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Journal of Applied Mathematics, Islamic Azad University of Lahijan, Vol.7, No.4(27), Winter 2011, pp 65-70 ISSN: 2008-594x Legendre Wavelets for Solving Fractional Differential Equations M. Soleymanivaraki *1, Hossein Jafari 2, M. Arab.Firoozjaee1 1Department 2Islamic of Mathematics, University of Mazandaran, Babolsar, Iran Azad University, Ayatollah Amoli Branch, Amol, Iran Received: 8 May 2010 Accepted: 16 November 2010 Abstract In this paper, we develop a framework to obtain approximate numerical solutions to ordinary differential equations (ODEs) involving fractional order derivatives using Legendre wavelets approximations. The continues Legendre wavelets constructed on [0, 1] are utilized as a basis in collocation method. Illustrative examples are included to demonstrate the validity and applicability of the technique. Keywords: Legendre Wavelet, Fractional Differential Equations, Collocation Method. 1 Introduction Ordinary differential equations involving fractional order derivatives are used to model a variety of systems, of which an important engineering application lies in viscoelastic damping [1, 8, 3]. Another important application of fractional derivatives lies in control theory [6]. Linear ODEs with fractional order derivatives or integrals are studied using Laplace [1] and Fourier transforms [8]. Nonlinear ODEs involving fractional order derivatives can be solved numerically using several method [9, 11]. In this paper, we present a numerical technique to solve ODEs with fractional order derivatives using legendre wavelet. There are many different types of definitions of fractional calculus. For example, the Riemann-Liouville integral operator [12] of order α is defined by 1 x f (t ) 0, t 0 dt I f ( x) ( ) ( x t )1 (1) 0 0. f ( x) and its fractional derivative of order α (α ≥ 0) is normally used: ( Dl f )( x) ( d m m ) (I f )( x), dx ( 0, m 1 m), *Corresponding author E-mail address: [email protected] 65 (2) H.Jafari et al, Legendre Wavelets for Solving Fractional Differential Equations where n is an integer. For Riemann-Liouvilles definition, one has I xv (v 1) x v (v 1 ) (3) The Riemann-Liouville integral operator plays an important role in the development of the theory of fractional derivatives and integrals. However, it has some disadvantages for fractional differential equations with initial and boundary conditions. Therefore, we adopt here Caputo’s definition [3, 10], which is a modification of Rieman-Liouville definition: x 1 f ( m ) (t ) 0 ( x t ) m1 dt ( m ) ( D f )( x) m f ( x) x m ( 0, m 1 n, ) m. (4) where m is an integer. Caputo’s integral operator has a useful property [3, 10]: m 1 ( I D f )( x) f ( x) k 0 f ( k ) (0 ) k x , k! ( x 0, m 1 m.) (5) where n is an integer. In this study, fractional differentiation is understood in the Caputo sense because of its applicability to real–world problems. Wavelets theory is a relatively new and an emerging area in mathematical research. It has been applied in a wide range of engineering disciplines; particularly, wavelets are very successfully used in signal analysis for waveform representations and segmentations, time–frequency analysis and fast algorithms for easy implementation [4]. Wavelets permit the accurate representation of a variety of functions and operators. Moreover, wavelets establish a connection with fast numerical algorithms [2]. In the present article, we are concerned with the application of Legendre wavelets to the numerical solution of FODEs. The method consists of converting of ordinary differential equations involving fractional order to algebraic equations and expanding the solution by Legendre wavelets with unknown coefficients. The article is organized as follow: In Section 2, we describe the basic formulation of wavelets and Legendre wavelets required for our subsequent development. Section 3 is devoted to the solution of FODEs using integral operator matrix and product operator matrix and Legendre wavelets. In Section 4, considering numerical examples our numerical finding reported and the accuracy of the proposed scheme is demonstrate. 2 The Properties of Legendre wavelets 2.1 Wavelets and Legendre wavelets Wavelets constitute a family of functions constructed from the dilation and translation of a single function called the mother wavelet. When the dilation parameter a and the translation parameter b vary continuously we have the following family of continuous wavelets as [7]: 66 Journal of Applied Mathematics, Islamic Azad University of Lahijan, Vol.7, No.4(27), Winter 2011, pp 65-70 1 a.b (t ) a 2 ( t b ) a a , b R, a 0 (6) If we restrict the parameters a and b to discrete values as a a0 k , b nb0 a0 k , a0 1, b0 0 and n , and k positive integers, we have the following family of discrete wavelets: k k ,n (t ) a0 2 (a0 k t nb0 ), Where k ,n (t ) form a wavelet basis for L2 (R) . In particular, when a 0 2 and b0 1 then k , n (t ) forms an orthonormal basis [7]. nm (t ) (k , n, m, t ) have Legendre wavelets four arguments; n 2n 1, n 1,2,3,...,2 k 1 , k can assume any positive integer, m is the order for Legendre polynomials and t is the normalized time. They are defined on the interval [0, 1) as follows: 1 k2 m 2 L m (2k t n), 2 nm (t) 0, for n 1 n 1 t k k 2 2 , (7) otherwise Where m 0,1,2,..., M 1, n 1,2,...,2 k 1 . 1 is for orthonormality, the dilation parameter is a 2 k and the 2 translation parameter b n 2 k . Here, Lm (t ) are the well–known Legendre polynomials of order m which are defined on the interval [−1, 1], and can be determined with the aid of the following recurrence formulae: The coefficient m L0 (t ) 1, L1 (t ) t , Lm 1 (t ) ( 2m 1 m )tLm (t ) ( ) Lm 1 (t ), m 1 m 1 m 1,2,3,... The set of { nm (t )] are composed an orthonormal system for L2 [0,1]. [13] 2.2 Function approximation A function f (t ) defined over [0, 1) may be expanded by Legendre wavelet series as f (t) c nm nm (t), (8) n 1 m 0 where 67 H.Jafari et al, Legendre Wavelets for Solving Fractional Differential Equations cnm f (t ), nm (t ) . (9) In (9), . denotes the inner product. If the infinite series in Eq. (8) is truncated, then Eq.(8) can be written as 2 k 1 M 1 f (t ) cnm nm (t ) C T (t ), (10) n 1 m 0 where C and (t ) are 2 k 1 M 1 matrices given by C(t ) [c10 , c11 ,..., c1M 1 , c20 ,..., c2M 1 ,..., c2k 1 o ,..., c2k 1 M 1 ]T (11) (t ) [ 10 (t ), 11 (t ),..., 1M 1 (t ), 20 (t ),..., 2M 1 (t ),..., 2k 1 o (t ),..., 2k 1 M 1 (t )]T (12) 3 Solution of fractional differential equations Consider the initial value problem given in Eq. (13). D x(t ) Nx(t ) g (t ) 0, (13) where N is the nonlinear operator and D is fractional derivative. In order to use Legendre wavelets, we first approximate x(t ) as x(t ) C T (t ) (15) where C and (t ) are defined similarly to Eqs. (11) and (12). Then, we have: D C T (t ) NC T (t ) g (t ), (16) we now collocate Eq. (16) at 2 k 1 M points xi as D C T (t i ) NC T (t i ) g (t i ) (17) We have chosen arbitrary collocation points. By solving equations system (17), we obtained solution for Eq.(13). 4 Illustrative examples We applied the method presented in this paper and solved various examples: 4.1 Example 1 Now, let us consider the Bagley-Torvik equation that governs the motion of a rigid plate immersed in a Newtonian fluid [1]. 68 Journal of Applied Mathematics, Islamic Azad University of Lahijan, Vol.7, No.4(27), Winter 2011, pp 65-70 3 2 2 d x(t ) d x(t ) (18) x(0) 1, x ' (0) 1 x(t ) 1 t 2 3 dt 2 dt Following Diethelm and Ford [5]. We applied the method presented in this paper and solved Eq. (18) with M 3 and K 3 For this equation we find: c1, 0 =0.5625, c1,1 =0.0360844, c1, 2 =0. c 2, 0 =0.6875, c 2 ,1 =0.0360844, c 2, 2 =0. c3, 0 =0.8125, c 3,1 =0.0360844, c3, 2 =0. c 4, 0 =0.9375, c 4 ,1 =0.0360844, c 4, 2 =0. Now, we get: x(t ) c1,0 1,0 (t ) c1,1 1,1 (t ) c1, 2 1, 2 (t ) c2,0 2,0 (t ) c2,1 2,1 (t ) c2, 2 2, 2 (t ) c3,0 3,0 (t ) c3,1 3,1 (t ) c3, 2 3, 2 (t ) c4,0 4,0 (t ) c4,1 4,1 (t ) c4, 2 4, 2 (t ) 1 t which is the exact solution. Example 4.2 Next, following Momani and Odibat [11], we consider the composite fractional oscillation equation 1 2 d x(t ) dt 1 2 x(t ) t 2 2 t 1.5 0 , (2.5) (19) with the initial conditions: We applied the method presented in this paper and solved Eq. (19) with M 3 and K 2 For this equation we find: c1, 0 =0.0589256, c1,1 =0.051031, c1, 2 =0.0131762 c 2, 0 =0.412479, c 2 ,1 =0.153093, c 2, 2 =0.0131762 Now, we get: x(t ) c1,0 1,0 (t ) c1,1 1,1 (t ) c1, 2 1, 2 (t ) c 2,0 2,0 (t ) c 2,1 2,1 (t ) c 2, 2 2, 2 (t ) t which is the exact solution. 5 Conclusions In this work, we solved fractional differential equations by using Legendre wavelets in Collocation method. Legendre wavelets are well behaved basic functions that are orthonormal on [0, 1]. Application of the wavelets allows the creation of more effective and faster algorithms than the ordinary ones. Illustrative examples are included to demonstrate the validity and applicability of the technique. 69 H.Jafari et al, Legendre Wavelets for Solving Fractional Differential Equations References [1] Bagley, R. L. and Torvik, P. J., Fractional calculus: A different approach to the analysis of viscoelastically damped structures, AIAA Journal 21(5), (1983), 741-748. [2] Beylkin G, R. Coifman, V. Rokhlin, Fast wavelet transforms and numerical algorithms, I, Commun. Pure Appl. Math. 44 (1991) 141–183. [3] Caputo M. Linear models of dissipation whose Q is almost frequency independent. Part II. J Roy Austral Soc 13 (1967); 529–39. [4] Chui. C.K, Wavelets: A Mathematical Tool for Signal Analysis, SIAM, Philadelphia, PA,1997. [5] Diethelm K.J, Ford NJ. Numerical solution of the Bagley-Torvik equation. Bit 42: (2004); 490-507. [6] Debnath L., Recent applications of fractional calculus to science and engineering, International Journal of Mathematics and Mathematical Sciences (54), (2003), 34133442. [7] Gu J.S, Jiang W.S, The Haar wavelets operational matrix of integration, Int. J. Syst. Sci. 27 (1996) 623-628. [8] Gaul L., Klein, P., and Kempfle, S., Impulse response functions of an oscillator with fractional derivative in damping description, Mechanics R. Comm. 16(5), 1989, 297-305. [9] Kai Diethelm and Guido Walz, Numerical solution of fractional order differential equations by extrapolation, Numerical Algorithms 16 (1997) 231-253. [10] Luchko Yu, Gorenflo R. An operational method for solving fractional differential equations with the Caputo derivatives. Acta Math Vietnamica 24(2): (1999); 207-33. [11] Momani S, Odibat Z. Analytical approach to linear fractional partial differential equations arising in fluid mechanics. Phys Lett A 355:(2006); 271–9. [12] Podlubny I. Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. New York: Academic Press; 1999. [13] Razzaghi M, Yousefi S, The Legendre wavelets operational matrix of integration, International Journal of Systems Science, 32(2001)495–502. 70