A Numerical Method for Delayed Fractional-Order Differential Equations: Based on G-L Definition

Transcription

1 Appl. Math. If. Sci. 7, No. 2L, (213) 525 Applied Mathematics & Iformatio Scieces A Iteratioal Joural A Numerical Method for Delayed Fractioal-Order Differetial Equatios: Based o G-L Defiitio Zhe Wag 1,2, Xia Huag 3 ad Jiapig Zhou 4 1 College of Iformatio Sciece ad Egieerig, Shadog Uiversity of Sciece ad Techology, Qigdao 26659, Chia 2 School of Automatio, Naig Uiversity of Sciece ad Techology, Naig 2194, Chia 3 Key Laboratory of Robotics ad Itelliget Techology, College of Iformatio ad Electrical Egieerig, Shadog Uiversity of Sciece ad Techology, Qigdao 26659, Chia 4 School of Computer Sciece, Ahui Uiversity of Techology, Ma asha 2432, Chia Received: 19 Oct. 212, Revised: 4 Ja. 213, Accepted: 6 Ja. 213 Published olie: 1 Ju. 213 Abstract: I this paper, a umerical method for oliear fractioal-order differetial equatios with costat or time-varyig delay is devised. The order here is a arbitrary positive real umber, ad the differetial operator is the Grüwald-Letikov derivative. The detailed error aalysis for this algorithm is give, meawhile, the covergece of the iteratio algorithm is proved. Compared with the exact aalytical solutio, a umerical example is provided to illustrate the effectiveess of the proposed method. Keywords: Delay, fractioal-order differetial equatios, Grüwald-Letikov derivative 1 Itroductio Fractioal calculus is a old mathematical problem, ad maily developed as a pure mathematics problem for early three ceturies [1]-[3]. Though havig a log history, it was ot used i physics ad egieerig for a log period. However, i the last few decades, fractioal calculus bega to attract icreasig attetio of scietists from the viewpoit of applicatio [3]-[6]. Fractioal derivative has prove to be a very suitable tool for the descriptio of memory ad hereditary properties of various materials ad processes. I the fields of cotiuous-time modelig, may researchers poited out that fractioal derivative is very useful i liear viscoelasticity, acoustics, rheology, polymeric chemistry, etc [7, 12]. Nowadays, the mathematical theories ad practical applicatios of these operators are well established, ad their applicabilities to sciece ad egieerig are beig cosidered as a attractive topic. The developmet of effective ad well-suited methods for umerically solvig FDEs has draw more ad more attetio over the last few years. Several umerical methods based o Caputo or Riema-Liouville defiitio have bee proposed ad aalyzed [13]-[32]. For istace, based o the predictor-corrector scheme, Diethelm et al itroduced Adams-Bashforth-Moulto algorithm [17,18,19], ad meawhile some error aalyses ad a extesio of Richardso extrapolatio were also preseted to improve the umerical accuracy. These techiques elaborated i [17, 18, 19] take advatage of the fact that the FDEs ca be reduced to Volterra type itegral equatios. Ad therefore, oe ca apply the umerical schemes for Volterra type itegral equatios to fid the solutio of FDEs, the readers ca refer to [2]-[24] ad the literature cited therei for more details. The techique preseted i [17, 18, 19] has bee further aalyzed ad exteded to multi-term [25] ad multi-order systems [26]. Li et al. [32] studied the error aalysis of the fractioal Adams method for fractioal-order ordiary differetial equatios i more geeral case. Deg [14] obtaied a good umerical approximatio by combiig the short memory priciple with the predictor-corrector approach. However, i practice, delay is very ofte ecoutered i differet techical systems, such as automatic cotrol, biology ad hydraulic etworks, ecoomics, log trasmissio lies, etc. Cosequetly, delayed differetial equatios are used to describe such kids of dyamical systems. I recet years, delayed FDEs begi to arouse the attetio of a umber of researchers [33]-[35]. It is well kow that fidig robust ad stable umerical Correspodig author huagxia.qd@gmail.com c 213 NSP Natural Scieces Publishig Cor.

2 526 Z. Wag et al.: A umerical method for delayed fractioal-order differetial equatios: based o G-L defiitio methods for solvig the delayed FDEs is debatable ad has become a ope research questio. To the best of our kowledge, there are very few works devoted i the literature to this problem so far. Sice the Riema-Liuville derivative ad the Grüwald-Letikov derivative are the same i effect [3], this gives a approximate method for fractioal order derivatives. Therefore, this paper will study the umerical algorithm for the delayed FDEs based o the Grüwald-Letikov defiitio. The rest of this paper is orgaized as follows: I Sectio 2, basic defiitio ad prelimiaries i fractioal calculus are preseted. I Sectio 3, the umerical scheme is devised i the case of costat ad time-varyig delay, meawhile, the covergece of the proposed algorithm is proved. The iteratio algorithm is desiged i Sectio 4. Numerical simulatios are performed i Sectio 5 to illustrate the effectiveess of the proposed scheme. Fially, some cocludig remarks are reported i Sectio 6. 2 Basic defiitio ad prelimiaries There are several differet defiitios of fractioal itegratio ad differetiatio till ow [3]. The most frequetly used are the Grüwald-Letikov (G-L) defiitio ad the Riema-Liuville (R-L) defiitio. For a wide class of fuctios, the two defiitios G-L ad R-L are equivalet. I this paper, the Grüwald-Letikov defiitio will be used. Defiitio 1.[3]. The Grüwald-Letikov fractioal derivative of order α is defied as follows ad α t y(t) = lim h [ t a h ] = ( 1) ( α ) y(t h), where [ ] deotes the iteger part, α R is the order of the derivative, a is the iitial time, ad h is the samplig time. I [36], Lubich proposed a secod order umerical method for the fractioal derivative of order α ad α t y(t) 1 where ad = 1 2πi [ t a h ] = x 1 y(t h), (1) 2π W 2 (x) 1 dx = W x +1 2 (e iϕ )e iϕ dϕ, 2π W 2 (x) = ( 3 2 2x + x2 2 )α. 3 Formulatio of the umerical method for delayed FDEs The mai aim of this paper is to study a umerical scheme for the approximate solutio of delayed FDEs. For this purpose, we cosider delayed FDEs described as follows ad α t y(t) = f (t,y(t),y(t τ)), (a t b,m 1 < α m) (2) y(t) = φ(t), t a, where α is the order of the differetial equatio, φ(t) is the iitial value, ad m is a iteger. By the Grüwald- Letikov defiitio, the discrete form of system (3) ca be writte as h α = y(t ) + O(h 2 ) = f (t,y(t ),y(t τ)). (3) Now, the key problem is to establish the approximatio to the delayed term y(t τ), which cotais two cases, discussed as below. Case I: whe τ is costat It ca be see that, for ay positive costat τ, t τ may ot be a grid poit t for ay. Suppose (m+δ)h = τ, ad δ < 1. Whe δ =, y(t τ) ca be approximated by y m, i f > m, y(t τ) φ(t τ), i f m, ad whe < δ < 1, y(t τ) caot be calculated directly. By virtue of Taylor expasio ad the umerical differetiatio techique, we have y(t τ) = y(t (m + δ)h) = y(t m 1 + (1 δ)h) = y(t m 1 ) + y (t m 1 )(1 δ)h + O(h 2 ) (4) y(t m 1 ) + [ y(t m) y(t m 1 ) + O(h 2 )](1 δ)h h +O(h 2 ) = (1 δ)y(t m ) + δy(t m 1 ) + O(h 2 ). Let v be the approximatio to y(t τ), ad the umerical approximatio for the computatio of y(t τ) are proposed as follows v = y(t τ) (1 δ)y m + δy m 1. (5) It ca be see from (5) that, if m >, the umerical equatio is explicit, ad thus it ca be computed directly. However, whe m = ad δ 1, i.e., τ < h, the first term i the right-had side of the above equatio is (1 δ)y. It still eeds to predict, for this case, v is calculated as v = (1 δ) + δy 1 where l = 1,2, is the iteratio umber. Case II: whe τ is time varyig c 213 NSP Natural Scieces Publishig Cor.

3 Appl. Math. If. Sci. 7, No. 2L, (213) / If τ is time varyig, i.e., τ = τ(t), the the approximatio i this case is more complicated. Let v is the approximatio to y(t τ), ad the liear iterpolatio of y at poit t = t τ(t ) is employed to approximate the delay term. Let τ(t ) = (m + δ )h, where m is a positive iteger ad δ [,1), the v = (1 δ )y m + δ y m 1. (6) It should be oted that whe τ is costat, for give h ad τ, at the begiig of the programme, we ca udge whether m = or m >. However, whe τ is time varyig, m is also time varyig, that is, at oe momet it may equal to, but at aother momet it may greater tha. If m =, the first term i the right-had side of (6) still eeds to predict, otherwise, whe m >, it does ot eed. So i each step of the computig, it eeds to test weather m = or ot firstly, the computatio of the first term i the right-had side of Eq.(6) depeds o whether it requires predictig or ot i differet steps. By the above discussios, the umerical scheme for the FDEs (3) ca be depicted as: = y = f (t,y,v ), (7) (1 δ)y m + δy m 1, i f > m,m >, v = (1 δ) + δy 1, i f > m,m =, φ(t τ), i f m, where l = 1, 2, is the iteratio umber ad = 1,2,,N. Theorem 1.Suppose the fuctio f (t, y, u) satisfies the followig Lipschitz coditios (8) f (t,y 1,u) f (t,y 2,u) L 1 y 1 y 2, (9) f (t,y,u 1 ) f (t,y,u 2 ) L 2 u 1 u 2, (1) the for the fractioal order differetial equatio (3), the local trucatio error of the umerical scheme (7), (8) is O(h 2+α ). Proof. Sice the umerical scheme (7) ca be writte as y + we have =1 y = f (t,y,v ), (11) (y y(t )) + y(t ) + =1 y (12) = [ f (t,y,v ) f (t,y,y(t τ)) + f (t,y(t ),y(t τ)). (13) Suppose y = y(t ), =,1,, 1, the Eq.(13) ca be rewritte as (y y(t )) + = y(t ) = (14) f (t,y(t ),y(t τ)) + [ f (t,y,v ) f (t,y,y(t τ)). From Lubich umerical expressio of fractioal derivative (1), we have 1 ω(α) (y y(t )) + α D α t y(t) = [ f (t,y,v ) f (t,y,y(t τ))] (15) + f (t,y(t ),y(t τ)) + O(h 2 ), Elimiatig the idetical term i the above equatio, we have 1 ω(α) (y y(t )) = f (t,y,v ) f (t,y,y(t τ)) + O(h 2 ). (16) From the assumptios (9) ad (1), we get ( L ) y y(t ) O(h 2+α ). (17) Thus, for sufficiet small h, the local trucatio error is O(h 2+α ), which completes the proof. 4 Desig of the umerical algorithm It is obvious that(7), (8) is a implicit oliear algebraic equatio with respect to y. I order to solve y, we costruct the iteratio algorithm as follows: = 1 v (l 1) = ω α [ f (t,y (l 1) ) =1 y ], (18) (1 δ)y m + δy m 1, i f > m,m >, (1 δ)y (l 1) m + δy m 1, i f > m,m =, (19) φ(t τ), i f m, y = φ(a),y () = 1, (2) l = 1,2,, = 1,2,,N. (21) where l is the iteratio umber. If y (l 1) < ε (ε is the give error, e.g. ε = 1 6 ), we would cosider y as. Theorem 2.Suppose the fuctio f (t, y, u) satisfies the Lipschitz coditios (9), (1), the the iteratio algorithm (18)-(21) is coverget. c 213 NSP Natural Scieces Publishig Cor.

4 528 Z. Wag et al.: A umerical method for delayed fractioal-order differetial equatios: based o G-L defiitio Proof. Sice y (l 1) (22) = f (t,y (l 1) f (t,y (l 1) + f (t,y (l 2) L 1 y (l 1) L 1 y (l 1) ) f (t,y (l 2) ) f (t,y (l 2) ) f (t,y (l 2) + L 2 v (l 1) (L 1 + L 2 ) y (l 1) y (l 2) ( ) l 1 L y (1) y (),v (l 2),v (l 2) ) ) ) v (l 2) + L 2 (1 δ) y (l 1) ( the for sufficiet small h satisfyig ) L < 1, whe l, we have y (l 1), that is, the iteratio algorithm (18)-(21) is coverget. Now, we summarize the computig procedure for the iteratio algorithm as follows (1) For give iitial value y = φ(a), time legth h, ad, = 1, let t = a + h, compute. (2) If m >, break. Otherwise, compute τ = τ(t ), ω (α), the tolerate error ε, N = b a h sum = m ω (α) y. =1 (3) Let l = 1,y (l 1) = c, compute v. (4) Compute m = [sum f (t,y (l 1),v )]/. (5) If y (l 1) < ε, the y =, = + 1, ad retur to the step (2). Otherwise, tur ito the ext step. (6) Let l = l + 1, compute v,,v )]/ m = [sum f (t,y (l 1) step (5). 5 A umerical example, ad retur to the I this sectio, the followig delayed FDE is cosidered: Dt α y(t) = 2 Γ (3 α) t2 α 1 Γ (2 α) t1 α + 2τt τ 2 τ y(t) + y(t τ), α (,1), y(t) =, t. Notice that the exact solutio to this equatio is y(t) = t 2 t. I accordace with delay τ beig costat or time-varyig, the umerical results are displayed i Table 1 ad Table 2 respectively, where E A deotes the absolute umerical error ad E R deotes the relative umerical error. From the umerical results we ca see that the computig errors are i geeral acceptable for egieerig. Table 1. Numerical solutio, exact solutio ad the error estimate at time t = T whe h = 1/2 T y(t ) y E A E R Table 2. Error behavior at time t = T with aalytical value wit =.9,τ =.1e 1t T E h = 1/1 h = 1/2 h = 1/4 2 E A e e e-3 E R e e e-3 1 E A 3.599e e e-3 E R e e e-5 2 E A e e e-3 E R e e e-5 6 Coclusios I this paper, a umerical algorithm is formulated based o Grüwald-Letikov derivative for fractioal-order differetial equatios with time delay. The error aalysis of the umerical scheme is carried out, meawhile, a umerical example with costat delay ad time varyig delay is proposed to testify the effectiveess of the proposed scheme. This algorithm ca be used ot oly i the simulatio of delayed fractioal-order differetial equatios but also i the simulatio of delayed fractioal-order cotrol systems. Ackowledgemet This work was supported by the Natioal Natural Sciece Foudatio of Chia uder grats 61478, , ad the Natural Sciece Foudatio of Jiagsu Provice uder grat BK Refereces [1] P. L. Butzer, U.Westphal, A Itroductio to Fractioal Calculus, World Scietific, Sigapore, 2. [2] S. M. Keeth, R. Bertram, A Itroductio to the Fractioal Calculus ad Fractioal Differetial Equatios, Wiley- Itersciece Publicatio, US, [3] I. Podluby, Fractioal Differetial Equatios, New York: Academic, c 213 NSP Natural Scieces Publishig Cor.

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