ISSN 749-3889 (prin), 749-3897 (online) Inernaional Journal of Nonlinear Science Vol.23(27) No.3, pp.8-92 Numerical Soluion of Fuzzy Fracional Differenial Equaions by Predicor-Correcor Mehod T. Jayakumar, T. Muhukumar, D. Geehamani Deparmen of Mahemaics Sri Ramakrishna Mission Vidyalaya College of Ars and Science Coimbaore-64 2, Tamilnadu, India (Received 26 Ocober 25, acceped 9 January 27) Absrac:In his paper we sudy a numerical mehod for fuzzy fracional differenial equaions using Caupo fracional derivaive by an Predicor-Correcor mehod. In addiion, his mehod is illusraed by solving some numerical examples. Keywords: Fuzzy sysem; Fuzzy Fracional Differenial Equaions; Predicor-Correcor Mehod Inroducion Agarwal e, al. [] have aken an iniiaive o inroduce he concep of soluion for Fuzzy Fracional Differenial Equaions(FFDEs). This conribuion has moivaed several auhors o esablish some resuls on he exisence and uniqueness of soluion [2]. Allahviranloo e, al. [3] derived he explici soluion of FFDEs using he Riemann-Liouville H-derivaive. Recenly, Salahshour e, al. [22] applied fuzzy Laplace ransforms [4] o solve FFDEs. Basically, he proposed ideas are a generalizaion of he heory and soluion of fuzzy differenial equaions [6, 7, 8, 4, 5, 23]. However, he auhors considered FFDEs 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 paricular Ahmad e, al. [5] have discussed numerical soluion of FFDEs Euler mehod using Zadeh s exension principle. The heory and applicaion of fracional differenial equaions under boh ypes of fracional derivaives have been discussed by many auhors [9,, 2, 3, 6, 7, 8, 9, 2, 24]. The srucure of his paper is organized as follows. In secion 2. we bring definiions o fuzzy valued funcions. In secion 3 we define fuzzy fracional differenial sysems. In secions 4 and 5, we presen he soluion of fuzzy fracional differenial equaions analyically and numerically using Predicor-Correcor mehod. The proposed algorihm is illusraed by solving some examples in secion 6. 2 Preliminaries In his secion, some definiions and basic conceps which will be used in his paper. Le I = [, ] R be as compac inerval and le E n denoe he se of all u : R n I such ha u saisfies he following condiions (i) u is normal ha is here exiss an x R n such ha u(x ) =, (ii) u is fuzzy convex, (iii) u is upper semiconinuous, (iv) [u] = cl{x R n : u(x) > } is compac. Then, from (i) - (iv), i follows ha he level se [u] P k (R) n for all. If g : R n R n R n is a funcion, { hen using } Zadeh s exension principle [ we ] can exend g o E n E n ) E n by he equaion g(u, v)(z) = sup min u(x), u(y) I is well known ha g (u, v) = g ([u], [v]. For all z=g(x,y) u, v E n,, and coninuous funcion g. Furher we have Corresponding auhor. jayakumar.hippan68@gmail.com,jayakumar.hippan68@gmail.com, vmuhukumar@gmail.com: Copyrigh c World Academic Press, World Academic Union IJNS.27.6.5/963
82 Inernaional Journal of Nonlinear Science, Vol.23(27), No.3, pp. 8-92 [u + v] = ([u] + [v] ), [ku] = k[u]. where k R. The real numbers can be embedded in E n by he rule c ĉ() where, for =c, ĉ() =, elsewhere. Definiion A real funcion x(), >, is said o be in he space C µ, µ R, if here exis a real number ρ > µ, such ha x() = p x (), where x () C(, ) and i is said o be in he space C n µ if and only if x n C µ, n N. Definiion 2 The Capuo fracional derivaive of x of order q > wih a is defined as c D q ax() = for n < q n, n N, a, x C n. Γ(n q) a ( s) n q x n (s)ds, Two basic properies of he Capuo fracional derivaive are as follows: (i) Le x C n, n N. Then c D q ax, q n, is well defined and c D q ax C, (ii) Le n < q n, n N, and x C n µ, µ. Then n Ia( q c Da)x() q = x() k= x (k) (a) The Laplace ransform of he Capuo fracional derivaive is given by ( a)k. n L { c Dax()} q = s q x(s) s q k x k (), n < q n. k= There exis a relaion beween he Riemann-Liouville fracional derivaive and Capuo fracional derivaive, n c Da+x() q = Da+x() q x k (a) Γ(k q + ) ( a)k q. k= Theorem Le f(x) C F [, a] L F [, a], be a fuzzy valued funcion. The Riemann-Liouville inegral of he f(x), based on is cu represenaion can be expressed as follows: [ ] [ ] J q f(x) = J q f (x), J q f (x),, where J q f (x) = x f () Γ(q) (x ) q d x, q R +, J q f (x) = Γ(q) x f () (x ) q d x, q R +. 3 Fuzzy Cauchy Problem Consider Fuzzy Fracional Iniial Value Problem (FFIVP) c Da x() q = f(, x()), < q, > a, x( ) = x. () IJNS email for conribuion: edior@nonlinearscience.org.uk
T. Jayakumar e.al: Numerical Soluion of Fuzzy Fracional Differenial Equaions by Predicor-Correcor Mehod 83 where x() is a fuzzy funcion of, f(, x()) is a fuzzy funcion of he crisp variable and he fuzzy variable x(), c Da x() q is he fuzzy Caupo fracional derivaive of x() and x( ) = x is a riangular or a riangular shaped fuzzy number. Therefore we have a fuzzy Cauchy problem. We denoe he fuzzy funcion x() by x() = [x(; ), x(; )]. I means ha he -level se of x() for [, T ] is [ x( ) [ ] ] = x( ; ), x( ; ), [ x() [ ] ] = x(; ), x(; ), (, ]. By using he exension principle of Zadeh s we have he membership funcion ( ) { } f, x() (s) = sup x()(τ) s = f(, τ), s R, (2) ( ) so f, x() is a fuzzy number. From his i follows ha [ [ ( ) ( )] f(, x() )] = f, x(;, f, x(;, (, ], (3) where ( ) { } f, x(); = min f(, u) u [x(; ), x(; )], ( ) { } f, x(); = max f(, u) u [x(; ), x(; )]. (4) 4 Analyical Soluion of Fuzzy Fracional Differenial Equaions Consider he following fracional differenial equaions c D q ax() = f(, x()), x( ) = x, (5) where f : [, x()] R R is a real valued funcion, x R, and q (, ]. If q =, hen (5) 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 q a x() = f(, x()), x( ) = x, (6) where x F (R). If q (, ]. If q =, hen (6) becomes an fuzzy differenial equaion. In order o find he soluion of (6), we firs find he soluion of (5). Taking Laplace ransform on boh sides of (5), we ge I follows ha Ł [ c D q ax()] = L [f(, x())]. (7) s q L {x()} x( )s q = L [f(, x())], L [x()] = m(s). (8) Then by aking he inverse Laplace ransform o (8), we have x() = L [m(s)] = g(, q, x ), (9) for [, T ] and x R. In order o find he soluion of (6), we fuzzify (9) using Zadeh s exension principle. Hence we have which is he soluion of (6). x() = g(, q, u) u (x, x ). () IJNS homepage: hp://www.nonlinearscience.org.uk/
84 Inernaional Journal of Nonlinear Science, Vol.23(27), No.3, pp. 8-92 Theorem 2 Le G be an open se in R and [ x ] F (R) G. Suppose ha f is coninuous and ha for each q (, ) and each x G here exis a unique soluion g(, q, x ) of he problem (5) and ha g(, q, x ) is coninuous in G for each [, T ] fixed. Then, here exis a unique fuzzy soluion x() = g(, q, u) u (x, x ) of he problem (6). Theorem 3 If X : [, T ] F (R) is a fuzzy soluion of (6) and denoing [ x()] = [x (; ), x (; )] for [, ] hen (i) [ x()] is compac subse of R, (ii) [ x()] 2 [ x()] for 2, (iii) [ x()] = n= [ x()] n for any nondecreasing sequence n in [, ]. Theorem 4 If x() = g(, q, x ) is obained by using Theorem 2 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 2, hen is membership funcion has he following form: sup x (), if y range(g), x g x()(y) = (,q,y), if y / range(g). I follows ha x(; ) = min{g(, q, u) u [x(; ), x(; )]}, x(; ) = max{g(, q, u) u [x(; ), x(; )]}, () for [, ]. I is obvious ha x(; ) x(; ). This holds for all [, ). This Complees he proof. 5 The Predicor-Correcor Algorihm for Fuzzy Fracional Differenial Equaions In his secion, we show he Predicor-Correcor algorihm of he following FFIVP c Da x() q = f(, x()), <, > a, x() = x. (2) where x() is a fuzzy funcion of, f(, x()) is a fuzzy funcion of he crisp variable and he fuzzy variable x(), c D q a x() is he fuzzy Caupo derivaive x() and x( ) = x is a riangular or a riangular shaped fuzzy fuzzy number. Using he Laplace ransformaion formula for he Caupo fracional derivaive. n L{ c Da} q = s q x(s) s q k x ( k)(), n < q < n. (3) k= from (2), we have s q x(s) n k= sq k x ( k)() = F (s, x(s), x(s)), s q x(s) n k= sq k x ( k)() = G(s, x(s), x(s)), (4) IJNS email for conribuion: edior@nonlinearscience.org.uk
T. Jayakumar e.al: Numerical Soluion of Fuzzy Fracional Differenial Equaions by Predicor-Correcor Mehod 85 or applying he inverse Laplace ransform gives x(s) = s q F (s, x(s), x(s)) + n k= a k x ( k)(), x(s) = s q F (s, x(s), x(s)) + n k= a k x ( k)(), (5) x() = q k= x k k + Γ(q) ( τ) q f(τ, x(τ), x(τ))dτ, where he fac x() = q k= { L c Da x() q = L Γ(µ) x k k + Γ(q) ( τ) q f(τ, x(τ), x(τ))dτ, (6) } { } x(τ) µ dτ = L ( τ) µ Γ(µ) x() = s µ x(s), and L{ µ } = s µ Γ(µ). are used. The approximaion is based on he equivalen from of hr Volerra inegral equaion (6). A fracional Adams Predicor-Correcor approach was firsly developed by [4] o numerically solve he problem (6). Using he sandard quadraure echniques for he inegral in (6), denoe g(τ) = f(τ, x(τ)), he inegral is replaced by he rapezoidel quadrraure formula a he poin n+ n+ ( n+ τ) q g(τ)dτ n+ ( n+ τ) q g n+ (τ)dτ, (7) where g n+ is he piecewise linear inerpolaion of g wih nodes j, j =,, 2,..., n +. Afer some elemenary calculaions, he righ hand side of (7) gives n+ ( n+ τ) q g n+ (τ)dτ = h q n+ a j,n+ g( j ), (8) q(q + ) where he uniform mesh is used and h is he sep size. And if we use he produc recangle rule, he righ hand of (8) can be wrien as where and n+ j= ( n+ τ) q g n+ n+ (τ)dτ = b j,n+ g( j ), (9) n q+ (n q)(n + ) q, if j = a j,n+ = (n j 2) q+ 2(n j + ) q+ + (n j) q+, if j n, if j = n + j= b j,n+ = hq q [(n + j)q (n j) q, if j n + ]. Then he predicor and correcor formula for solving (6) are given, respecively, by x p h ( n+) = q k= k n+ x ( k) + Γ(q) n b j,n+ F ( j, x h ( j ), x h ( j )), j= (2) IJNS homepage: hp://www.nonlinearscience.org.uk/
86 Inernaional Journal of Nonlinear Science, Vol.23(27), No.3, pp. 8-92 and x p h ( n+) = q k= x p h ( n+) = q k= k n+ x ( k) + Γ(q) n b j,n+ G( j, x h ( j ), x h ( j )), (2) j= k n+ x ( k) + Γ(q+2) n j= F ( j, x h ( j ), x h ( j )) + Γ(q+2) n j= a j,n+f ( j, x h ( j ), x h ( j )) x p h ( n+) = q k= k n+ x ( k) + Γ(q+2) n j= G( j, x h ( j ), x h ( j )) + Γ(q+2) n j= a j,n+f ( j, x h ( j ), x h ( j )) (22) (23) The approximaion accuracy of he scheme (2)-(22) is O(h min[2,q+] ). Now we make some improvemens for scheme (2)-(22). We modify he approximaion of (7) as n+ ( n+ τ) ( q ) g(τ)dτ n ( n+ τ) ( q ) g n (τ)dτ + n+ ( n+ τ) ( q ) g n (τ)dτ (24) where g n is he piecewise linear inerpolaion for g wih nodes and knos chosen a j, j =,, 2,..., n. Then using he sandard quadraure echnique,he righ hand of (24) can be wrien as where n ( n+ τ) ( q ) g n (τ)dτ + b j,n+ = n+ ( n+ τ) ( q ) g n (τ)dτ = a j,n+, if j n. h q q(q + ) 2 q+, if j = n if n > b, = q +, if n > Hence, his algorihm for he predicor sep can be improved as x p h ( n+) = q k= k n+ x ( k) + γ(2 q) n bj,n+, g( j ) (25) j= n b j,n+ F ( j, x h ( j ), x h ( j )), j= x p h ( n+) = q k= k n+ x ( k) + γ(2 q) n b j,n+ G( j, x h ( j ), x h ( j )), (26) j= The new predicor-correcor approach (26)and (22) has numerical accuracy O(h min[2,2q+] ). compuaional cos can be reduced, for < q, if we reformulae (26) and (22) as x + hq Γ(q+) F (, x h ( ), x h ( )), if n = x p h ( n+) = x + hq Γ(q+2) (2q+ )F ( n, x h ( n ), x h ( n )), Obiviously half of he (27) x p h ( n+) = + hq n Γ(q+2) j= a j,n+f ( j, x h ( j ), x h ( j )), if n. x + x + hq Γ(q+) G(, x h ( ), x h ( )), if n = hq Γ(q+2) (2q+ )G( n, x h ( n ), x h ( n )), + hq n Γ(q+2) j= a j,n+g( j, x h ( j ), x h ( j )), if n. (28) IJNS email for conribuion: edior@nonlinearscience.org.uk
T. Jayakumar e.al: Numerical Soluion of Fuzzy Fracional Differenial Equaions by Predicor-Correcor Mehod 87 Table : The approximae soluion bypredicor-correcor mehod o he FFIVP(3) - x(; ) for q =.5...2.3.4.5.6.7.8.9..83.73.534.894.2255.265.2975.3336.3696.457.447.2.38.362.459.4499.4938.5377.587.6256.6695.735.7574.3.547.5987.653.78.7534.85.8566.98.9597 2.3 2.628.4.7849.8444.939.9634 2.229 2.824 2.49 2.24 2.269 2.324 2.3798.5 2.38 2.6 2.739 2.249 2.398 2.3777 2.4457 2.536 2.585 2.6495 2.774.6 2.3 2.388 2.4652 2.5422 2.692 2.6963 2.7733 2.853 2.9274 3.44 3.84.7 2.677 2.6946 2.785 2.8684 2.9553 3.423 3.292 3.26 3.33 3.3899 3.4769.8 2.933 3.29 3.267 3.2244 3.322 3.498 3.575 3.653 3.73 3.87 3.984.9 3.2855 3.395 3.546 3.64 3.7236 3.833 3.9426 4.522 4.67 4.272 4.387. 3.674 3.7966 3.99 4.45 4.64 4.2865 4.489 4.534 4.6539 4.7764 4.8988 and x p h ( n+) = x + hq Γ(q+2) (F (, x p h ( ), x p h ( )) + qf (, x p h ( ), x p h ( ))) if n = ( +x + hq Γ(q+2) F (n+, x p h ( n+)) + (2 q+ 2)F ( n, x h ( n ), x h ( n )) ) (29) + hq n Γ(q+2) j=o a j,n+f ( j, x h ( j ), x h ( j )) if n x p h ( n+) = x + hq Γ(q+2) (G(, x p h ( ), x p h ( )) + qg(, x p h ( ), x p h ( ))) if n = ( +x + hq Γ(q+2) G(n+, x p h ( n+)) + (2 q+ 2)F ( n, x h ( n ), x h ( n )) ) (3) + hq n Γ(q+2) j=o a j,n+g( j, x h ( j ), x h ( j )) if n 6 Numerical Examples Example 5 Consider he following FFIVP c D q x() = x(), [, ], x() = (.75 +.25,.25.25), <. (3) where q (, ), >. By using (27)(28)(29) and (3) wih N=, we ge he approximae soluion as x(;)=4.8988. The exac soluion is given by where E q ( q ) = k= x(; ) = (.75 +.25)E q ( q ), x(; ) = (.25.25)E q ( q ), ( q ) k Γ(kq + ) = ( q ) k (kq)!. k= The approximae soluion by predicor-correcor mehod are ploed a [, ] and q=.5. (see ables - 2 and figure ) The exac and he approximae soluions by predicor-correcor mehod are compared and ploed a = and q=.5. (see ables - 4 and figure 2) IJNS homepage: hp://www.nonlinearscience.org.uk/
88 Inernaional Journal of Nonlinear Science, Vol.23(27), No.3, pp. 8-92 Table 2: The approximae soluion by predicor-correcor mehod o he FFIVP(3) - x(; ) for q =.5...2.3.4.5.6.7.8.9..629.639.5859.5679.5498.538.538.4958.4778.4597.447.2.977.955.9332.92.8892.8673.8453.8233.84.7794.7574.3 2.327 2.2949 2.269 2.2433 2.276 2.98 2.66 2.42 2.44 2.886 2.628.4 2.6773 2.6476 2.678 2.588 2.5583 2.5286 2.4988 2.469 2.4393 2.496 2.3798.5 3.57 3.23 2.9892 2.9552 2.922 2.8873 2.8533 2.893 2.7853 2.754 2.774.6 3.4666 3.428 3.3896 3.35 3.325 3.274 3.2355 3.97 3.585 3.2 3.84.7 3.95 3.868 3.8246 3.78 3.7376 3.6942 3.657 3.673 3.5683 3.52 3.3.4769.8 4.3969 4.348 4.299 4.254 4.25 4.527 4.38 4.55 4.6 3.9572 3.984.9 4.9283 4.8735 4.888 4.764 4.793 4.6545 4.5998 4.545 4.492 4.4355 4.387. 5.52 5.4499 5.3887 5.3275 5.2662 5.25 5.438 5.825 5.23 4.96 4.8988 Table 3: The exac soluion o he FFIVP(3) - x(; ) for q =.5...2.3.4.5.6.7.8.9..6726.654.6354.669.5983.5797.56.5425.5239.553.4868.2 2.239 2.4.9789.9564.9339.95.889.8665.844.825.799.3 2.372 2.3448 2.385 2.292 2.2658 2.2394 2.23 2.867 2.64 2.34 2.77.4 2.7338 2.734 2.673 2.6426 2.623 2.589 2.555 2.52 2.498 2.464 2.43.5 3.2 3.863 3.56 3.69 2.9822 2.9476 2.929 2.8782 2.8435 2.889 2.7742.6 3.5392 3.4998 3.465 3.422 3.389 3.3425 3.332 3.2639 3.2246 3.852 3.459.7 3.9938 3.9495 3.95 3.867 3.863 3.772 3.7276 3.6832 3.6388 3.5945 3.55.8 4.492 4.443 4.394 4.345 4.296 4.247 4.98 4.49 4.9 4.42 3.993.9 5.334 4.9774 4.925 4.8656 4.896 4.7537 4.6978 4.649 4.5859 4.53 4.474. 5.635 5.5725 5.599 5.4473 5.3847 5.322 5.2594 5.968 5.342 5.76 5.33 Table 4: The exac soluion o he FFIVP(3) - x(; ) for q =.5...2.3.4.5.6.7.8.9..5.522.894.2266.2637.39.338.3753.424.4496.4868.2.3493.3942.4392.4842.5292.574.69.664.79.754.799.3.588.6335.6862.7388.795.8442.8969.9496 2.23 2.55 2.77.4.8225.8833.944 2.48 2.655 2.263 2.87 2.2478 2.385 2.3693 2.43.5 2.86 2.5 2.293 2.2887 2.358 2.4274 2.4968 2.566 2.6355 2.748 2.7742.6 2.3594 2.438 2.567 2.5954 2.674 2.7527 2.833 2.9 2.9886 3.673 3.459.7 2.6626 2.753 2.84 2.9288 3.76 3.63 3.95 3.2838 3.3726 3.463 3.55.8 2.9935 3.932 3.93 3.2928 3.3926 3.4924 3.592 3.699 3.797 3.895 3.993.9 3.3556 3.4674 3.5793 3.69 3.83 3.948 4.267 4.385 4.254 4.3622 4.474. 3.7567 3.882 4.72 4.324 4.2576 4.3829 4.58 4.6333 4.7585 4.8838 5.33 IJNS email for conribuion: edior@nonlinearscience.org.uk
T. Jayakumar e.al: Numerical Soluion of Fuzzy Fracional Differenial Equaions by Predicor-Correcor Mehod 89 Predicor Correcor.9.8 Predicor Corrcor Mehod. Imp Eular o Predicor Corrcor Exac.8.7.6.6.4.2 r.5.4 4 3 y() 2 2.3.2. 3.5 4 4.5 5 5.5 6 6.5 y Figure : For h=. Figure 2: For h=. Example 6 Consider he following FFIVP c D q x() = x(), [, ], x() = (.75 +.25,.25.25), <. (32) where q (, ), >. By using (29) and (3) wih N=, we ge he approximae soluion as x(;)=.3924. The exac soluion is given by where E q ( q ) = x(; ) = (.75 +.25)E q ( q ), x(; ) = (.25.25)E q ( q ), k= ( q ) k Γ(kq + ) = k= ( q ) k (kq)!. The approximae soluion by predicor-correcor mehod are ploed a [, ] and q=.75. (see ables 5-6 and figure 3) The exac and he approximae soluions by predicor-correcor are compared and ploed a = and q=.75.(see ables 5-8 and figure 4) References [] R. P. Agarwal, V. Lakshmikanham and J. J. Nieo, On he concep of soluion for fracional differenial equaions wih uncerainy, Nonlinear Analysis: Theory, Mehods and Applicaions, 72(2):2859-2862. [2] S. Arshad and V. Lupulescu, On he fracional differenial equaions wih uncerainy, Nonlinear Analysis: Theory, Mehods and Applicaions,74(2):3685-3693. [3] T. Allahviranloo, S. Salahshour and S. Abbasbandy, Explici soluions of fracional differenial equaions wih uncerainy, Sof Compuing, 6(22):297-32. [4] T. Allahviranloo and M. B. Ahmadi, Fuzzy Laplace ransforms, Sof Compuing,4(2):235-243. IJNS homepage: hp://www.nonlinearscience.org.uk/
9 Inernaional Journal of Nonlinear Science, Vol.23(27), No.3, pp. 8-92 Table 5: The approximae soluion by predicor-correcor mehod o he FFIVP(32) - x(; ) for q =.75...2.3.4.5.6.7.8.9..933.92.97.93.89.8796.8693.8589.8486.8385.8279.2.8229.834.847.7955.7864.7772.768.7589.7498.746.735.3.7464.7334.725.769.786.74.692.6839.6757.6674.6592.4.676.6686.66.6536.646.6386.63.6235.66.685.6.5.626.647.678.69.594.587.582.5732.5663.5594.5525.6.5752.5688.5624.556.5496.5432.5368.534.524.577.53.7.535.529.5232.572.53.553.4994.4934.4875.485.4756.8.4999.4944.4888.4833.4777.4722.4666.46.4555.4499.4444.9.4689.4637.4585.4533.448.4429.4377.4325.4273.422.468..444.4365.436.4267.428.469.42.47.422.3973.3924 Table 6: The approximae soluion by improved Euler mehod o he FFIVP (22) in Example 7.2 - x(; ) for q =.75...2.3.4.5.6.7.8.9..629.646.6623.683.737.7244.745.7658.7865.872.8279.2.5486.5669.5852.635.628.64.6583.6766.949.732.735.3.4944.59.5273.5438.563.5768.5933.697.6262.6427.6592.4.457.4658.488.4958.58.5259.549.5559.579.586.6.5.444.4282.442.4558.4696.4835.4973.5.5249.5387.5525.6.3834.3962.49.428.4346.4473.46.4729.4857.4985.53.7.3567.3686.385.3924.442.46.428.4399.458.4637.4756.8.3333.3444.3555.3666.3777.3888.3999.4.4222.4333.4444.9.326.323.3335.3439.3543.3647.375.3856.396.464.468..2943.34.339.3237.3335.3433.353.3629.3727.3825.3924 Table 7: The exac soluion o he FFIVP(22) in Example 7.2 - x(; ) for q =.75...2.3.4.5.6.7.8.9..937.924.9.97.893.88.8696.8593.8489.8386.8282.2.824.85.858.7966.7875.7783.7692.76.759.747.7325.3.7428.7346.7263.78.798.76.6933.685.6768.6685.663.4.6773.6698.6623.6548.6472.6397.6322.6247.67.696.62.5.6228.658.689.62.595.5882.582.5743.5674.565.5536.6.5763.5699.5635.557.557.5443.5379.535.525.586.522.7.536.53.5242.582.523.563.53.4944.4884.4825.4765.8.59.4953.4898.4842.4786.473.4675.469.4564.458.4452.9.4698.4646.4594.4542.449.4437.4385.4333.428.4229.476..4422.4373.4324.4275.4225.476.427.478.429.398.393 IJNS email for conribuion: edior@nonlinearscience.org.uk
T. Jayakumar e.al: Numerical Soluion of Fuzzy Fracional Differenial Equaions by Predicor-Correcor Mehod 9 Table 8: The exac soluion o he FFIVP(22) in Example 7.2 - x(; ) for q =.75...2.3.4.5.6.7.8.9..622.648.6626.6833.74.7247.7454.766.7868.875.8282.2.5494.5677.586.643.6227.64.6593.6776.6959.742.7325.3.4952.57.5282.5447.562.5777.5943.68.6273.6438.663.4.455.4666.487.4967.58.5268.549.5569.572.587.62.5.452.429.4428.4567.475.4844.4982.52.5259.5397.5536.6.3842.397.498.4226.4354.4482.46.4738.4866.4994.522.7.3574.3693.382.393.45.469.4289.448.4527.4646.4765.8.3339.345.3562.3677.3785.3896.47.49.423.434.4452.9.332.3237.334.3445.355.3654.3759.3863.3968.472.476..2948.346.344.3243.334.3439.3538.3636.3734.3832.393 Predicor Correcor.9. Imp Eular Predicor Corrcor Mehod.8.8.7 o Predicor Corrcor Exac.6.6.4.2 r.5.4 4 3 y() 2 2.3.2..3.32.34.36.38.4.42.44.46 y Figure 3: For h=. Figure 4: For h=. IJNS homepage: hp://www.nonlinearscience.org.uk/
92 Inernaional Journal of Nonlinear Science, Vol.23(27), No.3, pp. 8-92 [5] M. Z. Ahamad, M. K. Hasan, Solving fuzzy fracional differenial equaions using Zadeh s exension principle,the Scienific World Journal,(23):-. [6] M. Z. Ahmad and M. K. Hasan, Numerical mehods for fuzzy iniial value problems under differen ypes of inerpreaion: a comparison sudy, in Informaics Engineering and Informaion Science, 252(2):275-288. [7] B. Bede, I. J. Rudas and A. L. Bencsik, Firs order linear fuzzy differenial equaions under generalized differeniabiliy, Informaion Sciences,77(27):648-662. [8] B. Bede and S. G. Gal, Generalizaions of he differeniabiliy of fuzzy - number - valued funcions wih applicaions o fuzzy differenial equaions, Fuzzy Ses and Sysems, 5(25):58-599. [9] K. Diehelm and N. J. Ford, Analysis of fracional differenial equaions, Journal of Mahemaical Analysis and Applicaions,265(22):229-248. [] V. S. Erurk and S. Momani, Solving sysems of fracional differenial equaions using differenial ransform mehod, Journal of Compuaional and Applied Mahemaics, 25(28):42-5. [] H. Jafari, H. Tajadodi and S. A. Hosseini Maikolai, Homoopy perurbaion pade echnique for solving fracional Riccai differenial equaions, The Inernaional Journal of Nonlinear Sciences and Numerical Simulaion, (2):27-276. [2] H. Jafari, H. Tajadodi, H. Nazari and C. M. Khalique, Numerical soluion of non-linear Riccai differenial equaions wih fracional order, The Inernaional Journal of Nonlinear Sciences and Numerical Simulaion,(2):79-82. [3] A. A. Kilbas, H. M. Srivasava and J. J. Trujillo, Theory and Applicaions of Fracional Differenial Equaions,Elsevier Science B. V, Amserdam, The Neherlands, 26. [4] O. Kaleva, Fuzzy differenial equaions, Fuzzy Ses and Sysems,24(987):3-37. [5] O. Kaleva, A noe on fuzzy differenial equaions, Nonlinear Analysis: Theory, Mehods and Applicaions, 64(26):895-9. [6] V. Lakshmikanham and R. N.Mohapara, Theory of Fuzzy Differenial Equaions and Applicaions, Taylor and Francis, London, UK, 23. [7] V. Lakshmikanham and A. S. Vasala, Basic heory of fracional differenial equaions,nonlinear Analysis: Theory, Mehods and Applicaions,69(28):2677-2682. [8] V. Lakshmikanham and S. Leela, Nagumo-ype uniqueness resul for fracional differenial equaions, Nonlinear Analysis: Theory, Mehods and Applicaions,7(29):2886-2889. [9] K. S. Miller and B. Ross, An Inroducion o he Fracional Calculus and Differenial Equaions, John Wiley and Sons, New York, NY, USA, 993. [2] Z. M. Odiba and S. Momani, An algorihm for he numerical soluion of differenial equaions of fracional order, Journal of Applied Mahemaics and Informaics,26(28):5-27. [2] I. Podlubny, Fracional Differenial Equaion, Academic Press, 999. [22] S. Salahshour, T. Allahviranloo and S. Abbasbandy, Solving fuzzy fracional differenial equaions by fuzzy Laplace ransforms,communicaions in Nonlinear Science and Numerical Simulaion,7(22):372-38. [23] S. Seikkala, On he fuzzy iniial value problem, Fuzzy Ses and Sysems,24(987):39-33. [24] S. Zhang, Monoone ieraive mehod for iniial value problem involving Riemann-Liouville fracional derivaives, Nonlinear Analysis: Theory, Mehods and Applicaions, 7(29):287-293. IJNS email for conribuion: edior@nonlinearscience.org.uk