Time Fractional Wave Equation: Caputo Sense

Transcription

1 Adv. Studies Theor. Phys., Vol. 6, 2012, no. 2, Time Fractional Wave Equation: aputo Sense H. Parsian Department of Physics Islamic Azad University Saveh branch, Saveh, Iran Abstract In this paper we propose a new approach in time fractional wave equation. We built fractional wave equation from fractional Lagrangian wave equation. In this approach the velocity of wave is v = τ 2α ρ, 0 < α<1. Then we solve fractional wave equation for special initial values by separation of variables method. We investigate this problem in aputo fractional derivative. Keywords: Fractional calculus, Wave equation, Fractional Euler-Lagrange equation 1 Introduction Fractional derivative integral related topic is a very attractive branch of mathematics for physicist engineers. In many papers, authors investigate the properties of fractional derivative integral operators such as [1, 2] the results are used for modelling different phenomena in physic, chemistry etc. The fractional Brownian motion, fractional Langevin, fractional Fokker-Planck fractional diffusion-wave equations attracted attention in the last few years [3, 4, 5]. In this research work we present a new approach in fractional wave equation. Almost all authors introduce investigate the fractional wave equation as 2 x u = ρ α 2 τ t u (1) α In which ρ τ are linear mass density tension respectively, u(x, t) is the amplitude vibrations 0 < α < 1[6]. In this model the velocity of wave is v = τ. We propose fractional wave equation from fractional ρ Lagrangian wave equation the result is a wave equation with the velocity v = τ 2α, 0 <α<1. It is suitable idea for anomalous materials. We ρ

2 96 H. Parsian organize this paper as follows. In section 2 provide the initial background, in section 3 we built fractional Lagrangian wave equation fractional wave equation in aputo sense. In section 4 provide an example in section 5 provide the conclusion. 2 Mathematical background 2.1 fractional operator Several type of fractional derivative integral have proposed [7, 8, 9]. We review some of them in brief. The left sided aputo fractional derivative a D α xf(x) = 1 x Γ(n α) a The right sided aputo fractional derivative ( d dt )n f(t) dt (2) (x t) x Dα b f(x) = 1 b ( d dt )n f(t) dt (3) Γ(n α) x (t x) The left iemann-liouville fractional derivative a Dα x f(x) = 1 Γ(n α) ( d x f(t) dx )n dt (4) a (x t) The right iemann-liouville fractional derivative x Dα b f(x) = 1 Γ(n α) ( d b f(t) dx )n dt (5) x (t x) In all definitions α is the order of derivative such that n 1 α<n.ifα is an integer, all these definitions reduce in to usual derivative. The left iemann-liouville fractional integral aix α f(x) = 1 x f(t) dt (6) Γ(α) a (x t) 1 α The right iemann-liouville fractional integral xib α f(x) = 1 b f(t) dt (7) Γ(α) x (t x) 1 α

3 Time fractional wave equation: aputo sense Some Properties If we choose x = λt, for ordinary derivative integral we have d n dn = λn (8) dtn dx n ait n = λa Ix n (9) λ n in which n is a positive integer number. For left aputo fractional derivative of order α [n 1,n) x = λt we have a Dα t = a I n α t D n t = λ α n λai n α x λ n D n x = λ α λa Dα x (10) also t Db α =( 1) n tib n α Dt n =( 1) n λ α n xi n α λb λ n D n x = λ α x D α λb (11) Lemma 1: For any integrable differentable f(t), x = λt α [n 1,n) we have: a Dα t = λ α λa Dα x t Db α = λ α x Dλb α (12) For left iemann-liouville fractional derivative of order α [n 1,n) we have dn x Dα b f(x) =( 1)n dx [ xi n α n b f(x)] dx n [( 1)n aib n α f(x) ( 1) α+1 aix n α f(x)] = dn = e iπα a D α xf(x) (13) Lemma 2: For any integrable differentable f(t) α [n 1,n) we have: x D α b f(x) =e iπα a D α xf(x) (14)

4 98 H. Parsian 3 Lagrangian wave equation 3.1 lassical Lagrangian wave equation The classical Lagrangian for a vibrating string is L = 1 (ρu 2 t 2 (x, t) τu2 x (x, t)) dxdt (15) In which ρ is linear mass density τ is the tension. u x (x, t) u t (x, t) are derivatives of u(x, t) respect to x t respectively. If we extremum Lagrangian wave equation, the result is one dimension classical wave equation. In which v 2 = τ ρ 2 u x = ρ 2 u (16) 2 τ t 2 v is the velocity of wave in the string. 3.2 Fractional Lagrangian wave equation We built the time-fractional Lagrangian for a vibrating string in the aputo,s sense as below L = 1 [ ρ ( 0 Dt α u(x, t)) 2 τ (D x u(x, t)) 2] dxdt (17) 2 in which D 1 = D. If we extremum fractional Lagrangian wave equation, the result is one dimension fractional wave equation in aputo sense. If we use η = vt ρ t D α t 0 ( 0 D α t u(x, t)) + τd 2 xu(x, t) = 0 (18) ρv 2α η Dα η 0 ( 0 Dα t u(x, η)) + τd2 xu(x, η) = 0 (19) This is a fractional wave equation if we choose ρv 2α = τ The velocity of wave is v = 2α τ ρ. η Dα η 0 ( 0 Dα η u(x, η)) + D2 xu(x, η) = 0 (20) 4 Example We choose initial values u(0,t)=u(1,t)=0 t 0 u(x, 0) = g(x) 0 x 1 α u(x, t) t α t=0 = f(x) 0 x 1 (21)

5 Time fractional wave equation: aputo sense 99 We solve by separate of variable method. If we X(0) = X(1) = 0, we have X(x) = sin(mπx). u(x, η) =X(x)T (η) (22) η D α η 0 ( 0 D α η T (η)) = m 2 π 2 T (η) (23) This is an ordinary fractional differential equation we have a general solution in [10]. If we choose We have from lemma 2 0 Dα η T (η) = T (η) = a n η nα (24) a n n=1 η Dα η 0 ( 0 Dα η T (η)) = Now we find the relation between the coefficients a n a n+2 = We find the coefficients of (24) in term of a 0 a 1 (nα)! (nα α)! ηnα α (25) (nα +2α)!e iπα a n+2 η nα (26) (nα)! (nα)! (nα +2α)! m2 π 2 e iπα a n (27) a 2n = 1 (2nα)! (m2 π 2 e iπα ) n a 0, n 0. (28) a 2n+1 = α! ((2n +1)α)! (m2 π 2 e iπα ) n a 1, n 0. (29) m 2n π 2n T m (vt) =b m (2nα)! cos(nπα)(vt)2nα +c m And the solution of fractional wave equation In which α!m 2n+1 π 2n+1 ((2n +1)α)! cos(nπα)(vt) (2n+1)α (30) u(x, t) = T m (vt) sin(mπx) (31) m=1 1 b m =2 sin(mπx)g(x)dx c m = sin(mπx)f(x)dx mπv α α! 0

6 100 H. Parsian 5 onclusion In research work we proposed a new approach in time fractional wave equation. We start from classical Lagrangian wave equation built fractional Lagrangian wave equation in aputo sense. The velocity of wave is v = 2α τ ρ the solution the velocity of wave reduce to the solution velocity of classical wave equation for α =1. eferences [1]. Hilfer J. Seybold, omputation of the generalized Mittag-Leffler function its inverse in the complex plane, Integral Transform Special Functions, 17, (2006), [2] H. Seybold. Hilfer, Numerical Algorithm for alculating the Generalized Mittag-Leffler Function, SIAM J. Numer Anal. 47, (2008), [3] E. Heinsalu, M. Patriarca, I. Goychuk, G. Schmid P. Hanggi, Phys. ev. E, 73, (2006), [4]. Hilfer, Exact Solutions for a lass of Fractal Time om Walks, Fractals, 3, (1995), 211. [5]. Hilfer L. Anton, Fractional Master Equations Fractal Time om Walks, Phys. ev. E, 51, (1995), 848. [6] T. Sev Z. Tomovski, General time fractional wave equation for a vibrating string, J. Phys. A: Math. Theor., 43, (2010). [7] K. S. Miller, B. oss, An Introduction to the Fractional alculus Fractional Diff. Eq., John Wiley Sons, New York, [8] I. Podlubny, Fractional Differential Equations, Academic Press, New York, [9] P. L. Butzer U. Westphal, An introduction to fractional calculus, in:. Hilfer (Ed.), Applications of Fractional alculus in Physics, World Scientific, New Jersey, [10] D. Baleanu J. J. Trujillo, On exact solutions of a class of fractional Euler-Lagrange equations, Nonlinear Dyn., 52, (2008), 199. eceived: April, 2011