It. J. Adv. Appl. Math. ad Mech. () (04) 83-9 ISSN: 347-59 Available olie at www.iaamm.com Iteratioal Joural of Advaces i Applied Mathematics ad Mechaics Numerical solutio of Bagley-Torvik equatio usig Chebyshev wavelet operatioal matrix of fractioal derivative Fakhrodi Mohammadi Departmet of Mathematics, Hormozga Uiversity, Badarabbas, Ira Research Article Received 04 July 04; accepted (i revised versio) 4 August 04 Abstract: MSC: I this paper Chebyshev wavelet ad their properties are employed for derivig Chebyshev wavelet operatioal matrix of fractioal derivatives ad a geeral procedure for formig this matrix is itroduced. The Chebyshev wavelet expasio alog with this operatioal matrix are used for umerical solutio of Bagley-Torvik boudary value problems. The error aalysis ad covergece properties of the Chebyshev wavelet method are ivestigated. 34K37 4C40 65T60 Keywords: Chebyshev wavelet Fractioal derivatives Operatioal matrix Bagley-Torvik equatio Tau method c 04 IJAAMM all rights reserved.. Itroductio The idea of derivatives of oiteger order was iitially appeared i a letter from Leibiz to L Hospital i 695. For three ceturies, studies of the theory of fractioal order were maily costrait to the field of pure theoretical mathematics, which were oly useful for mathematicias. I the last several decades, may researchers foud that derivatives of o-iteger order are very suitable for the descriptio of various physical pheomea such as dampig laws, diffusio process, etc. These fidigs ivoked the growig iterest of studies of the fractioal calculus i various fields such as physics, chemistry ad egieerig. For these reasos we eed reliable ad efficiet techiques for the solutio of fractioal differetial equatios [ 5]. The Bagley-Torvik equatio is a kid of fractioal differetial equatio that apeare i the studies o behavior of real material by use of fractioal calculus [6, 7]. It has may applicatios i egieerig ad applied scieces fields, for mor details see[3]. So, umerical solutio of Bagley-Torvik fractioal differetial equatio has attracted may attetio ad has bee studied by may authors. Several methods such as Adomia decompositio method [8, 9], HeâĂŹs variatioal iteratio method [0], Taylor collocatio method [] have bee used to solve this fractioal differetial equatio. Diethelm [] trasformed this equatio ito first-order coupled fractioal differetial equatio ad solved the problem with Adams predictor ad corrector approach. Podluby used successive approximatio method to solve the equatio ad recetly applied the matrix approach to discretizatio of fractioal derivatives for the same problem [3, 3]. Wavelets theory is a relatively ew ad a emergig area i mathematical research. As a powerful tool, wavelets have bee extesively used i sigal processig, umerical aalysis, ad may other areas. Wavelets permit the accurate represetatio of a variety of fuctios ad operators [4 6]. I this paper The Chebyshev wavelets are first itroduced, the by usig shifted Chebyshev polyomial ad their properties the operatioal matrix of derivative ad fractioal derivative are derived. The, applicatios of these operatioal matrices for solvig fractioal order Bagley-Torvik boudary value problems are described. Illustrative examples are give to demostrate the efficiecy Correspodig author. E-mail address: f.mohammadi6@hotmail.com 83
Numerical solutio of Bagley-Torvik equatio usig Chebyshev wavelet operatioal matrix of fractioal derivative ad capability of the proposed method. The error aalysis ad covergece properties of the Chebyshev wavelet method are ivestigated. The article is orgaized as follows: We begi by itroducig some ecessary defiitios ad mathematical prelimiaries of the fractioal calculus theory ad Chebyshev polyomials. I Sectio 3 the Chebyshev wavelet ad its operatioal matrix of derivatives ad fractioal derivative are derived ad geeral procedure for formig these matrices are itroduced. Fractioal order boudary value problems are itroduced i Sectio 4 ad the a method based o Chebyshev wavelet ad its operatioal matrices is establish for solvig this fractioal boudary value problems. Numerical examples are icluded i sectio 5. Fially, a coclusio is give i Sectio 6.. Basic defiitios There are various defiitios of fractioal itegratio ad differetiatio, such as Gruwald-Letikov s defiitio ad Riema-Liouville s defiitio. The Riema-Liouville derivative has certai disadvatages whe tryig to model real world pheomea with fractioal differetial equatios. Therefore, we shall itroduce a modified fractioal differetial operator proposed by Caputo i his work o the theory of viscoelasticity [3, 4]. Defiitio.. A real fuctio f (t ), t > 0, is said to be i the space C µ, µ if there exists a real umber p (> µ) ad a fuctio f (t )C [0, ) such that f (t ) = t p f (t ), ad it is said to be i the space C µ, N if f () C. Defiitio.. The Riema-Liouville fractioal itegratio operator of order α 0 of a fuctio f C µ,, is defied as: f (t ) = d f (t ) d t, α =, d t f (τ) d t (t τ) d τ, 0 < α <. α + Γ ( α) 0 () Defiitio.3. The fractioal derivative of order α > 0 i the Caputo sese is defied as f (t ) = d f (t ) t Γ ( α) 0 d t, α =, f () (τ) (t τ) d τ, t > 0, 0 < α <. α + () The useful relatio betwee the Riema-Liouvill operator ad Caputo operator is give by the followig expressio I α f (t ) = f (t ) f (k) (0 + ) t k, t > 0, ( < α ). (3) k! k=0 Also for the Caputo s derivative we have c = 0, i which c is a costat ad 0, < α, + t = Γ (+) Γ (+ α) t α, α, +, (4) we use the ceilig fuctio α to deote the smallest iteger greater tha or equal to α. For more details about fractioal calculus ad its properties see [3]... Shifted Chebyshev polyomials The sequece of Chebyshev polyomials {T (t )} =0 which are orthogoal with respect to the weight fuctio w (t ) = defied o the iterval [, ] ad they obey the followig recurrece relatio [0] t T 0 (t ) =, T (t ) = t, T (t ) = T (t ) T (t ),. for useig Chebyshev polyomials o the iterval [0, ] we defie the so-called shifted Chebyshev polyomials by itroducig the chage of variable t = x. So, the shifted Chebyshev polyomials T (x ) o [0,] ca be obtaied as 84 T (x ) = T (x ).
Fakhrodi Mohammadi / It. J. Adv. Appl. Math. ad Mech. () (04) 83-9 The orthogoality coditio for shifted polyomials is 0 γ m = T m (x )T (x ) πγm d x = 4, m =, (x ) 0, m,, m = 0,, m. (5) Lemma.. The aalytic form of the shifted Chebyshev polyomial T m (x ) of degree m is give by m T m (x ) = a mi x m i, (6) i =0 a mi = ( ) i m i m(m i )!. (7) (i )!(m i )! Proof. See [7]. 3. Chebyshev wavelets I this sectio, we itroduce Chebyshev wavelet ad its operatioal matrix. Chebyshev wavelets ψ m (x ) = ψ(k,, m, x ) are defied o the iterval [0, ) by k+ T ψ m (x ) = m ( k x ( + )), x + k k 0, o t he r w i s e, (8) π, m = 0 T m (x ) = π T m(x ), m > 0, ad = 0,,..., k, k, m Z M = {0,,,..., M } for a fixed positive iteger M. The Chebyshev wavelets {ψ m (x ) = 0,,..., k, m Z M } forms a orthoormal basis for L w [0, ] with respect to the weight fuctio w (x ) = w ( k+ x ( + )), i which w (x ) =. x Ay square itegralable fuctio f (x ) defied over [0, ] may be expaded by Chebyshev wavelets as f (x ) = c m ψ m (x ), (9) = m=0 If the ifiite series i (9) is trucated, the it ca be writte as k M f (x ) c m ψ m (x ) = C T Ψ(x ), (0) =0 m=0 C ad Ψ(x ) are k M colum vectors as C = c 00,..., c 0(M ) c 0,..., c (M ),..., c k 0,..., c k (M ) T, () Ψ(x ) = ψ 00,...,ψ 0(M ) ψ 0,...,ψ (M ),..., ψ k 0,...,ψ k (M ) T. Lemma 3.. By usig the shifted Chebyshev polyomials, ay compoet Ψ r (x ) of () ca be writte as: Ψ r (x ) = ψ m (x ) = k+ T m ( k x )χ πγ [ m k, + ], k r = M + m +, m = 0,..., M, = 0,..., k ad χ [, x [ χ [ k, + ](x ) =, + k ], k k 0, t he r w i s e. k, + k ] is the characteristic fuctio defied as: 85
Numerical solutio of Bagley-Torvik equatio usig Chebyshev wavelet operatioal matrix of fractioal derivative Now we preset a useful theorem about operatioal matrix of derivative for Chebyshev wavelets. Theorem 3.. Let Ψ(x ) be the Chebyshev wavelet vector defied i (), ad α > 0 (N < α N ) the we have Ψ(x ) = Ψ(x ), () is the ( k M ) ( k M ) operatioal matrix of fractioal derivative of order α, i the Caputo sese ad its (p,q )-th compoet is [ ] p q = 0, k+ i which m m i w α mq = M + p M + α, w α mq, M + α + p ( + )M. (3) b q a mi m i b q = (x χ [ k, + ]),Ψ q (x ) k w ad a m is defied i (7). Proof. ( ) m i k m i (4) For M + p M + α, we have 0 m α, ad cosequetly Ψ p (x ) = Ψ p (x ) = k+ T m (k x ) = 0, = 0,..., k, (6) for M + α + p ( + )M we have α m M, so Ψ p (x ) 0. By usig Lemma 3. we obtai Ψ p (x ) = k+ m a mi ( k x ) m i χ πγ [ m k, + ]. (7) k i =0 this fuctio is zero outside the iterval χ [ obtai Ψ p (x ) = k+ m m i m i a mi Now by apply Ψ p (x ) = k+ k, + o both sides of (8) we have: m m i m i a mi Eq. (9) ca be rewritte as Ψ p (x ) = k+ f (x ) = x χ [ + x + k d d t t k, + k m m i ] = (x t ) d t χ α + [ + k Γ ( α) a mi m i + k k (5) k ]. So by substitute Newto expasio of ( k x ) m i ito (7) we ( ) m i k m i x χ [ k, + k ( ) m i k m i x χ [ ], (8) k, + k ]. (9) ( ) m i k m i f (x ), (0) d d t t (x t ) α + d t χ [ k, + k ],], = 0,..., m i. () Here, by kowig that f (x ) is zero outside the iterval [ k, + k ], it is cosequece that Chebyshev wavelets expasio of this fuctio have oly compoets of basis Chebyshev wavelets Ψ(x ) that are o-zero i this iterval that yield: (+)M f (x ) = b q Ψ q (x ), = 0,..., m i, () q =M b q = (f (x ),Ψ q (x )). Now by substitute () ito (0), we obtai: Ψ (+)M p (x ) = k+ m m i m i b q a mi q =M ad this leads to desired results. ( ) m i k m i Ψ q (x ), (3) 86
Fakhrodi Mohammadi / It. J. Adv. Appl. Math. ad Mech. () (04) 83-9 Lemma 3.. For the iteger value α = Chebyshev wavelet operatioal matrix of derivative ca be expressed by d Ψ(x ) = D Ψ(x ) d t D is the k M operatioal matrix of derivative defied as follow F 0 0 0 F 0 D =........ 0 0 0 F (4) (5) I which F is M M matrix ad its (r, s )-th elemet is defied as follow F r, s = k+ m γ r γ s r =,..., M, s =,..., r a d (r + s ) o d d 0 o.w (6) Proof. It is a immediate cosequece of Theorem 3.. Corollary 3.. By usig Eq. (4) the operatioal matrix for -th derivative ca be derived as d Ψ(x ) d x = D Ψ(x ), (7) D is the -th power of matrix D. 3.. Error aalysis I this sectio we give the covergece properties ad error boud for the Chebyshev wavelets expasio. Theorem 3.. Ay square itegrable fuctio f (x ) i [0, ) with bouded secod derivative f (x ) B ca be expressed i terms of Chebyshev wavelets, ad the series coverges uiformly to f (x ), that is f (x ) = c m ψ m (x ), (8) = m=0 c m = (f (x ),ψ m (x )) ad (.,.) deotes the ier product o L w [0, ]. Proof. Please see [8]. Theorem 3.3. Suppose f (x ) be a cotiuous fuctio defied o [0, ), with bouded secod derivative f (x ) B, ad k M =0 m=0 c mψ m (x ) be its approximated value usig Chebyshev wavelets, the we have the followig accuracy estimatio σ M,k = B π 8 5 = k m=m (m ) 4 (9) σ k,m = k M f (x ) c m ψ m (x ) w (x )d x 0 =0 m=0 Proof. Please see [8]. 87
Numerical solutio of Bagley-Torvik equatio usig Chebyshev wavelet operatioal matrix of fractioal derivative 4. Applicatio ad results I this sectio, i order to show the high importace of operatioal matrix of derivative, we apply it to solve fractioal order Bagley-Torvikboudary value problems. These problems are cosidered because closed form solutios are available for them, or they have also bee solved usig other umerical schemes. This allows oe to compare the results obtaied usig this scheme with the aalytical solutio or the solutios obtaied usig other schemes. The geeral Bagley-Torvik boudary value problems of order have the form [9] A 0 D + A D 3 + A y (t ) = f (t ), t [0, T ], (30) subect to iitial-boudary coditios γ0 y (0) + γ y (0) = α 0, γ 3 y (T ) + γ 4 y (T ) = α, (3) A 0, A, A, α 0,γ 0, γ, γ 3,γ 4,α 0 ad α are costats with A 0 0, ad y L [0, T ]. The existece ad uiqueess of the exact solutio of the solutio for these problems are discussed i [3]. Here we itroduce a ew method based o Chebyshev wavelets expasio ad their operatioal matrices of derivatives. To solvig the Bagley-Torvik boudary value problems of the form (30) subect to the coditios (3) we approximate the y (t ) ad f (t ) by the Chebyshev wavelets as y (t ) C T Ψ(t ), f (t ) F T Ψ(t ) (3) C ad F are coefficiet vector of Chebyshev wavelets expasio for fuctios y (t ) ad f (t ). By usig theorems 3. we have D y (x ) C T D Ψ(x ), D 3 y (x ) C T D 3 Ψ(x ), (33) Substitutig Eqs. (37)i (30) the residual R (x ) ca be derived as R (x ) A 0 C T D + A C T D 3 + A C T Ψ(x ). (34) By usig typical tau method [0] we geerate k M liear equatios by applyig R k M (x ),Ψ (x ) = 0, = 0,,..., k M (35) Also, by substitutig boudary coditios we get γ0 C T Ψ(0) + γ C T D Ψ(0) = α 0, γ 3 C T Ψ(T ) + γ 4 C T D Ψ(T ) = α, (36) Eqs. (35) ad (36) together geerate k M set of liear equatios. These liear equatios ca be solved for ukow coefficiets of the vector C. By substitutig the derived vector C i Eq. (37) approximatio solutio y (x ) ca be obtaied. 5. Numerical experimets I this sectio we will cosider the three fractioal order Bagley-Torvik boudary value problems. We used the method described i the Sectio 4 for solvig these problems. The algorithms are performed by Maple with 30 digits precisio. Example 5.. Cosider the followig boudary value problem i the case of the ihomogeeous Bagley-Torvik equatio [9] D y (x ) + D 3 y (x ) + y (x ) = t + 4 t π + y (0) = 0, y (5) = 5. Where the exact solutio of this problem is y (x ) = x. We solve this fractioal boudary value problem by applyig the method described i Sectio 4 usig Chebyshev wavelets expasio ad its operatioal matrices of derivatives with M=4, k=. Usig (35) we obtai four liear equatios. Ad by applyig boudary coditio we have two liear equatios. By solvig this liear system we get the ukow vector C. By substitutig this vectori Eq. (37) we have the exact solutio. (37) 88
Fakhrodi Mohammadi / It. J. Adv. Appl. Math. ad Mech. () (04) 83-9 Table. The absolute errors for differet values of M ad k x (M, k) = (6, ) (M, k) = (6, ) (M, k) = (, ) (M, k) = (, ) 4. 0 3.4 0 4 3.9 0 6 6. 0 7 8 4.4 0 4 3.5 0 4 7.6 0 7 8.8 0 9.6 0 4 5.4 0 5 8.7 0 8 5.4 0 9 6.0 0 4 6.5 0 6 l9.5 0 8 7.4 0 0 Example 5.. Cosider the boudary value problem D 3 y (x ) + y (x ) = x 5 x 4 + 8 7 π x 3.5 64 5 π x.5 y (0) = 0, y () = 0. (38) the exact solutio of this problem is y (x ) = x 4 ( x ). We solve this fractioal boudary value problem by applyig the method described i Sectio 4 by usig Chebyshev wavelets expasio M = 6, k =. Similar to Example by solvig liear system derived for this problem we get the exact solutio. Example 5.3. Cosider the boudary value problem D y (x ) + D 3 y (x ) + y (x ) = 8ν(x ) 8ν(x ) y (0) = 0, y (0) =.48433, (39) I which ν(x ) is the Heaviside fuctio. This example was solved theoretically by Podluby [3] ad the compact form of the solutio is give by y (x ) = x 0 8G (x t )(ν(t ) ν(t ))d t (40) G is the fractioal Gree fuctio defied as ( ) k ( + k)! t G (t ) = k k!! Γ + k + (4) 3 k + k=0 =0 I Eq. (4) is the Gamma fuctio. Here we solve this problem usig Chebyshev wavelets method described i Sectio 4 with (M, k) = (6, ),(M, k) = (6, ),(M, k) = (, ) ad (M, k) = (, ). Fig. shows the approximate solutio ad theoretical solutio derived by Podluby[3] for (M, k) = (, ) i the iterval [0, 0]. The absolute errors for differet values i the iterval are show i the Table. From Table, we see that we ca achieve a good approximatio with the exact solutio by usiga few terms of Chebyshev wavelets. 6. Coclusio I this article a geeral formulatio for derivig the Chebyshev wavelet operatioal matrix of derivatives has bee derived. The a umerical method based o Chebyshev wavelets expasio, its operatioal matrix of fractioal order ad tau method is itroduced for approximate the solutio of Bagley-Torvik fractioal boudary value problems. Moreover, the covergece ad error aalysis for the proposed method is cosidered. 89
Numerical solutio of Bagley-Torvik equatio usig Chebyshev wavelet operatioal matrix of fractioal derivative Fig.. Approximate solutio ad exact solutio for Example 3 i the iterval [0, 0] Refereces [] K.S. Miller, B. Ross, A Itroductio to The Fractioal Calculus ad Fractioal Differetial Equatios, Wiley, New York, 993. [] K.B. Oldham, J. Spaier, The Fractioal Calculus, Academic Press, New York, 974. [3] I. Podluby, Fractioal Differetial Equatios, Academic Press, Sa Diego, 999. [4] A. Saadatmadi, M. Dehgha, A ew operatioal matrix for solvig fractioal-orderdifferetial equatios,comput. Math. Appl, 59 (00) 36-336. [5] Y. Li, W. Zhao, Haar wavelet operatioal matrix of fractioal order itegratio ad itsapplicatios i solvig the fractioal order differetial equatios,appl Math Comput, 6 (00) 76-85. [6] P. J. Torvik ad R. L. Bagley, O the appearace of the fractioal derivative i the behavior of real materials, Joural of Applied Mechaics 5. (984): 94-98. [7] R. L. Bagley ad P. J. Torvik, Fractioal calculus - A differet approach to the aalysis of viscoelastically damped structures, AIAA Joural.5 (983): 74-748. [8] Y. Hu, Y. Luo, ad Z. Lu, Aalytical solutio of the liear fractioal differetial equatio by Adomia decompositio method, Joural of Computatioal ad Applied Mathematics 5. (008): 0-9. [9] A. M. A. El-Sayed, I. L. El-Kalla, ad E. A. A. Ziada, Aalytical ad umerical solutios of multi-term oliear fractioal orders differetial equatios, Applied Numerical Mathematics 60.8 (00): 788-97. [0] A. Ghorbai ad A. Alavi, Applicatio of He s variatioal iteratio method to solve semidifferetial equatios of th order, Mathematical Problems i Egieerig 008 (008): -0. [] Y. Ceesiz, Y. Keski, ad A. Kuraz, The solutio of the Bagley-Torvik equatio with the geeralized Taylor collocatio method, Joural of the Frakli Istitute 347. (00): 45-466. [] K. Diethelm, N.J. Ford, Numerical solutio of the BagleyâĂŞTorvik equatio, BIT 4 (00) 490-507. [3] Podluby, Igor, Tomas Skovraek, ad Viagre Jara, Matrix approach to discretizatio of fractioal derivatives ad to solutio of fractioal differetial equatios ad their systems, Emergig Techologies & Factory Automatio, 009. ETFA 009. IEEE Coferece o. IEEE, 009. [4] A. Boggess, F. J. Narcowich, A first course i wavelets with Fourier aalysis, Joh Wiley & Sos, 00. [5] Mallat S. A wavelet tour of sigal processig. d ed. Academic Press, 999. [6] M. Razzaghi, S. Yousefi, Legedre wavelets operatioal matrix of itegratio, It. J. Syst. Sci. 3 (4) (00) 495-50. [7] M. M. Khader, O the umerical solutios for the fractioal diffusio equatio, Commu Noliear Sci Numer Simul., 6 (0) 535-54. [8] S. Sohrabi, Compariso Chebyshev wavelets method with BPFs method for solvig AbelâĂŹs itegral equatio, Ai Shams Egieerig Joural (0), 49-54. [9] Q. M. Al-Mdallal, M. I. Syam, M.N. Awar, A collocatio-shootig method for solvig fractioal boudaryvalue problems, Commu Noliear SciNumerSimulat 5 (00) 384-38. 90
Fakhrodi Mohammadi / It. J. Adv. Appl. Math. ad Mech. () (04) 83-9 [0] C. Cauto, M. Hussaii, A. Quarteroi, T. Zag, Spectral Methods i Fluid Dyamics, Spriger, Berli, 988. 9