Thông tin tài liệu
RIESZ POTENTIALS AS CENTRED DERIVATIVES 109 D c 1 f(t) = - 1 2(-)sin(/2) - + f() |t-| --1 sgn(t-)d (69) 5.3 On the existence of a inverse Riesz potential This means that we can define those potentials even for positive orders. However, we cannot guaranty that there is always an inverse for a given potential. The the- ory presented in section 4.1 allows us to state that: The inverse of a given potential, when existing, is of the same type: the The inverse of a given potential exists iff its order verifies | | < 1. The order of the inverse of an order potential is a - order potential. The inverse can be computed both by (33) [respectively (34)] and by (43) [respectively (44)]. This is in contradiction with the results stated in [10], about this subject and will have implications in the solution of differential equations involving centred derivatives. 5.4 An “analytic” derivative An interesting result can be obtained by combining (53) with (65) to give a com- plex function H D () = H D1 ()+iH D2 () (70) We obtain a function that is null for < 0. This means that the operator defined by (44) is the Hilbert transform of that defined in (43). The inverse Fourier trans- form of (70) is an “analytic signal” and the corresponding “analytic” derivative is given by the convolution of the function at hand with the operator: H (t) = D |t| --1 2(-)cos(/2) -i |t| --1 sgn(t) 2(-)sin(/2) (71) tials [10]. We can give this formula another aspect by noting that inverse of the type k (k = 1,2) potential is a type-k potential. This leads to a convolution integral formally similar to the Riesz–Feller poten- In current literature [7,10], the Riesz potentials are only defined for negative orders verifying -1 < < 0. However, our formulation is valid for every > -1. 1 2(-)cos(/2) = - (+1).sin() 2cos(/2) = - (+1) sin(/2) (72) and - 1 2(-)sin(/2) = (+1).sin() 2sin(/2) = (+1) cos(/2) (73) We obtain easily: H (t) = D - (+1) [] |t| --1 sin(/2) -i|t| --1 sgn(t)cos(/2) (74) that can be rewritten as H (t) = D i(+1) |t| --1 sgn(t)e i/2sgn(t) (75) This impulse response leads to the following potential: D D f(t) = (+1) - + f(t-) || --1 sgn()e i/2sgn() d (76) Of course, the Fourier transform of this potential is zero for < 0. Similarly, the function H D () = H D1 ()-iH D2 () (77) is zero for > 0. Its inverse Fourier transform is easily obtained, proceeding as above. 5.5 The integer order cases It is interesting to use the centred type 1 derivative with = 2M +1 and the type 2 with = 2M. For the first, /2 is not integer and we can use formulae (49) to (54). How- ever, they are difficult to manipulate. We found better to use (55), but we must - (2M+1)! (-1) M . We obtain finally FT -1 [|| 2M+1 ] = - (2M+1)! (-1) M |t| -2M-2 (78) and the corresponding impulse response: Ortigueira avoid the product (-).cos(/2), because the first factor is and the second is zero. To solve the problem, we use (72) to obtain a factor equal to 110 RIESZ POTENTIALS AS CENTRED DERIVATIVES 111 h D1 (t) = - (2M+1)! (-1) M |t| -2M-2 (79) Concerning the second case, = 2M, we use formula (65). As above, we have the product (-).sin(/2) that is again a .0 situation. Using (73) we ob- tain a factor (2M)! (-1) M . We obtain then: FT -1 [|| 2M sgn()] = sgn(t) (2M)!(-1) M |t| -2M-1 (80) and h (t) = D2 sgn(t) (2M)!(-1) M |t| -2M-1 (81) As we can see, the formulae (78) and (80) allow us to generalise the Riesz poten- tials for integer orders. However, they do not have inverse. 6 Conclusions We introduced a general framework for defining the fractional centred differences and consider two cases that are generalisations of the usual even and odd integer orders centred differences. These new differences led to centred derivatives simi- For those differences, we proposed integral representations from where we ob- tained the derivative integrals, similar to the ordinary Cauchy formula, by limit computations inside the integrals and using the asymptotic property of the quotient sion needing two branch cut lines to define a function. For the computation of those integrals we used a special path consisting of two straight lines lying immediately above and below the real axis. These computa- tions led to generalisations of the well known Riesz potentials. The most interesting feature of the presented theory lies in the equality be- tween two different formulations for the Riesz potentials. As one of them is based on a summation formula it will be suitable for numerical computations. To test the coherence of the proposed definitions we applied them to the com- plex exponential. The results show that they are suitable for functions with Fourier transform, meaning that every function with Fourier transform has a centred de- rivative. lar to the usual Grüwald–Letnikov ones. of two gamma functions. We obtained an integrand that is a multivalued expres- 112 Ortigueira Fifth EUROMECH Nonlinear Dynamics Conference, Eindhoven University of Technology, The Netherlands, August, 7–12. 2. Ortigueira MD, Coito F (2004) From Differences to Differintegrations, Fract. Calc. Appl. Anal. 7(4). 3. Ortigueira MD (2006) A coherent approach to non integer order derivatives, Signal Processing, special issue on Fractional Calculus and Applications. 4. Diaz IB, Osler TI (1974) Differences of fractional order, Math. Comput. 28 (125). 5. Ortigueira MD (2006) Fractional Centred Differences and Derivatives, to be presented at the IFAC FDA Workshop to be held at Porto, Portugal, 19–21 July, 2006. 6. Ortigueira MD (2005) Riesz potentials via centred derivatives submitted for publication in the Int. J. Math. Math. Sci. December 2005. 7. Okikiolu GO (1966) Fourier Transforms of the operator Hα, In: Proceedings of Cambridge Philosophy Society 62, 73–78. 8. Andrews GE, Askey R, Roy R (1999) Special Functions, Cambridge University Press, Cambridge. 9. Henrici P (1974) Applied and Computational Complex Analysis, Vol. 1. Wiley, pp. 270–271. 10. Samko SG, Kilbas AA, Marichev OI (1987) Fractional Integrals and Derivatives – Theory and Applications. Gordon and Breach Science, New York. 11. Ortigueira MD (2000) Introduction to Fractional Signal Processing. Part 2: Discrete-Time Systems, In: IEE Proceedings on Vision, Image and Signal Processing, No.1, February 2000, pp. 71–78. References 1. Ortigueira, MD, (2005) Fractional Differences Integral Representation and its Use to Define Fractional Derivatives, In: Proceedings of the ENOC-2005, Part 2 Classical Mechanics and Particle Physics ON FRACTIONAL VARIATIONAL PRINCIPLES Dumitru Baleanu 1 and Sami I. Muslih 2 1 Institute of Space Sciences, P.O. Box MG-36, R 76900, Magurele-Bucharest, baleanu@venus.nipne.ro 2 Department of Physics, Al-Azhar University, Gaza, smuslih@ictp.it Abstract The paper provides the fractional Lagrangian and Hamiltonian formula- tions of mechanical and field systems. The fractional treatment of constrained system is investigated together with the fractional path integral analysis. Fractional Schr¨odinger and Dirac fields are analyzed in details. Keywords S chr¨er 1 Introduction It has been observed that in physical sciences the methodology has changed from complete confidence on the tools of linear, analytic, quantitative mathe- techniques. applications in recent studies in various fields [6 E-mail: Department of Mathematics and Computer Sciences, Faculty of Arts andSciences, Ankara, E-mail: E-mail: Fractional calculus, fractional variational principles, fractional Lagrangian and Hamiltonian, fractional Schrödinger field, oing fractional Dirac field. Variational principles playanimportant role in physics, mathematics,and engi- neering science because they bring together a variety of fields, lead to novel results andrepresent a powerful tool of calculation. matical physics towards a combination of nonlinear, numerical,and qualitative – Derivativesandintegralsoffractionalorder[1 5]havefound manyappli- – 18].Several important results in numerical analysis [19], variousareasofphysics[5],andengineering – have been reported. For example, infieldsasviscoelasticity[20 22],electro- chemistry, diffusion processes[23],theanalysisisformulatedwithrespect respect to fractional-order derivativesandintegrals.Thefractionalderiva- tive accurately describes natural phenomena that occur in such common engineering problems as heat transfer, electrode/electrolytebehavior,and subthreshold nerve propagation [24]. Also,thefractionalcalculusfoundmany © 2007 Springer. in Physics and Engineering, 115 –126. 115 C¸ ankaya University, 06530 Turkey; dumitru@cankaya.edu.tr; Romania; Palestine; J. Sabatier et al. (eds.), Advances in Fractional Calculus: Theoretical Developments and Applications 2 116 classical mechanics [26]. Although many laws of nature can be obtained using certain functionals and the theory of calculus of variations, not all laws can be obtained by using this procedure. For example, almost all systems contain internal damping, describing the behavior of a nonconservative system [27]. For these reasons during the last decade huge efforts were dedicated to apply the fractional calculus to the variational problems [28 conservative and nonconservative systems [28 29]. By using this approach, one can obtain the Lagrangian and the Hamiltonian equations of motion for the nonconservative systems. The fractional variational problem of Lagrange was studied in [32]. A new application of a fractal concept to quantum physics has been reported in [33 34]. The issue of having equations from the use of a fractional Dirac equation of order 2/3 was investigated recently in [36]. Even more recently, the fractional calculus technique was applied to the constrained systems [37 38] and the path integral quantization of fractional mechanical systems with constraints was analyzed in [39]. The aim of this paper is to present some of the latest developments in the formulation are discussed for both discrete systems and field theory. The paper is organized as follows: Euler are presented and the fractional Schr¨odinger equation is obtained from a frac- tional variational principle. Section 4 is dedicated to the fractional Hamilto- nian analysis. Section 5 is dedicated to the fractional path integral of Dirac field. Finally, section 6 is devoted to our conclusions. within the variational principles is the possibility of defining the integration by parts as well as the fractional Euler Lagrange equations become the classical ones when α is an integer. In the following some basic definitions and properties of Riemann Liouville fractional derivatives are presented. many applications in recent studies of scalingphenomena[25]aswellasin yet traditional energy-based approach cannot be used to obtain equations – 31]. Riewe has applied the frac- tional calculus to obtain a formalism which can be used for describing both – – nonconservative – field of fractional variational principles. The fractional Euler Lagrange equa- – tions, the fractional Hamiltonian equations,and the fractional path integral – Lagrange equations for discrete systems are briefly reviewed in sec- tion 2. In section 3 the fractional EulerLagrange equations of field systems – field and nonrelativistic particle interacting with external electromagnetism – 2.1Riemann Liouville fractional derivatives – One of the main advantages of using Riemann Liouville fractional derivatives 2 Fractional Euler Lagrange Equations – – – Baleanu and variational principle was investigatedrecently in [35]. The simple solution of the Muslih ON FRACTIONAL VARIATIONAL PRINCIPLES 117 3 The left Riemann Liouville fractional derivative is defined as follows a D α t f(t)= 1 Γ (n − α) d dt n t a (t − τ) n−α−1 f(τ )dτ, (1) and the form of the right Riemann Liouville fractional derivative is given below t D α b f(t)= 1 Γ (n − α) − d dt n b t (τ − t) n−α−1 f(τ )dτ. (2) Here the order α fulfills n − 1 ≤ α<nand Γ represents the Euler’s gamma function. If α becomes an integer, these derivatives become the usual derivatives a D α t f(t)= d dt α , t D α b f(t)= − d dt α ,α=1, 2, (3) Let us consider a function depending on variables, x 1 ,x 2 , ··· x n .Apartial left Riemann Liouville fractional derivative of order α k ,0<α k < 1, in the -th variable is defined as [2] (D α k a k + f)(x)= 1 Γ (1 − α) ∂ ∂x k x k a k f(x 1 , ···,x k−1 ,u,x k+1 , ···,x n ) (x k − u) α k du (4) and a partial right Riemann Liouville fractional derivative of order α k has the form (D α k a k − f)(x)= 1 Γ (1 − α) ∂ ∂x k a k x k f(x 1 , ···,x k−1 ,u,x k+1 , ···,x n ) (−x k + u) α k du. (5) If the function is differentiable we obtain (D α k a k + f)(x)= 1 Γ (1 − α k ) [ f(x 1 , ···,x k−1 ,a k ,x k+1 , ···,x n ) (x k − a k ) α k ] + x k a k ∂f ∂u (x 1 , ···,x k−1 ,u,x k+1 , ···,x n ) (x k − u) α k du. (6) Many applications of fractional calculus amount to replacing the time deriva- tive in an evolution equation with a derivative of fractional order. For a given classical Lagrangian the first issue is to construct its fractional generalization. The fractional Lagrangian is not unique because there are sev- eral possibilities to replace the time derivative with fractional ones. One of – – f n − k − f 2.2Fractional EulerLagrange equations for mechanical systems − 4 118 the requirements is to obtain the same Lagrangian expression if the order α becomes 1. was considered as L t, q ρ , a D α t q ρ , t D β b q ρ ,whereρ =1, ···n.Let J[q ρ ] be a functional as given below b a L t, q ρ , a D α t q ρ , t D β b q ρ dt, (7) where ρ =1···n defined on the set of functions which have continuous Liouville fractional derivative of order β in [a, b] and satisfy the boundary conditions q ρ (a)=q ρ a and q ρ (b)=q ρ b . In [32] it was proved that a necessary condition for J[q ρ ]toadmitan extremum for given functions q ρ (t),ρ=1, ···,n is that q ρ (t)satisfiesthe ∂L ∂q ρ + t D α b ∂L ∂ a D α t q ρ + a D β t ∂L ∂ t D β b q ρ =0,ρ=1, ···,n. (8) 3 A covariant form of the action would involve a Lagrangian density L via S = Ld 3 xdt where L = L(φ, ∂ μ φ)andwithL = Ld 3 x. The classical covariant Euler Lagrange equation are given below ∂L ∂φ − ∂ μ ∂L ∂(∂ μ φ) =0. (9) Here φ denotes the field variable. In the following the fractional generalization of the above Lagrangian density is developed. Let us consider the action function of the form S = L φ(x), (D α k a k − )φ(x), (D α k a k + )φ(x),x d 3 xdt, (10) where 0 <α k ≤ 1anda k correspond to x 1 ,x 2 ,x 3 and respectively. Let us consider the ǫ finite variation of the functional S(φ), that we write with explicit dependence from the fields and their fractional derivatives, namely Δ ǫ S(φ)= [L(x μ ,φ+ ǫδφ, (D α k a k − )φ(x)+ǫ(D α k a k − )δφ,(D α k a k + )φ(x) n left RiemannLiouville fractional derivative of order α and right Riemann − − following fractional EulerLagrange equations − Fractional Lagrangian Treatment of Field Theory 3.1Fractional classical fields − t The most general case was investigated in [32], namely the fractional lagrangian + ǫ(D α k a k + )δφ) −L(x μ ,φ,(D α k a k − )φ(x), (D α k a k + )φ(x))]d 3 xdt. (11) Baleanu and Muslih [...]... Scalas E, Mainardi F (2002) Waiting-times and returns in high- frequency financial data: an empirical study Physica A, 3 14( 1 4) : 749 –755 Ortigueira MD (2003) On the initial conditions in continuous-time fractional linear systems Signal Processing, 83(11):2301–2309 Agrawal OP (20 04) Application of fractional derivatives in thermal analysis of disk brakes Nonlinear Dynamics, 38(1 4) :191–206 Tenreiro Machado. .. expansion (38) Pint (t) ∼ const t−κ (41 ) can be easily understood by substituting it into (38) Omitting the term ΔP we have for the singular part of the solution to (38): t−κ = (1/λT λa )(t/λT )−κ (42 ) that leads to (40 ) Comparing (40 ) to (26) we obtain γ = 2 + ln λa / ln λT μ = γ − 1 = κ = 1 + ln λa / ln λT (43 ) Substitutions of values (20) into (43 ) defines μ ≈ 1.88, γ ≈ 2.88 (44 ) in a good agreement... second case is called pseudochaos and the last case can be close to either chaos or to pseudochaos, depending on the situation Additional insight into chaos and pseudochaos is given in the review paper [1] It becomes clear that the last 127 J Sabatier et al (eds.), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, 127–138 © 2007 Springer 2 128 Zaslavsky... non-ergodic piece-wise affine maps of the torus Ergodic Theory Dynam Sys 21:959–999 Gutkin E, Katok A (1989) Weakly mixing billiards In: Holomorphic Dynamics, Lecture Notes in Mathematics 1 345 :163–176 Springer, Berlin FRACTIONAL KINETICS IN PSEUDOCHAOTIC SYSTEMS 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 137 1 1 Katok A (1987) The growth-rate for the number of singular and periodic-orbits... already in proceedings of IDETC/CIE 2005, the ASME 2005 InterInternational Design Engineering Technical Conference and Computers and information in Engineering Conference, September 24 28, 2005, Long Beach, California, USA This work was done within the framework of the Associateship Scheme of the Abdus Salam ICTP References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Miller KS, Ross B (1993) An Introduction... fractional derivatives in viscoelasticity J Sound Vibr., 2 84: 1239–1 245 Adolfsson K, Enelund M, Olsson P (2005) On the fractional order model of viscoelasticity Mechanic of Time-Dependent Mater, 9:15– 34 Metzler R, Joseph K (2000) Boundary value problems for fractional diffusion equations Physica A, 278:107–125 Magin RL (20 04) Fractional calculus in bioengineering Crit Rev Biom Eng., 32(1):1–1 04 Zaslavsky GM... where, n = 0, , N, n′ = 1, , N ′ 5 Fractional Path Integral Formulation In this section we define the fractional path integral as a generalization of the classical path integral for fractional field systems The fractional path integral for unconstrained systems emerges as follows K= dφ dπ α dπ β exp i d4 x π α a Dα φ + π β t Dβ φ − H t b (40 ) ON FRACTIONAL VARIATIONAL PRINCIPLES 123 5.1 Dirac field Lagrangian... Nevertheless, Laurent [4] solved the integral Eq (2) using fractional operators Today, a lot of different definitions for Abel’s integral equations are given in the literature and we have to distinguish them carefully Let us start with the following integral SEMI-INTEGRALS AND SEMI-DERIVATIVES IN PARTICLE PHYSICS 141 equation, which is also called Abel equation, with has variable lower integration limit... (41 ) By using (41 ) the generalized momenta become ¯ (πt− )ψ = ψγ 0 , (πt− )ψ = 0 ¯ (42 ) From (41 ) and (42 ) we construct the canonical Hamiltonian density as 2/3 ¯ ¯ HT = −ψ γ k Dk− ψ(x) + (m)2/3 ψ(x) +λ1 [(πt− )ψ − ψγ 0 ]+λ2 [(πt− )ψ ] (43 ) ¯ Making use of (43 ), the canonical equations of motion have the following forms 2/3 2/3 ¯ ¯ Dt+ (πt− )ψ = −(m)2/3 ψ(x) − Dk− γ k ψ(x), (44 ) = −(m)2/3 ψ(x) − (45 )... (2001) Discrete-time fractional order controllers Fract Calc Appl Anal., 4( 1) :47 –68 Lorenzo CF, Hartley TT (20 04) Fractional trigonometry and the spiral functions Nonlinear Dynamics, 38(1 4) :23–60 Baleanu D, Avkar T (20 04) Lagrangians with linear velocities within RiemannLiouville fractional derivatives Nuovo Cimento, B119:73–79 12 126 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 . second FRACTIONAL KINETICS IN PSEUDOCHAOTIC SYSTEMS © 2007 Springer. 127 J. Sabatier et al. (eds.), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering,. for fractional diffusion equations. Physica A, 278:107–125. 24. Magin RL (20 04) Fractional calculus in bioengineering. Crit. Rev. Biom. Eng., 32(1):1–1 04. 25. Zaslavsky GM (2002) Chaos, fractional. and returns in high- frequency financial data: an empirical study. Physica A, 3 14( 1 4) : 749 –755. 13. Ortigueira MD (2003) On the initial conditions in continuous-time fractional linear systems.
Ngày đăng: 12/08/2014, 16:21
Xem thêm: Sabatier Agrawal Machado Advances in Fractional Calculus Episode 4 doc
