Numerical Solution of Fuzzy Fractional Differential Equations by Predictor-Corrector Method

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1 ISSN (prin), (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 /963

2 82 Inernaional Journal of Nonlinear Science, Vol.23(27), No.3, pp [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 for conribuion: edior@nonlinearscience.org.uk

3 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://

4 84 Inernaional Journal of Nonlinear Science, Vol.23(27), No.3, pp 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 for conribuion: edior@nonlinearscience.org.uk

5 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://

6 86 Inernaional Journal of Nonlinear Science, Vol.23(27), No.3, pp 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 for conribuion: edior@nonlinearscience.org.uk

7 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 = 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() = ( ,.25.25), <. (3) where q (, ), >. By using (27)(28)(29) and (3) wih N=, we ge he approximae soluion as x(;)= The exac soluion is given by where E q ( q ) = k= x(; ) = ( )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://

8 88 Inernaional Journal of Nonlinear Science, Vol.23(27), No.3, pp Table 2: The approximae soluion by predicor-correcor mehod o he FFIVP(3) - x(; ) for q = Table 3: The exac soluion o he FFIVP(3) - x(; ) for q = Table 4: The exac soluion o he FFIVP(3) - x(; ) for q = IJNS for conribuion: edior@nonlinearscience.org.uk

9 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 r y() y Figure : For h=. Figure 2: For h=. Example 6 Consider he following FFIVP c D q x() = x(), [, ], x() = ( ,.25.25), <. (32) where q (, ), >. By using (29) and (3) wih N=, we ge he approximae soluion as x(;)= The exac soluion is given by where E q ( q ) = x(; ) = ( )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): [2] S. Arshad and V. Lupulescu, On he fracional differenial equaions wih uncerainy, Nonlinear Analysis: Theory, Mehods and Applicaions,74(2): [3] T. Allahviranloo, S. Salahshour and S. Abbasbandy, Explici soluions of fracional differenial equaions wih uncerainy, Sof Compuing, 6(22): [4] T. Allahviranloo and M. B. Ahmadi, Fuzzy Laplace ransforms, Sof Compuing,4(2): IJNS homepage: hp://

10 9 Inernaional Journal of Nonlinear Science, Vol.23(27), No.3, pp Table 5: The approximae soluion by predicor-correcor mehod o he FFIVP(32) - x(; ) for q = Table 6: The approximae soluion by improved Euler mehod o he FFIVP (22) in Example x(; ) for q = Table 7: The exac soluion o he FFIVP(22) in Example x(; ) for q = IJNS for conribuion: edior@nonlinearscience.org.uk

11 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 x(; ) for q = Predicor Correcor.9. Imp Eular Predicor Corrcor Mehod o Predicor Corrcor Exac r y() y Figure 3: For h=. Figure 4: For h=. IJNS homepage: hp://

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