shown ha (F(R), D) is a complee meric space and he following properies are well known: (i) D(A C,BC)=D(A,B), A,B,C F(R), (ii) D(λA, λb) = λ D(A, B), A, B F(R) and λ R, (iii) D(A B, C D) D(A, C) D(B, D), A, B, C, D F(R). 3. Fuzzy Fracional Differenial Equaions In his secion, we presen analyical and numerical soluions of fuzzy fracional differenial equaions. 3.. Analyical Soluion of Fuzzy Fracional Differenial Equaions. Firs, le us consider he following fracional differenial equaion: c D β ax () =f(, x ()), x( )=x, (9) where f:[,t] R R is a real-valued funcion, x R, andβ (, ]. Ifβ=,hen(9) becomes an ordinary differenial equaion. Assume ha he iniial value is replaced by a fuzzy number; hen we have he following fuzzy fracional differenial equaion: c D β ax () =f(, X ()), X( )=X, () where X F(R) and β (, ].Ifβ=,hen()becomes a fuzzy differenial equaion. In order o find he soluion of (), we firs find he soluion of (9). By aking he Laplace ransform on boh sides of (9), we ge I follows ha L [ c D β ax ()] = L [f (, x ())]. () s β L {x ()} x( )s β = L [f (, x ())]. ()

4 The Scienific World Journal Assume ha afer simplifying (), we ge L [x ()] =m(s). (3) Then by aking he inverse Laplace ransform o (3), we have x () = L [m (s)] =g(,β,x ), () for [,T]and x R. In order o find he soluion of (), we fuzzify () using Zadeh s exension principle. Hence, we have X () = g (, β, X ), (5) which is he soluion of (). This procedure can be shown precisely in he following heorem. Theorem. Le G be an open se in R and [X ] F(R) G. Supposehaf is coninuous and ha for each β (, ] and each x Ghere exiss a unique soluion g(, β, x ) of he problem (9) and ha g(, β, ) is coninuous in G for each [,T]fixed. Then, here exiss a unique fuzzy soluion X() = g(, β, X ) of he problem (). Proof. The proof of his heorem uses he basic idea found in [6]. Since f is coninuous, here exiss a unique soluion g(, β, x ). This soluion is well defined and coninuous in G, for each [,T] fixed. Then, from Theorem 8, wehave g(, β, ) : F(G) F(R), whichisconinuousandwell defined. Therefore, here exiss a unique fuzzy soluion of he form X() = g(, β, X ) for he problem (). Remark. The exisence of X() is guaraneed by Theorem 8 since x() is coninuous. Moreover, in order o have valid level ses, X() should saisfy he following Saking Theorem [58]. Theorem. If X:[,T] F(R) is a fuzzy soluion of () and denoing [X()] =[x (), x ()] for [,],hen (i) [X()] is nonempy compac subse of R; (ii) [X()] [X()] for ; (iii) [X()] = n= [X()] n for any nondecreasing sequence n in [, ]. In he following resul, we will show ha x () and x () do no inerchange a all [, ). Theorem 3. If X() = g(,β,x ) is obained by using Theorem and [X()] =[x (), x ()] for [,],hen x () and x () do no inerchange a all [, ). Proof. We know ha X() is obained by Zadeh s exension principle hrough Theorem ; hen is membership funcion has he following form: I follows ha x () = min {g(,β,u) u [x,,x, ]}, x () = max {g(,β,u) u [x,,x, ]} (7) for [,].Iisobviousha x () x (). (8) This holds for all [, ).Thiscompleesheproof. In general, he soluion of () may no be found analyically. Therefore, a numerical mehod mus be proposed. 3.. Numerical Soluion of Fuzzy Fracional Differenial Equaions. Fracional Euler mehod under he Capuo fracional derivaive has been proposed in [59]. However, in order o approximae he soluion of fuzzy fracional differenial equaions, he fracional Euler mehod has o be exed in he fuzzy seing. In his case, Zadeh s exension principle plays an imporan role. Le x() be he soluion of (9). The firs wo erms of fracional Taylor series for x() a i can be wrien as [59] x ( i ) x( i ) c D β x ( i ) From (9), we have x ( i ) x( i ) h β Γ(β). (9) h β Γ(β) f ( i,x( i )). (3) Le w i x( i ); hen we have he following fracional Euler mehod [59]: w i =w i h β Γ(β) f( i,w i ), (3) for i =,,,...,N.Leg(β, h, i,w i ) = w i (h β /Γ(β ))f( i,w i );hen(3)becomes w i =g(β, h, i,w i ). (3) To approximae he soluion of (), (3) will be fuzzified using Zadeh s exension principle. We hen obain he following fuzzy fracional Euler mehod: W i =g(β, h, i,w i ) (33) for i =,,...,N,andW i F(R). Please noe ha if β=, hen his fuzzy fracional Euler mehod becomes fuzzy Euler mehod as proposed in [7]. The membership funcion of g in (33) can be defined as g(β,h, i,w i )(y) X () (y) = { sup X (x), if y range (g), x g { (,β,y) {, if y range (g). (6) = { sup W i (x), if y range (g), x g { (β,h, i,y) {, if y range (g). (3)

5 The Scienific World Journal 5 Inpu: Fracional Euler Equaion, Fuzzy Iniial Value, β,, N and N Descriise {,,..., n }; h N N ; for j =#() do w (, j) a (j) (b a); w (,j) c (j) (c b); for i=: N do for j=:#() do w (i, j) min(g(β, h, i,w (i, j)), g(β, h, i,w (i, j))); w (i, j) max(g(β, h, i,w (i, j)), g(β, h, i,w (i, j))); Oupu: w and w Algorihm : Fuzzy fracional Euler mehod for a linear problem. In general, he compuaion of (3) is no an easy ask. By using he concep of -cus, (3) can be calculaed as follows: w i, w i, = min {(u h = max {(u h β Γ(β) f( i,u)) u [w i,,w i, ]}, β Γ(β) f( i,u)) u [w i,,w i, ]}. (35) The opimisaion problems in (35) are performed as follows. (a) If g(β, h, i,u)is increasing or decreasing on he inerval [w i,,w i, ], hen he opimal soluions are obained a he poins of ha inerval (see Algorihm ). (b) If g(β, h, i,u) is nonmonoone, we firs spli he inerval [w i,,w i, ] ino several subinervals and hen solve he opimisaion problems on he subinervals. By aking he minimum and maximum of all heresuls,weobainheopimalsoluionsonhe inerval [w i,,w i, ]. These procedures are given in Algorihm.. Numerical Examples In his secion, we presen hree examples for solving fuzzy fracional differenial equaions. Example. Consider he following linear fuzzy fracional differenial equaion: This problem is a generalisaion of he following fracional differenial equaion: c D β x () =x(), x () =x, (37) where β (, ], >,andx is a real number. In order o find he soluion of (36), we firs find he soluion of (37). By aking he Laplace ransform on boh sides of (37), we have We hen obain Simplifying (39), we ge L [ c D β x ()] = L [x ()]. (38) s β L {x ()} x( )s β = L {x ()}. (39) L {x ()} = x s β s β. () By aking he inverse Laplace ransform o (), we obain x () =x L { sβ }, () s β which finally has he following soluion: x () =x E β ( β ), () where E β ( ) is he Miag-Leffler funcion defined as E β (z) =, β >. (3) Γ(βk) k= z k c D β X () =X(), X () =X, (36) By using Zadeh s exension principle o ()inrelaionox, we obain X () =X E β ( β ), () where β (, ], >,andx is any riangular fuzzy number. which is he soluion of (36).

6 6 The Scienific World Journal Inpu: Fracional Euler Equaion, Fuzzy Iniial Value, β,, N and N Descriise { n, n,..., }; h N N ; for j=#() do x (, j) a (j) (b a); x (, j) c (j) (c b); w (, :) fliplr(x (, :)); w (, :) fliplr(x (, :)); for i=:ndo for j=:#() do w (i, j) min {g(β,h, i,u) u [w (i, ), w (i, )]}; w (i, j) max {g(β, h, i,u) u [w (i, ), w (i, )]}; for i=:ndo for j=:#() do p (i, j) min {g(β,h, i,u) u [w (i, j), w (i, j )]}; q (i, j) w (i, j ); r (i, j) min {g(β, h, i,u) u [w (i, j ), w (i, j)]}; w (i, j) min([p (i, j), q (i, j), r (i, j)]); p (i, j) max {g(β, h, i, u) u [w (i, j), w (i, j )]}; q (i, j) w (i, j ); r (i, j) max {g(β, h, i, u) u [w (i, j ), w (i, j)]}; w (i, j) max([p (i, j), q (i, j), r (i, j)]); Oupu: w and w Algorihm : Fuzzy fracional Euler mehod for a nonlinear problem. Le [X()] = [x (), x ()] and X = (,, 3), where [X ] = [, 3 ] for [,]; hen he firs en erms of () can be expressed as follows: x x () = () ( () = (3 ) ( β Γ(β) β Γ(β) 3β Γ(3β) β Γ(β) 5β Γ(5β) 6β Γ(6β) 7β Γ(7β) 8β Γ(8β) 9β Γ(9β) ), β Γ(β) β Γ(β) 3β Γ(3β) β Γ(β) 5β 6β Γ(5β) Γ(6β) 7β Γ(7β) 8β Γ(8β) Γ(9β) ). (5) Clearly, (5) are he valid -cus of he soluion of (36). For numerical approximaion, we se [, ] and N =. ByusingAlgorihm, he resuls for differen values of β are ploed in Figure. Fromhegraphs,we can see ha if β approaches, he approximae soluions will approach he approximae soluion of fuzzy differenial equaion. Numerical values a =fordifferen values of β are lised in Table. Example. Consider he following linear fuzzy fracional differenial equaion: 9β c D β X () = X(), X () =X. The nonfuzzy problem associaed wih (6)is c D β x () = x(), x () =x. (6) (7)

7 The Scienific World Journal 7 Table : Numerical soluions of Example wih differen values of β. x β () β=.6 β=.8 β= () xβ () xβ () xβ () xβ () Afer simplifying, we ge X().5.5 L {x ()} = x s β s β. (5) By aking he inverse Laplace ransform o (5), we obain (a).5 x () =x L { sβ }, (5) s β 3 X() (b).5.5 which finally has he following soluion: x () =x E β ( β ), (5).5 where E β ( ) is he Miag-Leffler funcion. Using Zadeh s exension principle o (5) inrelaionox,weobainhe soluion of (6) as follows: 3 X() (c) Figure : The numerical soluion of (36)for(a)β =.6,(b)β =.8, and (c) β=. In order o find he soluion of (6), we firs find he soluion of (7). By aking he Laplace ransform on boh sides of (7), we have I follows ha.5.5 L { c D β x ()} = L {x ()}. (8) s β L {x ()} x( )s β = L {x ()}. (9) X () =X E β ( β ). (53) Le [X()] =[x (), x ()] and X = (, 3, ) where [X ] = [, ]for [,]; hen he firs en erms of (5)can be expressed as follows: x () = () ( β Γ(β) β Γ(β) 3β Γ(3β) β Γ(β) 6β Γ(6β) 7β Γ(7β) 8β Γ(8β) 9β Γ(9β) ), 5β Γ(5β)

8 8 The Scienific World Journal Table : Numerical soluions of Example wih differen values of β. x β () β=.6 β=.8 β= () xβ () xβ () xβ () xβ () x () = ( ) ( β Γ(β) β Γ(β) 3β Γ(3β) β Γ(β) 6β Γ(6β) 7β Γ(7β) 8β 9β 5β Γ(5β) Γ(8β) Γ(9β) ). (5).5 3 X() (a) Clearly, (5) arehevalid-cus of he soluion of (6). By using Algorihm wih he same inerval and inerval N as in Example,henumericalsoluionsof(6)fordifferen values of β are ploed in Figure. Again, we can see ha he numerical soluions will approach he numerical soluion of fuzzy differenial equaion as β increases o. Numerical soluions a =for differen values of β are lised in Table. 3 X() (b).5.5 In he following example, we provide deailed procedures for solving a nonlinear fuzzy fracional differenial equaion. Example 3. Le us consider he following problem: c D β X () = cos (X), X () =X, (55) where β (, ], =[,5],andX is any riangular fuzzy number. Tosolvehisproblem,weuseAlgorihm. Firs,le [X ] =[x,,x, ]. We discreise up o poins, which are =< =.< < =.LeX =W ;henwe have w 9 w, =g(β,h,,w, )=g(β,h,,w, )=w,,, = min [ [ u [w 9 min g(β,h,,u),w,,w, ],, min g(β,h, u [w,u)],,,w 9, ] ].5 3 X() (c) Figure : The numerical soluions of (6) for(a)β =.6, (b)β=.8,and(c)β=. w 9, = max [ [ w.5.5 max g (β, h,,u),w,,w, ],, max g (β, h, u [w,u)],,,w 9, ] ]. u [w 9 N, = min [ min g(β,h, u [w N,,w N,u),w N, ] N,, N, = max [ max g(β,h, u [w N,,w N,u),w N, ] w N,, min u [w N,,w max u [w N,,w N, ] g(β,h, N,u)], N, ] g(β,h, N,u)], (56)

9 The Scienific World Journal 9 Table 3: Numerical soluions of Example 3 wih differen values of β. x β () β=.6 β=.8 β= () xβ () xβ () xβ () xβ () X() 3 X() 3 5 (a) 3 5 see ha he numerical soluions approach o he numerical soluion of fuzzy differenial equaion as β approaches. Numerical soluions a = 5a differen values of β are lised in Table Conclusions In his paper, we have sudied a fuzzy fracional differenial equaion and presened is soluion using Zadeh s exension principle. The classical fracional Euler mehod has also been exed in he fuzzy seing in order o approximae he soluions of linear and nonlinear fuzzy fracional differenial equaions. Final resuls showed ha he soluion of fuzzy fracional differenial equaions approaches he soluion of fuzzy differenial equaions as he fracional order approaches he ineger order..5 3 X() (b) (c) 3 5 Figure 3: The approximae soluions of (55) for(a) β =.6, (b) β =.8,and(c)β=. where g(β,h, i,u)=u h β Γ(β) cos ( iu), u [w j i,,w j i, ] (57) for i=,,...,nand j=,,...,. Le X = (, π/, π) and N = ; hen hese procedures will resul in he approximae soluions of (55) a differen values of β as ploed in Figure 3. Fromhegraphs,wecan Acknowledgmens This research was suppored by he Shor Term Gran (STG) of Universii Malaysia Perlis (UniMAP) under he Projec code 9-39 and parially suppored by Research Acculuraion Gran Scheme (RAGS) under he Projec code References [] G. Jumarie, Table of some basic fracional calculus formulae derived from a modified Riemann-Liouville derivaive for nondiffereniable funcions, Applied Mahemaics Leers, vol., no.3,pp ,9. [] J. J. Trujillo, M. Rivero, and B. Bonilla, On a Riemann-Liouville generalized Taylor s formula, Mahemaical Analysis and Applicaions,vol.3,no.,pp.55 65,999. [3] G. A. Anasassiou, Opial ype inequaliies involving Riemann- Liouville fracional derivaives of wo funcions wih applicaions, Mahemaical and Compuer Modelling, vol.8,no.3-, pp. 3 37, 8. [] Y. S. Liang, Box dimensions of Riemann-Liouville fracional inegrals of coninuous funcions of bounded variaion, NonlinearAnalysis:Theory,Mehods&Applicaions,vol.7,no., pp. 3 36,.

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