SOLVING MULTI-TERM ORDERS FRACTIONAL DIFFERENTIAL EQUATIONS BY OPERATIONAL MATRICES OF BPs WITH CONVERGENCE ANALYSIS

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1 Roanian Reports in Physics Vol. 65 No. 2 P SOLVING ULI-ER ORDERS FRACIONAL DIFFERENIAL EQUAIONS BY OPERAIONAL ARICES OF BPs WIH CONVERGENCE ANALYSIS DAVOOD ROSAY OHSEN ALIPOUR HOSSEIN JAFARI 2 DUIRU BALEANU 34 Departent of atheatics Ia Khoeini International University P.O. Box Qazvin Iran E-ails: rostay@haya.ut.ac.ir.alipour2323@gail.co 2 Departent of atheatics University of azandaran P.O. Box Babolsar Iran E-ail: afari@uz.ac.ir 3 Departent of atheatics and Coputer Science Canaya University 653 Anara urey 4 Institute of Space Sciences P.O.BOX G-23 RO-7725 agurele-bucharest Roania E-ail: duitru@canaya.edu.tr Received July Abstract. In this paper we present a nuerical ethod for solving a class of fractional differential equations (FDEs). Based on Bernstein Polynoials (BPs) basis new atrices are utilized to reduce the ulti-ter orders fractional differential equation to a syste of algebraic equations. Convergence analysis is shown by several theores. Illustrative exaples are included to deonstrate the validity and applicability of this ethod. Key words: Bernstein polynoials fractional differential equations operational atrix Caputo derivative convergence analysis.. INRODUCION Fractional differential equations (FDEs) are generalizations of ordinary differential equations to an arbitrary (non-integer) order. hese equations have attracted considerable interest because of their ability to odel coplex phenoena. Also we observe that these equations capture nonlocal relations in space and tie with power-law eory ernels. here are several (nonequivalent) definitions of the fractional derivative in widespread use and we choose to focus on one particular for (the so-called Caputo version ) in this paper []. he use of fractional orders differential operators and integral operators in atheatical odels has becoe increasingly widespread in recent years (see for exaple Refs. [2 3] and the references therein). Several fors of fractional differential equations have been proposed in standard odels and there has been significant interest in developing nuerical schees for their solution [2 2].

2 2 Solving ulti-ter orders fractional differential equations 335 In this paper we focus on the ulti-ter orders fractional differential equations as follows: with the initial conditions β () = D yt () = a() td yt () + a() t yt () + f() t () i y () = bi i= (2) where >β >β 2 > >β > are constant and denotes the sallest integer a () t f () t (as an input signal) are nown greater than or equal to. Also { } = functions and unnown function yt () is as an output response. Now we use the initial conditions to reduce proble () and (2) to a proble with zero initial conditions. herefore we define yt () = yt ˆ() + zt () (3) where y ˆ( t) is a nown function that satisfied the initial conditions (2) and z (t) is a new unnown function. Substituting (3) in () and (2) we have the following initial-value proble where g (t) is nown function: with the initial conditions β (4) = D zt () = a() td zt () + a() tzt () + gt () () i z () = i=. he rest of this paper is as follows. In Section 2 we present soe definitions and preliinaries in fractional calculus and BPs. hen we approxiate functions by using BPs and we discuss convergence analysis. Also we obtain BPs operational atrix for product in Section 3. We present an operational atrix for fractional integration by BPs in Section 4. In Section 5 we apply BPs for solving linear ulti-order fractional differential equation. In Section 6 we discuss on the convergence of the proposed ethod. In Section 7 nuerical exaples are siulated to deonstrate the high perforance of the proposed ethod. Finally Section 8 concludes our wor in this paper. (5)

3 336 Davood Rostay ohsen Alipour Hossein Jafari Duitru Baleanu 3 2. BASIC DEFINIIONS In this section soe basic definitions and properties of the fractional calculus and BPs are briefly. Definition 2. []. he Rieann-Liouville fractional integral operator of order of a function f C µ µ is defined as I f t t x f x x t Γ ( ) I f t t ( ) = ( ) ( )d > > () = f(). t (6) Definition 2.2 []. he fractional derivative of f () t in the Caputo sense is defined as n n t n ( n) D f() t = I D f() t = ( t x) f ( x)d x Γ( n ) (7) n for n < n n Ν t > f C. n If n < n n Ν and f C µ µ then [3 5]). D I f() t = f() t (8) n ( ) + = x 2. I D f( t) = f( t) f ( ) t >. (9)! In the following we define the Bernstein polynoials (BPs) of -th degree on the interval [] as follows: i i Bi ( x) = x ( x) i= () i B ( x) B ( x) B ( x) is a coplete oreover we now that set { } basis in Hilbert space L 2 [ ]. herefore for each polynoial of degree we can write We recall that [6] Px ( ) = cb ( x) () i i i= i i i+ Bi ( x) = ( ) x = i

4 4 Solving ulti-ter orders fractional differential equations 337 for i=. Defining Φ ( x) = B ( x) B ( x) B ( x) we obtain where the eleents of the invertible atrix A ( ai ) and ( x) = x x Φ ( x) = A ( x) (2) + = are defined as i = i i ( ) i ai+ + = i i i = (3) i>. = and y L 2 [ ] 2 S is a coplete subset of [ ] Suppose that S Span { B B B }. Since S is a finite diensional and closed subspace then L. herefore y has the unique best approxiation out of S such as s S that is c i i= we have the unique coefficients { } where c= Q y( x) Φ ( x)d x such that (see [7]) i i i= yx ( ) s( x) = cb ( x) = cφ ( x) (4) + and Q ( Qi ) i = = i Qi+ + = B i ( x) B ( x)d x= i =. (5) 2 (2 + ) i+ LEA 2.3. Suppose that the function y : [ ] R continuously differentiable.. ie y C + ([ ]) { } ( ) is + ties and S = Span B B B. If c B be the best approxiation y out of then ˆ K y c B 2 (6) L [ ] ( + )! 2+ 3 S

5 338 Davood Rostay ohsen Alipour Hossein Jafari Duitru Baleanu 5 =. Also if y C ([ ] ) where ˆ ( + ) K ax y ( x) x [ ] vanishes. Proof. See [8]. then the error bound 3. BPS PRODUC OPERAIONAL ARIX In this section siilar to [6] we obtain a forula for BPs operational atrix of product. oreover we survey the error distribution in this approxiation. Suppose that c= c ( + ) is an arbitrary vector. Now we obtain the atrix Cˆ = C ˆ ( + ) ( + ) where Fro (2) we have [6] c Φ ( x) Φ ( x) Φ ( x) C. (7) ˆ ( ) c Φ ( x) Φ ( x) = c Φ ( x) ( x) A = = cb i i ( x) cxb i i ( x) cx i Bi ( x) A. i= i= i= (8) hus we define e i = e i e i e i then by (4) we can write x Bi ( x) = e i Φ ( x) + Ei i = (9) where E i is the approxiation error. So we get [6] ( ( ( )) ( )d ) e = Q x B x Φ x x = i i Q i = i+ i+ + i+ + i =. hen we have ( ) ( i ( )) ( ) cx i Bi x = ci e B x + Ei =Φ x V+ c+ ce i i i= i= = i= (2)

6 6 Solving ulti-ter orders fractional differential equations 339 where V + ( = ) is an ( + ) ( + ) atrix that has vectors e i ( i = ) for each colun s. If we define C = [ Vc V2c V+ c] fro (8) and (2) we can write where c Φ ( x) Φ ( x) =Φ ( x) CA + E P EP = ce i i ce i i ce i i A. (2) i= i= i= herefore we obtain the operational atrix of product C ˆ = CA. LEA 3.. If E P is the approxiation error for product in (7) then we have EP as. Proof. Fro (9) and Lea 2.3 we have E as for i =. herefore fro (2) it is clear that E as. P i 4. BPS OPERAIONAL ARIX FOR HE FRACIONAL INEGRAION Now we want to the operational atrix for the fractional integration. We can write: I Φ () t = t Φ() t t (22) Γ ( ) where denotes the convolution product. Fro (2) we have We can get ( ) ( ) t Φ () t = t A () t = A t () t. (23) t ( t) t t t t t ( ) I I t I t!! +! + = =Γ = where ( ) ( ) =Γ( ) t t t = (24) Γ+ ( ) Γ+ ( 2) Γ+ ( + ) =Γ( ) D D + + and ( + ) are as follows:

7 34 Davood Rostay ohsen Alipour Hossein Jafari Duitru Baleanu 7 i! i= Di = + + Γ+ ( i + ) i = = t t t. (25) i + i Now we need to approxiate t ( i= ) with respect to BPs by using (4). herefore we have + i t = E Φ () t + E (26) where i i i E is the approxiation error for i = and E = Q E where Ei = [ Ei Ei Ei ] + i! Γ ( i+ ++ ) Ei = t B ()d t t =! Γ ( i+ ++ 2) i= and =. Now we suppose E is an ( + ) ( + ) atrix that has vector E i ( i= ) for ith colun s. herefore we can write I Φ () t = ADE Φ () t + E I where E I = AD E E E. (27) Finally we obtain I Φ () t F Φ (). t (28) where F = ADE is called the Bernstein polynoials operational atrix of fractional integration. LEA 4.. If E I is the approxiation error for fractional integration in (28) then we have E as. I Proof. Fro (26) and Lea 2.3 we have E i as for i =. herefore fro (27) it is clear that E as. I i i 5. BPS FOR LINEAR ULI-ORDER FRACIONAL DIFFERENIAL EQUAION By using (4) the input signal g() t D zt () a () t ( = ) in (4) ay be expanded as follows:

8 8 Solving ulti-ter orders fractional differential equations 34 g t G Φ t (29) () () D z t C Φ t (3) () () a t A Φ t (3) () () where G A are nown ( + ) colun vectors and C is an unnown ( + ) colun vector. Fro (3) and (28) we have β β β ( ) D z() t = I D z() t I C Φ () t β C I t C F β t = Φ () Φ (). Now by substituting (29)-(3) and (32) into (4) we obtain ( β ) Φ = Φ Φ + Φ Φ + Φ = C () t A () t () t F C A () t () t F C G (). t (32) hen fro (7) we have ( ) ( ) ( ) ˆ A Φ x Φ x =Φ x A (33) where Aˆ ( = ) is a ( + ) ( + ) atrix. herefore by (33) we get ˆ ( β ) Φ ˆ = Φ +Φ + Φ = C () t () t A F C () t A F C G (). t (34) Finally we have the following linear syste: ( β ) ˆ ˆ Ι A F + A F C = G (35) = that by solving this linear syste we can obtain the vector C. hen we can get herefore fro (3) we have zt = I D zt C I Φ t C FΦ t (36) () () () (). yt yt ˆ + C FΦ t (37) () () (). 6. CONVERGENCE ANALYSIS In this section we investigate the convergence analysis for the ethod presented in section 5.

9 342 Davood Rostay ohsen Alipour Hossein Jafari Duitru Baleanu 9 By using (9) proble (4) change to the following proble β = D zt () = a() ti D zt () + a() ti D zt () + gt (). (38) By taing ut ( ) = D zt ( ) we obtain the following fractional integral equation β (39) = ut () = a () ti ut () + a() ti ut () + gt (). If we use the approxiation ut () C Φ () t then the proble (39) fro space C [] reduce to the following proble in space S ( β ) ( ) = C Φ () t = a () t C I Φ () t + a () t C I Φ () t + g(). t (4) Now siilar to heore in [6] we can propose the following heore. * HEORE 6.. Suppose that u () t C [ ] is the exact solution of the Eq. (39) and µ = J[ u ] = inj[ u] where u S hen we have β (4) = J[ u] = u() t a () t I u() t a () t I u() t g() t. * µ as ( ie.. u () t u () t as ). Proof. Since * * u () t is solution of the proble (39) we have Ju [ ] =. herefore we can write in Ju [ ] =. u C [ ] Also µ + µ because S S +. For that ε = exist C [ ] such J [ ] < ε. (42) Since J[ u ] is the continuous functional on C [ ] for all ε > then exist a δε ( ) such that

10 Solving ulti-ter orders fractional differential equations 343 u <δ( ε ) J[ u] J[ ] <ε. (43) Since polynoials space is dense in the space C [ ] the set of Bernstein polynoials on [ ] for a basis for the Space C [ ] and for sufficiently large v S exist such that v <δ( ε ) therefore fro (43) we have Fro (44) and (42) we get On the other hand we have herefore using (45) and (46) we obtain Jv [ ] J [ ] <ε. (44) Jv [ ] < J [ ] +ε < 2 ε. (45) Jv [ ] µ. (46) µ < 2 ε. (47) Now obviously such li µ =. herefore the proof is coplete. * HEORE 6.2. Suppose that u () t C [ ] is the exact solution of the Eq. (39) and u () t S is the obtained solution of Eq. (34). hen we have u t u t as. * () () Proof. Substituting (29) (3) and (28) in (4) we have C Φ () t = A Φ () t + e F Φ () t + E C + = (48) + Φ () + Φ () + + Φ () +. ( ) ( β ) A t e ( F t E C) ( G t eg ) ( ) ( ) Fro Lea 2.3 e ( = ) e as and using Lea 4. g E ( = ) as. So we can observe that as increases Eq. (48) gets close to Eq. (4). Now by taing = we propose the following proble that is gets close to (4) as increases ( β ) = C Φ () t = A Φ () t Φ () t F C + A Φ () t Φ () t F C + G Φ (). t (49) hen by (7) the Eq. (49) reduce to the following equation

11 344 Davood Rostay ohsen Alipour Hossein Jafari Duitru Baleanu ( ˆ ( ) ) ( ˆ β ) Φ = Φ + + Φ + + Φ = C () t () t A E F C () t A E F C G (). t (5) he Eq. (5) gets to (49) as because fro Lea 3. E ( = ) as. hen by taing = and deleting E ( = ) in (5) we get the Eq. (34). Obviously if u ( t ) is solution of Eq. (4) then we have u u as. * On the other hand fro heore 6. we obtained u u as. * herefore we can write u u as and proof is coplete. 7. NUERICAL EXAPLES In this section we apply our ethod to solve the following exaples. We define y () t and yt () for the approxiate solution and the exact solution in proble () respectively. EXAPLE 7.. We consider the coposite fractional oscillation equation 2 D yt () + D yt () + yt () = 8 < t with the initial conditions y() = y () =. Fig. Plot of y () t for different in Exaple 7..

12 2 Solving ulti-ter orders fractional differential equations 345 able Nuerical results for =.5 in Exaple 7. with coparison to Refs. [92] t BPs (ours) BPFs [2] AD [] FD [9] Exact able 2 Nuerical results for =.5 in Exaple 7. with coparison to Refs. [92] t BPs (ours) BPFs [2] AD [] FD [9] Exact his proble was solved in [92] for =.5 and =.5. We copare the nuerical results with Refs. [92] that are given in able and 2 (where the exact solution refers to the closed for series solution presented in [] by taing N = 3 ). Also Fig. shows the plot of y () t for different. We see that our ethod is very effective and accuracy of approxiate solutions in this ethod are in high agreeent with results obtained using the FD and better than those obtained using the BPFs and AD EXAPLE 7.2. Consider the equation D y() t = t y() t + 4 t t π < t with the following initial conditions y() = y () =.

13 346 Davood Rostay ohsen Alipour Hossein Jafari Duitru Baleanu 3 Fig. 2 Plot of y2() t y4() t and yt () for Exaple 7.2. t able 3 Absolute error for different in Exaple We now that the exact solution is yt () = t. he obtained results of BPs for = 2 4 are reported in able 3 and are plotted in Fig. 2. We observe that our ethod is very effective. EXAPLE 7.3. We consider the following linear fractional differential equation D y() t t D y() t td y() t ty() t = 6 πt 8 t t t π 5 3 < t subect to y() = y () =. With the exact solution yt () = π t. able 4 is reported the absolute errors for different values of t and Fig. 3 shows the absolute error for our ethod.

14 4 Solving ulti-ter orders fractional differential equations 347 Fig. 3 Plot of absolute error function for = in Exaple 7.3. able 4 Our results for = in Exaple 7.3 t Absolute error EXAPLE 7.4 []. Consider the equation 2 () () 2 ad y t + b t D y() t + c() t Dy() t + e() t D y() t + () t y() t = f () t < t and 2 bt () 2 et () 2 2 t f() t = a t c() t t t + () t 2 with the Γ( 3 2) Γ(3 ) 2 initial conditions y() = 2 y () =. 2 t We now that the exact solution is yt () = 2. For =.5 2 = and a = b( t) = t c( t) = t e( t) = t ( t) = t the obtained results of BPs are reported in able 5 and are plotted in Fig 4. We observe that our solutions are in perfect agreeent with the exact solutions.

15 348 Davood Rostay ohsen Alipour Hossein Jafari Duitru Baleanu 5 Fig. 4 Plot of yt () and y () t in Exaple 7.4. able 5 Our results for = in Exaple 7.4 t Absolute error CONCLUSION In this paper we proposed a nuerical solution for the linear ulti-ter orders fractional differential equation by the operational atrices of BPs. We get operational atrices of the product and fractional integration. hen by using these atrices we reduced the linear ulti-ter orders fractional differential equation to a syste of algebraic equations that can be solved easily. Finally nuerical exaples are siulated to deonstrate the high perforance of proposed ethod. We see that the obtained results are in good agreeent with the existing ones in open literature and it is shown that the technique introduced here is robust accurate and easy to apply. Acnowledgeents. his wor is partially supported by the Scientific and echnical Research Council of urey.

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