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The intrinsic damping of the fractional oscillator
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Physica A 329 (2003) 29 – 34
www.elsevier.com/locate/physa
The intrinsic damping of the fractional oscillator Ali To!ghi Faculty of Basic Sciences, Department of Physics, University of Mazandaran, P.O. Box 47415-453, Babolsar, Iran Received 28 March 2003
Abstract We obtain analytical expressions for the time rate of change of the potential energy, the kinetic energy and the total energy of a fractional oscillator in terms of the products of Mittag–Le,er functions. We propose a de!nition for the intrinsic damping force of this oscillator. We obtain a general expression for this damping force. An expression for this damping force in the asymptotic limit (!t → 0) is also obtained. c 2003 Elsevier B.V. All rights reserved. PACS: 03.20.+i Keywords: Fractional oscillation; Intrinsic damping
1. Introduction Some of the most important laws of physics are expressed in terms of certain differential equations. For instance, Newton’s second law of motion involves the !rst derivative of the momentum with respect to time, or equivalently it involves the second derivative of the position vector with respect to the variable time. A new trend is to consider the derivative of the type d x=dt where is non-integer. The motivation for this generalization is that it creates a lot of new possibilities. Furthermore, using fractional di:erential equation, one could explain some complex or anomalous phenomena [1]. In the realm of classical mechanics a simple problem is the so-called fractional oscillator. An early analysis of this problem is given in Ref. [2]. In this work they start from a fractional di:erential equation. A recent study of the fractional oscillator is based on the fractional integral equation [3]. E-mail address: [email protected] (A. To!ghi). c 2003 Elsevier B.V. All rights reserved. 0378-4371/$ - see front matter doi:10.1016/S0378-4371(03)00598-3
30
A. To.ghi / Physica A 329 (2003) 29 – 34
One of the characteristic features of a simple fractional oscillator is the decay of the amplitude of the oscillation. In this paper we will explore the issue of the intrinsic damping of the fractional oscillator. In Section 2 we review the solution of the fractional oscillator. In Section 3 we study the time rate of change of the potential energy, the kinetic energy and the total energy of the fractional oscillator. In Section 4 we suggest a de!nition of the intrinsic damping force of this system. Then we obtain an expression for this damping force. In Section 5, with the help of Laplace transform method we calculate this force in the asymptotic limit !t → 0. 2. Simple fractional oscillator The di:erential equation of a simple harmonic oscillator is d2 x m 2 + kx = 0 : (1) dt The di:erential equation of a simple fractional oscillator is d x m + kx = 0; 1 ¡ 6 2 : (2) dt We note that the dimension of the parameter m in Eq. (2) is MT −2 . Therefore, the parameter ! de!ned by k ! = (3) m has the dimension T −1 . So the fractional di:erential equation of the system is [2] d x (4) + ! x = 0 1 ¡ 6 2 : dt A detailed investigation of the fractional oscillator using an integral equation of motion is given in Ref. [3]. In our notation the appropriate integral equation of motion of this system is t ! x(t) = x(0) + x˙ (0)t − (t − t )−1 x(t ) dt ; (5)
() 0 where again 1 ¡ 6 2. In our notation the basic results of Ref. [3] are follows: x(t) = x0 E; 1 (z) ; p=m
d =2 x = −mx0 ! t =2 E; 1+=2 (z) ; dt =2
(6) (7)
1 1 2 p2 1 (8) kx + = kx2 (E; 1 (z))2 + mx02 !2 t (E; 1+=2 (z))2 ; 2 2m 2 0 2 where z = −! t and the system starts at rest from x(0) = x0 , and the generalized Mittag–Le,er function is de!ned by [4]. ∞ zn ¿0 (9) E; (z) =
(n + ) EF =
n=0
and is an arbitrary complex parameter.
A. To.ghi / Physica A 329 (2003) 29 – 34
31
3. Time rate of change of x, p, UF , KF , EF From Eq. (6) we obtain dx dE; 1 (z) = x0 : dt dt Using the identity [5] 1 + zE; + (z) E; (z) =
()
(10)
(11)
we obtain d z d E; +1 (z) + z E; +1 (z) : (12) E; 1 (z) = dt dt t Now we utilize the following relation [5] d E; +1 (z) (13) E; (z) = E; +1 (z) + z dz to perform the derivative on the right-hand side of Eq. (12). The !nal result is dx (14) = −x0 ! t −1 E; (−! t ) : dt For = 2, from Ref. [4] we have sinh(z 1=2 ) : (15) z 1=2 So in this case, from Eq. (14) we get dx = −!x0 sin(!t) dt the familiar expression from elementary mechanics. Now we consider the time rate of change of the potential energy of fractional oscillator, namely, dx dUF d 1 2 = kx = kx : (16) dt dt 2 dt E2; 2 (z) =
Substituting Eqs. (6) and (14) in Eq. (16) we obtain dUF (17) = −kx02 ! t −1 E; 1 (−! t )E; (−! t ) : dt Next we compute the time rate of change of the generalized momentum p, From Eq. (7) we get dp =2−1 d E; 1+=2 (z) + t =2 E; 1+=2 (z) : (18) = −mx0 ! t dt 2 dt Next, we use Eq. (12) to do the derivative of the right-hand side of Eq. (18). The !nal result is dp = −kx0 t =2−1 E; =2 (−! t ) : (19) dt
32
A. To.ghi / Physica A 329 (2003) 29 – 34
The time rate of change of the kinetic energy is d kF p dp = : dt m dt Substituting Eqs. (7) and (19) in Eq. (20) we obtain dKF = kx02 ! t −1 E; 1+=2 (−! t )E; =2 (−! t ) : dt
(20)
(21)
From Eq. (17) and (21) we obtain the following expression for the time rate of change of total energy of the fractional oscillator: dEF = −kx02 ! t −1 [E; 1 (−! t )E; (−! t ) − E; 1+=2 (−! t )E; =2 (−! t )] : dt (22) We notice that for = 2, The expression inside the bracket of Eq. (22) is identically zero. 4. The intrinsic damping force Fid In this section we investigate the intrinsic damping of the fractional oscillator. For the simple harmonic oscillator ( = 2) the total energy is conserved and x(t) exhibits permanent oscillation. However, in the fractional oscillator the total energy EF is not conserved and x(t) has some attenuated oscillation. The leading asymptotic behaviour of Mittag–Le,er function is [2] E; 1 (−! t ) ∼
(!t)−
(1 − )
as !t → ∞ :
Therefore, x(t) ∼ x0
(!t)−
(1 − )
as !t → ∞ :
This behaviour is the so called algebraic decay of x(t) as !t → ∞. The motion of the fractional oscillator is composed of two parts [5]. First, the oscillator makes a !nite number of damped oscillation. Next, it has a monotonic algebraic decay in the asymptotic region. We suggest the following relation for the intrinsic damping force Fid of the fractional oscillator: d =2 x d =2 EF = Fid =2 : =2 dt dt
(23)
In this paper we adopt the following de!nition for the fractional derivative. Let be a positive but non-integer number. We can always !nd a positive integer m such that m − 1 ¡ ¡ m. Then, t f(m) () d 1 d f = ; m − 1 ¡ ¡ m; (24) dt
(m − ) 0 (t − )+1−m
A. To.ghi / Physica A 329 (2003) 29 – 34
33
where f(m) denotes the mth order derivative of the function f(t). This de!nition is the so-called Caputo [6] derivative. From Eqs. (7), (23) and (24) we obtain the following expression for Fid : t kx0 0 −1 [E; 1 (y)E; (y) − E; 1+=2 (y)E; =2 (y)](t − )−=2 d ; (25) Fid =
(1 − =2)t =2 E; 1+=2 (z) where y = −! and z = −! t . We note that when = 2 dEF =dt = 0 so Fid is identically zero in this case. Moreover, at the turning points p=d x=2 dt =2 =0, therefore, EF = 12 kx2 . But x is a local maximum at these points, so d x=dt = 0 and dEF =dt = 0. Hence d =2 E=dt =2 = 0 at these points. Therefore, Fid is well de!ned everywhere. 5. Asymptotic form of Fid From Eq. (9) and in the asymptotic limit !t → 0 we obtain E; 1 (−! t ) = 1 −
(!t)2 (!t) + + ···
(1 + ) (1 + 2)
E; 1+=2 (−! t ) =
(!t) 1 − + ··· :
(1 + =2) (1 + 3=2)
and
Hence
(26)
x(t) = x0 1 −
(!t) (!t)2 + + ··· ;
(1 + ) (1 + 2) 1 (!t) =2 p = −mx0 ! t − + ··· :
(1 + =2) (1 + 3=2)
(27)
Therefore, EF = 12 kx02 [1 + c1 ! t + c2 !2 t 2 + · · · ] ;
(28)
where c1 =
1 2 − ; 2 ( (1 + =2))
(1 + )
c2 =
1 2 2 + : −
(1 + 2) ( (1 + ))2
(1 + 3=2)
Using the method of Laplace transform, we get d =2 m
(1 + m) t = t m−=2 ; dt =2
(1 + m − =2)
m¿0 :
(29)
So 1 dEF=2 = kx02 ! t =2 [B1 + B2 ! t + · · · ] ; =2 dt 2
(30)
34
A. To.ghi / Physica A 329 (2003) 29 – 34
where B1 =
(1 + ) C1 ;
(1 + =2)
B2 =
(1 + 2) C2 :
(1 + 3=2)
Finally from Eq. (23), we obtain the following expression for Fid , namely: 1 Fid = − kx0 (1 + =2)[B1 + (!t) B2 + · · · ] : (31) 2 In conclusion, we would like to make the following remarks. In the case of = 2, when we take the time derivative of the energy we obtain dEH d2 x dx m 2 + kx : (32) = dt dt dt But from our equation of motion (1), the expression inside the bracket is zero. So dEH =dt = 0. However, for 1 ¡ ¡ 2 this factorization does not occur. We have tried to give a simple discussion of the intrinsic damping of a fractional oscillator. Interested reader is encouraged to consult Refs. [7–9] for a more sophisticated treatment of a closely related subject. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
R. Metzler, J. Klafter, Phys. Rep. 39 (2000) 1–77. F. Mainardi, Chaos, Solitons & Fractals 7 (9) (1996) 1461–1477. B.N. Narahari Achar, J.W. Hanneken, T. Enck, T. Clarke, Physica A 297 (2001) 361–367. F. Mainardi, R. GorenOo, J. Comput. Appl. Math. 118 (2000) 283–299. R. GorenOo, F. Mainardi, Fractional calculus: integral and di:erential equations of fractional order, in: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, Springer, Wein, 1977, pp. 223–276. M. Caputo, F. Mainardi, Riv. Nuovo Cimento (Ser.II) 1 (1971) 161–198. F. Reiwe, Phys. Rev. E 53 (1996) 1890–1899. F. Reiwe, Phys. Rev. E 55 (1997) 3581–3592. O.P. Agrawal, Am. Soc. Mech. Eng. J. Appl. Mech. 68 (2001) 339–341.
Physica A 329 (2003) 29 – 34
www.elsevier.com/locate/physa
The intrinsic damping of the fractional oscillator Ali To!ghi Faculty of Basic Sciences, Department of Physics, University of Mazandaran, P.O. Box 47415-453, Babolsar, Iran Received 28 March 2003
Abstract We obtain analytical expressions for the time rate of change of the potential energy, the kinetic energy and the total energy of a fractional oscillator in terms of the products of Mittag–Le,er functions. We propose a de!nition for the intrinsic damping force of this oscillator. We obtain a general expression for this damping force. An expression for this damping force in the asymptotic limit (!t → 0) is also obtained. c 2003 Elsevier B.V. All rights reserved. PACS: 03.20.+i Keywords: Fractional oscillation; Intrinsic damping
1. Introduction Some of the most important laws of physics are expressed in terms of certain differential equations. For instance, Newton’s second law of motion involves the !rst derivative of the momentum with respect to time, or equivalently it involves the second derivative of the position vector with respect to the variable time. A new trend is to consider the derivative of the type d x=dt where is non-integer. The motivation for this generalization is that it creates a lot of new possibilities. Furthermore, using fractional di:erential equation, one could explain some complex or anomalous phenomena [1]. In the realm of classical mechanics a simple problem is the so-called fractional oscillator. An early analysis of this problem is given in Ref. [2]. In this work they start from a fractional di:erential equation. A recent study of the fractional oscillator is based on the fractional integral equation [3]. E-mail address: [email protected] (A. To!ghi). c 2003 Elsevier B.V. All rights reserved. 0378-4371/$ - see front matter doi:10.1016/S0378-4371(03)00598-3
30
A. To.ghi / Physica A 329 (2003) 29 – 34
One of the characteristic features of a simple fractional oscillator is the decay of the amplitude of the oscillation. In this paper we will explore the issue of the intrinsic damping of the fractional oscillator. In Section 2 we review the solution of the fractional oscillator. In Section 3 we study the time rate of change of the potential energy, the kinetic energy and the total energy of the fractional oscillator. In Section 4 we suggest a de!nition of the intrinsic damping force of this system. Then we obtain an expression for this damping force. In Section 5, with the help of Laplace transform method we calculate this force in the asymptotic limit !t → 0. 2. Simple fractional oscillator The di:erential equation of a simple harmonic oscillator is d2 x m 2 + kx = 0 : (1) dt The di:erential equation of a simple fractional oscillator is d x m + kx = 0; 1 ¡ 6 2 : (2) dt We note that the dimension of the parameter m in Eq. (2) is MT −2 . Therefore, the parameter ! de!ned by k ! = (3) m has the dimension T −1 . So the fractional di:erential equation of the system is [2] d x (4) + ! x = 0 1 ¡ 6 2 : dt A detailed investigation of the fractional oscillator using an integral equation of motion is given in Ref. [3]. In our notation the appropriate integral equation of motion of this system is t ! x(t) = x(0) + x˙ (0)t − (t − t )−1 x(t ) dt ; (5)
() 0 where again 1 ¡ 6 2. In our notation the basic results of Ref. [3] are follows: x(t) = x0 E; 1 (z) ; p=m
d =2 x = −mx0 ! t =2 E; 1+=2 (z) ; dt =2
(6) (7)
1 1 2 p2 1 (8) kx + = kx2 (E; 1 (z))2 + mx02 !2 t (E; 1+=2 (z))2 ; 2 2m 2 0 2 where z = −! t and the system starts at rest from x(0) = x0 , and the generalized Mittag–Le,er function is de!ned by [4]. ∞ zn ¿0 (9) E; (z) =
(n + ) EF =
n=0
and is an arbitrary complex parameter.
A. To.ghi / Physica A 329 (2003) 29 – 34
31
3. Time rate of change of x, p, UF , KF , EF From Eq. (6) we obtain dx dE; 1 (z) = x0 : dt dt Using the identity [5] 1 + zE; + (z) E; (z) =
()
(10)
(11)
we obtain d z d E; +1 (z) + z E; +1 (z) : (12) E; 1 (z) = dt dt t Now we utilize the following relation [5] d E; +1 (z) (13) E; (z) = E; +1 (z) + z dz to perform the derivative on the right-hand side of Eq. (12). The !nal result is dx (14) = −x0 ! t −1 E; (−! t ) : dt For = 2, from Ref. [4] we have sinh(z 1=2 ) : (15) z 1=2 So in this case, from Eq. (14) we get dx = −!x0 sin(!t) dt the familiar expression from elementary mechanics. Now we consider the time rate of change of the potential energy of fractional oscillator, namely, dx dUF d 1 2 = kx = kx : (16) dt dt 2 dt E2; 2 (z) =
Substituting Eqs. (6) and (14) in Eq. (16) we obtain dUF (17) = −kx02 ! t −1 E; 1 (−! t )E; (−! t ) : dt Next we compute the time rate of change of the generalized momentum p, From Eq. (7) we get dp =2−1 d E; 1+=2 (z) + t =2 E; 1+=2 (z) : (18) = −mx0 ! t dt 2 dt Next, we use Eq. (12) to do the derivative of the right-hand side of Eq. (18). The !nal result is dp = −kx0 t =2−1 E; =2 (−! t ) : (19) dt
32
A. To.ghi / Physica A 329 (2003) 29 – 34
The time rate of change of the kinetic energy is d kF p dp = : dt m dt Substituting Eqs. (7) and (19) in Eq. (20) we obtain dKF = kx02 ! t −1 E; 1+=2 (−! t )E; =2 (−! t ) : dt
(20)
(21)
From Eq. (17) and (21) we obtain the following expression for the time rate of change of total energy of the fractional oscillator: dEF = −kx02 ! t −1 [E; 1 (−! t )E; (−! t ) − E; 1+=2 (−! t )E; =2 (−! t )] : dt (22) We notice that for = 2, The expression inside the bracket of Eq. (22) is identically zero. 4. The intrinsic damping force Fid In this section we investigate the intrinsic damping of the fractional oscillator. For the simple harmonic oscillator ( = 2) the total energy is conserved and x(t) exhibits permanent oscillation. However, in the fractional oscillator the total energy EF is not conserved and x(t) has some attenuated oscillation. The leading asymptotic behaviour of Mittag–Le,er function is [2] E; 1 (−! t ) ∼
(!t)−
(1 − )
as !t → ∞ :
Therefore, x(t) ∼ x0
(!t)−
(1 − )
as !t → ∞ :
This behaviour is the so called algebraic decay of x(t) as !t → ∞. The motion of the fractional oscillator is composed of two parts [5]. First, the oscillator makes a !nite number of damped oscillation. Next, it has a monotonic algebraic decay in the asymptotic region. We suggest the following relation for the intrinsic damping force Fid of the fractional oscillator: d =2 x d =2 EF = Fid =2 : =2 dt dt
(23)
In this paper we adopt the following de!nition for the fractional derivative. Let be a positive but non-integer number. We can always !nd a positive integer m such that m − 1 ¡ ¡ m. Then, t f(m) () d 1 d f = ; m − 1 ¡ ¡ m; (24) dt
(m − ) 0 (t − )+1−m
A. To.ghi / Physica A 329 (2003) 29 – 34
33
where f(m) denotes the mth order derivative of the function f(t). This de!nition is the so-called Caputo [6] derivative. From Eqs. (7), (23) and (24) we obtain the following expression for Fid : t kx0 0 −1 [E; 1 (y)E; (y) − E; 1+=2 (y)E; =2 (y)](t − )−=2 d ; (25) Fid =
(1 − =2)t =2 E; 1+=2 (z) where y = −! and z = −! t . We note that when = 2 dEF =dt = 0 so Fid is identically zero in this case. Moreover, at the turning points p=d x=2 dt =2 =0, therefore, EF = 12 kx2 . But x is a local maximum at these points, so d x=dt = 0 and dEF =dt = 0. Hence d =2 E=dt =2 = 0 at these points. Therefore, Fid is well de!ned everywhere. 5. Asymptotic form of Fid From Eq. (9) and in the asymptotic limit !t → 0 we obtain E; 1 (−! t ) = 1 −
(!t)2 (!t) + + ···
(1 + ) (1 + 2)
E; 1+=2 (−! t ) =
(!t) 1 − + ··· :
(1 + =2) (1 + 3=2)
and
Hence
(26)
x(t) = x0 1 −
(!t) (!t)2 + + ··· ;
(1 + ) (1 + 2) 1 (!t) =2 p = −mx0 ! t − + ··· :
(1 + =2) (1 + 3=2)
(27)
Therefore, EF = 12 kx02 [1 + c1 ! t + c2 !2 t 2 + · · · ] ;
(28)
where c1 =
1 2 − ; 2 ( (1 + =2))
(1 + )
c2 =
1 2 2 + : −
(1 + 2) ( (1 + ))2
(1 + 3=2)
Using the method of Laplace transform, we get d =2 m
(1 + m) t = t m−=2 ; dt =2
(1 + m − =2)
m¿0 :
(29)
So 1 dEF=2 = kx02 ! t =2 [B1 + B2 ! t + · · · ] ; =2 dt 2
(30)
34
A. To.ghi / Physica A 329 (2003) 29 – 34
where B1 =
(1 + ) C1 ;
(1 + =2)
B2 =
(1 + 2) C2 :
(1 + 3=2)
Finally from Eq. (23), we obtain the following expression for Fid , namely: 1 Fid = − kx0 (1 + =2)[B1 + (!t) B2 + · · · ] : (31) 2 In conclusion, we would like to make the following remarks. In the case of = 2, when we take the time derivative of the energy we obtain dEH d2 x dx m 2 + kx : (32) = dt dt dt But from our equation of motion (1), the expression inside the bracket is zero. So dEH =dt = 0. However, for 1 ¡ ¡ 2 this factorization does not occur. We have tried to give a simple discussion of the intrinsic damping of a fractional oscillator. Interested reader is encouraged to consult Refs. [7–9] for a more sophisticated treatment of a closely related subject. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
R. Metzler, J. Klafter, Phys. Rep. 39 (2000) 1–77. F. Mainardi, Chaos, Solitons & Fractals 7 (9) (1996) 1461–1477. B.N. Narahari Achar, J.W. Hanneken, T. Enck, T. Clarke, Physica A 297 (2001) 361–367. F. Mainardi, R. GorenOo, J. Comput. Appl. Math. 118 (2000) 283–299. R. GorenOo, F. Mainardi, Fractional calculus: integral and di:erential equations of fractional order, in: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, Springer, Wein, 1977, pp. 223–276. M. Caputo, F. Mainardi, Riv. Nuovo Cimento (Ser.II) 1 (1971) 161–198. F. Reiwe, Phys. Rev. E 53 (1996) 1890–1899. F. Reiwe, Phys. Rev. E 55 (1997) 3581–3592. O.P. Agrawal, Am. Soc. Mech. Eng. J. Appl. Mech. 68 (2001) 339–341.