Sabatier Agrawal Machado Advances in Fractional Calculus Episode 4 doc

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.. Ortigueira, MD, 2005 Fractio

RIESZ POTENTIALS AS CENTRED DERIVATIVES 109

Dc1f(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 ory presented in section 4.1 allows us to state that:

the-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 plex function

com-HD( ) = HD1( )+iHD2( ) (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 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:

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

DDf(t) = ( +1)

-+ f(t- ) | |- -1 sgn( )ei /2sgn( )d (76)

Of course, the Fourier transform of this potential is zero for < 0 Similarly, the function

HD( ) = HD1( )-iHD2( ) (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) ever, they are difficult to manipulate We found better to use (55), but we must

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

M We obtain then:

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 tween two different formulations for the Riesz potentials As one of them is based

be-on a summatibe-on formula it will be suitable for numerical computatibe-ons

To test the coherence of the proposed definitions we applied them to the 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

com-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

applications in recent studies in various fields [6

E-mail:

Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences,

Ankara,E-mail:

engi-matical physics towards a combination of nonlinear, numerical, and qualitative

–Derivatives and integrals of fractional order [1 5] have found many appli-

–18] Several importantresults in numerical analysis [19], various areas of physics [5], and engineering

–have been reported For example, in fields as viscoelasticity [20 22], electro-chemistry, diffusion processes [23], the analysis is formulated with respectrespect to fractional-order derivatives and integrals The fractional deriva-tive accurately describes natural phenomena that occur in such commonengineering problems as heat transfer, electrode/electrolyte behavior, andsubthreshold nerve propagation [24] Also, the fractional calculus found many

conservative and nonconservative systems [28 29] By using this approach,one can obtain the Lagrangian and the Hamiltonian equations of motion forthe nonconservative systems.

The fractional variational problem of Lagrange was studied in [32] A newapplication of a fractal concept to quantum physics has been reported in

fractional Dirac equation of order 2/3 was investigated recently in [36] Evenmore recently, the fractional calculus technique was applied to the constrainedsystems [37 38] and the path integral quantization of fractional mechanicalsystems 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 tional variational principle Section 4 is dedicated to the fractional Hamilto-nian analysis Section 5 is dedicated to the fractional path integral of Diracfield Finally, section 6 is devoted to our conclusions

frac-within the variational principles is the possibility of defining the integration byparts as well as the fractional Euler Lagrange equations become the classicalones when α is an integer

In the following some basic definitions and properties of Riemann Liouvillefractional derivatives are presented

many applications in recent studies of scaling phenomena [25] as well as in

yet traditional energy-based approach cannot be used to obtain equations

–31] Riewe has applied the tional calculus to obtain a formalism which can be used for describing both

One of the main advantages of using Riemann Liouville fractional derivatives

2 Fractional Euler Lagrange Equations

–

––

Baleanu and

variational principle was investigated recently in [35] The simple solution of the

Muslih

ON FRACTIONAL VARIATIONAL PRINCIPLES 117

The left Riemann Liouville fractional derivative is defined as follows

aDαtf (t) = 1

Γ (n − α)

ddt

aDαtf (t) =

ddt

In [32] it was proved that a necessary condition for J[qρ] to admit anextremum for given functions qρ(t), ρ = 1, · · · , n is that qρ(t) satisfies the

∂L

∂φ− ∂μ ∂L

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

where 0 < αk ≤ 1 and ak correspond to x1, x2, x3 and respectively Let

us consider the ǫ finite variation of the functional S(φ), that we write withexplicit dependence from the fields and their fractional derivatives, namely

ΔǫS(φ) =

[L(xμ, φ + ǫδφ, (Dαk

ak−)φ(x) + ǫ(Dαk

ak−)δφ, (Dαk

ak+)φ(x)

nleft Riemann Liouville fractional derivative of order α and right Riemann− −

following fractional Euler Lagrange equations−

Fractional Lagrangian Treatment of Field Theory

3.1 Fractional classical fields

ON FRACTIONAL VARIATIONAL PRINCIPLES 119

We develop the first term in the square brackets, which is a function on

ǫ, as a Taylor series in ǫ and we retain only the first order By using (11) weobtain

∂φδφ +

{(Dαk

After taking the limit limǫ−→0ΔǫS(φ)ǫ we obtain the fractional EulerLagrange equations as given before

∂L

∂φ+

{(Dαk

4

4 4

4

We observe that if αk→ 1 the usual Schr¨odinger equation is obtain.

In the following we briefly review Riewe’s formulation of fractional tion of Lagrangian and Hamiltonian equations of motion The starting point

generaliza-is the following action

S =

 b a

Here the generalized coordinates are defined as

qnr = (aDαt)nxr(t), Qrn′ = (tDαb)n′xr(t), (22)and r = 1, 2, , R represents the number of fundamental coordinates, n =

0, , N, is the sequential order of the derivatives defining the generalized ordinates q, and n′ = 1, , N′ denotes the sequential order of the derivatives

co-in definition of the coordco-inates Q

A necessary condition for S to posses an extremum for given functions

xr(t) is that xr(t) fulfill the Euler Lagrange equations

3.2 Fractional Schr¨odinger equation

, respectively As a result the Euler−

4 Fractional Hamiltonian Formulations

4.1 Riewe approach

−

Dumitru and Muslih

ON FRACTIONAL VARIATIONAL PRINCIPLES 121

∂H

∂qr N

vari-4.2 Fractional Hamiltonian formulation of constrained systems

¯

L = L + λmΦm(t, q10, · · · , qR0, qrn, Qrn′), (33)where λmrepresents the Lagrange multiplier and L(qnr, Qrn′, t)

Using (32),the canonical Hamiltonian becomes

The other set of equations of motion are given by

∂qr n

In this section we define the fractional path integral as a generalization of theclassical path integral for fractional field systems The fractional path integralfor unconstrained systems emerges as follows

5 Fractional Path Integral Formulation

Dumitru and Muslih

ON FRACTIONAL VARIATIONAL PRINCIPLES 1239

HT = − ¯ψ$

γkD2/3k

−ψ(x) + (m)2/3ψ(x)%

+λ1[(πt−)ψ− ¯ψγ0]+λ2[(πt−)ψ¯] (43)Making use of (43), the canonical equations of motion have the followingforms

K =

d(πt−)ψ d(πt−)ψ¯ dψ d ¯ψδ[(πt−)ψ− ¯ψγ0]δ[(πt−)ψ¯]

× exp i



d4x'(πt−)ψD2/3t− ψ + (πt−)ψ¯D2/3t− ψ − H¯ ( (50)Integrating over (πα−)ψ and (πα−)ψ¯, we arrive at the result

6 Conclusions

tions of motion for both discrete and field theories As an example the tional Schr¨odinger equation for a single particle moving in a potential V (x)was obtained from a fractional variational principle The fractional Hamil-tonian was constructed by using the Riewe’s formulation and the extension

frac-of Agrawal’s approach for the case frac-of fractional constrained systems was sented The classical results are recovered under the limit α → 1 The existencefractional Lagrangians make the notion of fractional mechanical constrainedsystems not an easy notion to be defined Therefore we have to take into ac-For a given fractional constrained mechanical system a Poisson bracket wasdefined and it reduces to the classical case under certain limits The fractionalpath integral approach was analyzed and the fractional actions for Dirac’s fieldwere found We mention that in this manuscript the fractional path integralformulation represents the fractional generalization of the classical case Westress on the fact the fractional path integral formulation depends on thedefinitions of the momenta and the fractional Hamiltonian

pre-5.2 Nonrelativistic particle interacting with external

Let us consider the Lagrangian for a nonrelativistic particle of mass m and

We have presented the fractional extensions of the usual Euler Lagrange equa-−

of various definitions of fractional derivatives and the nonlocality property of

count the nonlocality property during the fractional quantization procedure

and nonrelativistic particle interacting with external electromagnetism field

Dumitru and Muslih

ON FRACTIONAL VARIATIONAL PRINCIPLES 125

Acknowledgments

Dumitru Baleanu would like to thank O Agrawal and J A Tenreiro Machadofor interesting discussions Sami I Muslih would like to thank the Abdus SalamInternational Center for Theoretical Physics, Trieste, Italy, for support andhospitality during the preliminary preparation of this work The authors wouldlike to thank ASME for allowing them to republish some results which werepublished already in proceedings of IDETC/CIE 2005, the ASME 2005 Inter-International Design Engineering Technical Conference and Computers andinformation in Engineering Conference, September 24 28, 2005, Long Beach,California, USA This work was done within the framework of the AssociateshipScheme of the Abdus Salam ICTP

7 Silva MF, Machado JAT, Lopes AM (2005) Modelling and simulation of artificial locomotion systems; Robotica, 23(5):595–606

8 Mainardi F (1996) Fractional relaxation-oscillation and fractional diffusion-wave phenomena Chaos, Solitons and Fractals, 7(9):1461–1477

9 Zaslavsky GM (2005) Hamiltonian Chaos and Fractional Dynamics Oxford

University Press, Oxford

10 Mainardi F (1996) The fundamental solutions for the fractional diffusion-wave equation Appl Math Lett., 9(6)23–28

11 Tenreiro Machado JA (2003) A probabilistic interpretation of the fractional order differentiation Fract Calc Appl Anal., 1:73–80

12 Raberto M, Scalas E, Mainardi F (2002) Waiting-times and returns in high- frequency financial data: an empirical study Physica A, 314(1–4):749–755

13 Ortigueira MD (2003) On the initial conditions in continuous-time fractional linear systems Signal Processing, 83(11):2301–2309

14 Agrawal OP (2004) Application of fractional derivatives in thermal analysis of disk brakes Nonlinear Dynamics, 38(1–4):191–206

15 Tenreiro Machado JA (2001) Discrete-time fractional order controllers Fract Calc Appl Anal., 4(1):47–68

16 Lorenzo CF, Hartley TT (2004) Fractional trigonometry and the spiral functions Nonlinear Dynamics, 38(1–4):23–60

17

Podlubny I (1999) Fractional Differential Equations Academic Press, New York

Liouville fractional derivatives Nuovo Cimento, B119:73–79

Baleanu D, Avkar T (2004) Lagrangians with linear velocities within

20 Blutzer RL, Torvik PJ (1996) On the fractional calculus model of viscoelastic behaviour J Rheology, 30:133–135

21 Chatterjee A (2005) Statistical origins of fractional derivatives in viscoelasticity J Sound Vibr., 284:1239–1245

22 Adolfsson K, Enelund M, Olsson P (2005) On the fractional order model of viscoelasticity Mechanic of Time-Dependent Mater, 9:15–34

23 Metzler R, Joseph K (2000) Boundary value problems for fractional diffusion equations Physica A, 278:107–125

24 Magin RL (2004) Fractional calculus in bioengineering Crit Rev Biom Eng., 32(1):1–104

25 Zaslavsky GM (2002) Chaos, fractional kinetics, and anomalous transport Phys Rep., 371(6):461–580

26 Rabei EM, Alhalholy TS (2004) Potentials of arbitrary forces with fractional derivatives Int J Mod Phys A, 19(17–18):3083–3092

27 Bauer PS (1931) Dissipative dynamical systems I Proc Natl Acad Sci., 17:311–314

28 Riewe F (1996) Nonconservative Lagrangian and Hamiltonian mechanics Phys Rev., E53:1890–1899

29 Riewe F (1997) Mechanics with fractional derivatives, Phys Rev E 55:3581–3592

30 Klimek M (2001) Fractional sequential mechanics-models with symmetric fractional derivatives Czech J Phys., 51, pp 1348–1354

31 Klimek M (2002) Lagrangean and Hamiltonian fractional sequential mechanics Czech J Phys 52:1247–1253

32 Agrawal OP (2002) Formulation of Euler – Lagrange equations for fractional

variational problems J Math Anal Appl., 272:368–379

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AND ITS APPLICATIONS

zero Lyapunov exponents Mixed dynamics means an alternation of the finitecase is called pseudochaos and the last case can be close to either chaos or topseudochaos, depending on the situation Additional insight into chaos andpseudochaos is given in the review paper [1] It becomes clear that the last

New York University, 251 Mercer Street, New York, NY 10012;

E-mail:

(i.e., dynamics is random but the Lyapunov exponent is zero), and the

corre-or random, corre-or mixed Chaotic dynamics means the existence of a nonzeroLyapunov exponent Random dynamics means nonpredictable motion withtime Lyapunov exponent between almost zero and nonzero values The 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, 127–138

two cases correspond to a random dynamics that cannot be described bythe processes of the Gaussian or Poissonian, or similar types with all finitemoments A more adequate description of chaos and pseudochaos corresponds

to the process of the L´evy type, with infinite second and higher moments, duefractional kinetic equation (FKE) was introduced in [2–4] in which the ideas

of L´evy flights and fractal time [5] were applied to the specific characteristic

of the randomness generated by the instability of the dynamics, rather than

by the presence of external random forces

A typical FKE has the form

∂βF (y, t)

∂tβ = D∂

αF (y, t)

∂|y|α , (0 < β ≤ 1, 0 < α ≤ 2) (1)where F (y, t) is the probability density function, and fractional derivativescould be of arbitrary type, specifically depending on the physical situation

of the initial-boundary conditions, etc More discussions on this subject anddifferent modifications of (1) can be found in [6] The general type of literaturerelated to the FKE is fairly large (see references in [1] and [7]) This work will

be restricted to specific dynamical systems

The most important issue of application of (1) to the dynamical systems

is that exponents (α, β) are defined by the dynamics only and, in some way,they characterize the local property of instability of trajectories This provides

a possibility to find the values of (α, β) from the first principles, and this will

be the subject of this paper where the dynamics in some rectangular billiardswill be considered, and a review of some previous results, as well as new ones,will be presented

Consider a standard definition of the finite-time Lyapunov exponent σt [8]:

σt= 1

where d(t) is a distance between two trajectories started in a very small main A, such that d0 ≤ diam A The function σt is fairly complicated anddepends on the choice of A in the full phase space Γ and on d0 To simplify theapproach one can consider a coarse-graining (smoothing) of σtover arbitrarysmall volume δΓ (A) → 0 Consider the measure