Abstract
Fractional calculus becomes a powerful tool used to investigate complex phenomena from various fields of science and engineering. In this context, the researchers paid a lot of attention for the fractional dynamics. However, the fractional modeling is still at the beginning of its developing. The aim of this chapter is to present some new results in the area of fractional dynamics and its applications.
Chapter PDF
References
Agrawal O.P., 2002, Formulation of Euler-Lagrange equations for fractional variational problems, Journal of Mathematical Analysis and Applications, 272, 368–379.
Agrawal O.P., 2004, A general formulation and solution scheme for fractional optimal control problems, Nonlinear Dynamics, 38, 191–206.
Agrawal O.P., 2006, Fractional variational calculus and the transversahty conditions, Journal of Physics A:Mathematical and General, 39, 10375–10384.
Agrawal O.P., 2007, Generalized Euler-Lagrange equations and transversality conditions for FVPs in terms of the Caputo derivative, Journal of Vibration and Control, 13, 1217–1237.
Agrawal O.P. and Baleanu D., 2007, Hamiltonian formulation and a direct numerical scheme for fractional optimal control problems, Journal of Vibration and Control, 13, 1269–1281.
Alvarez-Gaume L. and Barbon J.L.F., 2001, Gauge theory on a quantum phase space, International Journal of Modern Physics A, 16, 1123–1134.
Baleanu D., 2004, Constrained systems and Riemann-Liouville fractional derivative, Proceedings of 1st IFAC Workshop on Fractional Differentiation and its Applications, Bordeaux, France, July 19-21, 597–601.
Baleanu D., 2006, Fractional Hamiltonian analysis of irregular systems, Signal Processing, 86, 2632–2636.
Baleanu D., 2008, New applications of fractional variational principles, Reports on Mathematical Physics, 61, 199–206.
Baleanu D., 2009, About fractional quantization and fractional variational principies, Communications in Nonlinear Science and Numerical Simulation, 14, 2520–2523.
Baleanu D. and Agrawal O.P., 2006, Fractional Hamilton formalism within Caputo’s derivative, Czechoslovak Journal of Physics, 56, 1087–1092.
Baleanu D. and Avkar T., 2004, Lagrangians with linear velocities within Riemann Liouville fractional derivatives, Nuovo Cimento B, 119,73–79.
Baleanu D. and Trujillo J.J., 2008, On exact solutions of a class of fractional Euler Lagrange equations, Nonlinear Dynamics, 52,331–335.
Baleanu D. and Muslih S.I., 2005a, About fractional supersymmetric quantum mechanics, Czechoslovak Journal of Physics, 55, 1063–1066.
Baleanu D. and Muslih S.I., 2005b, Lagrangian formulation of classical fields within Riemann-Liouville fractional derivatives, Physica Scripta, 72, 119–121.
Baleanu D., Defterli O. and Agrawal O.P., 2009, A central difference numerical scheme for fractional optimal control problems, Journal of Vibration and Controls 15, 583–597.
Baleanu D., Golmankhaneh A.K. and Golmankhaneh A.K., 2009a, The dual action of fractional multi time Hamilton equations, International Journal of Theoretical Physics, 48, 2558–2569.
Baleanu D., Golmankhaneh A.K. and Golmankhaneh A.K., 2009b, Fractional Nambu mechanics, International Journal of Theoretical Physics, 48, 1044–1052.
Baleanu D., Maaraba T. and Jarad F., 2008a, Fractional variational principles with delay, Journal of Physics A:Mathematical and Theoretical, 41, Art. No. 315403.
Baleanu D., Muslih S.I. and Rabei E.M., 2008b, On fractional Euler-Lagrange and Hamilton equations and the fractional generalization of total time derivative, Nonlinear Dynamics, 53, 67–74.
Baleanu D., Muslih S.I. and Tas K., 2006, Fractional Hamiltonian analysis of higher order derivatives systems, Journal of Mathematical Physics, 47, Art. No.103503.
Barbosa R.S., Machado J.A.T. and Ferreira I.M., 2004, Tuning of PID controllers based on Bode’s ideal transfer function, Nonlinear Dynamics, 38, 305–321.
Birell N.D. and Davies P.C.W., 1982, Quantum Field in Curved Space, Cambridge University Press, Cambridge
Caputo M., 2001, Distributed order differential equations modelling dielectric induction and diffusion, Fractional Calculus and Applied Analysis, 4, 421–442.
Carpinteri A. and Mainardi F. (Eds), 1997, Fractals and Fractional Calculus in Continuum Mechanics, Springer, Berlin
Chen Y.Q., Vinagre B.M. and Podlubny I., 2004, Continued fraction expansion approaches to discretizing fractional order derivatives — an expository review, Nonlinear Dynamics, 38, 155–170.
Dinç; E. and Baleanu D., 2004a, Application of the wavelet method for the simultaneous quantitative determination of benazepril and hydrochlorothiazide in their mixtures, Journal of AOAC International, 87, 834–841.
Dinç; E. and Baleanu D., 2004b, Multicomponent quantitative resolution of binary mixtures by using continuous wavelet transform, Journal of AOAC International, 87, 360–365.
Dinç; E. and Baleanu D., 2006, A new fractional wavelet approach for simultaneous determination of sodium and sulbactam sodium in a binary mixture, Spectrochimica Acta Part A, 63, 631–638.
Dinç, E. and Baleanu D., 2007, A review on the wavelet transform applications in analytical chemistry, Mathematical Methods in Engineering, Springer, Eds. K. Tas, J.A. Tenreiro Machado, D. Baleanu, pp. 265–285.
Dinç; E. and Baleanu D., 2010, Fractional wavelet transform for the quantitative spectral resolution of the composite signals of the active compounds in a two-component mixture, Communications in Nonlinear Science and Numerical Simulation, 15, 812–818.
Dinç; E., Baleanu D. and Ustundag O., 2003, An approach to quantitative two-component analysis of a mixture containing hydrochlorothiazide and spirono-lactone in tablets by one-dimensional continuous Daubechies and biorthogonal wavelet analysis of UV-spectra, Spectroscopy Letters, 36, 341–355.
Dong J.P. and Xu M.Y., 2008, Space-time fractional Schrödinger equation with time-independent potentials, Journal of Mathematical Analysis and Applications, 344, 1005–1017.
Fogleman M.A., Fawcett M.J. and Solomon T.H., 2001, Lagrangian chaos and correlated Lévy flights in a non-Beltrami flow: Transient versus-long term transport, Physical Review E, 63, 020101–1.
Gelfand I.M. and Shilov G.E., 1964, Generalized Functions, Vol.I, Properties and Operators, Accademic Press, New York.
Gitman D.M. and Tytin I.V., 1990, Quantization of fields with constraints, Springer, Berlin.
Gomis J. and Mehen T., 2000, Space-time noncommutative field theories and unitarity, Nuclear Physics B, 591, 265–276.
Gomis J., Kamimura K. and Llosa J., 2001, Hamiltonian formalism for space-time noncommutative theories, Physical Review D, 63, 045003.
Gomis J., Kamimura K., Ramirez T. and Gomis J., 2004, Physical degrees of freedom of non-local theories, Nuclear Physics B, 696, 263–275.
Gorenflo R. and Mainardi F., 1997, Fractional Calculus: Integral and Differential; Equations of Fractional Orders, Fractals and Fractional Calculus in Continuum Mechanics, Springer, Wien and New York
Hardy G.H., 1945, Riemann’s form of Taylor’s series, Journal of the London Mathematical Society, 20, 48–56.
Heymans N. and Podlubny I., 2006, Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives, Rheologica Acta, 45, 765–771.
Hilfer R., 2000, Applications of Fractional Calculus in Physics, World Scientific Publishing Company, Singapore.
Jesus I.S. and Machado J.A.T., 2008, Fractional control of heat diffusion systems, Nonlinear Dynamics, 54, 263–282.
Jumarie G., 2009, From Lagrangian mechanics fractal in space to space fractal Schrödinger’s equation via fractional Taylor’s series, Chaos Solitons and Fractals, 41, 1590–1604.
Kilbas A.A., Srivastava H.H. and Trujillo J.J., 2006, Theory and Applications of Fractional Differential Equations, Volume 204 (North-Holland Mathematics Studies) Elsevier Science, Amsterdam.
Klimek M., 2001, Fractional sequential mechanics — models with symmetric fractional derivative Czechoslovak Journal of Physics, 51, 1348–1354.
Klimek M., 2002, Lagrangean and Hamiltonian fractional sequential mechanics, Czechoslovak Journal of Physics, 52, 1247–1253.
Lakshmikantham V., Leela S. and Vasundhara Devi J., 2009, Theory of Fractional; Dynamic Systems, Cambridge Scientific Publishers Ltd., Cambridge.
Lim S.C. and Teo L.P., 2009, The fractional oscillator process with two indices, Journal of Physics A:Mathematical and Theoretical, 42, Art. No. 065208.
Lim S.C. and Muniandy S.V., 2004, Stochastic quantization of nonlocal fields, Physics Letters A, 324, 396–405.
Llosa J. and Vives J., 1994, Hamiltonian-formalism for nonlocal Lagrangians, Journal of Mathematical Physics, 35, 2856–2877.
Lorenzo C.F. and Hartley T.T., 2004, Fractional trigonometry and the spiral functions, Nonlinear Dynamics, 38, 23–34.
Magin R.L., 2000, Fractional Calculus in Bioengineering, Begell House Publisher, Connecticut.
Magin R., Feng X. and Baleanu D., 2009, Solving the fractional order bloch equation, Concepts in Magnetic Resonance Part A, 34A, 16–23.
Magin R.L., Abdullah O., Baleanu D. and Zhou X.H.J., 2008, Anomalous diffu-I sion expressed through fractional order differential operators in the Bloch-Torrey equation, Journal of Magnetic Resonance, 190, 255–270.
Mainardi F., 1996, Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos, 7, 1461–1476.
Mainardi F., Luchko Yu. and Pagnini G., 2001, The fundamental solution of the space-time fractional diffusion equation, Fractional Calculus and Applied Analysis, 4, 153–192.
Maraaba T.A., Baleanu D. and Jarad F., 2008a, Existence and uniqueness theorem for a class of delay differential equations with left and right Caputo fractional derivatives, Journal of Mathematical Physics, 49, Art. No. 083507.
Maraaba T.A., Jarad F. and Baleanu D., 2008b, On the existence and the uniqueness theorem for fractional differential equations with bounded delay within Caputo derivatives, Science in China Series A-Mathematics, 51, 1775–1786.
Metzler R., SChick W., Kilian H.G. and Nonennmacher T.F., 1995, Relaxation in filled polymers: A fractional calculus approach, Journal of Chemical Physics; 103, 160137–1.
Miller K.S. and Ross B., 1993, An Introduction to the Fractional Integrals and Derivatives-Theory and Applications, John Wiley and Sons Inc., New York
Momani S., 2006, A numerical scheme for the solution of multi-order fractional differential equations, Applied Mathematics and Computation, 182, 761–786.
Muslih S.I. and Baleanu D., 2005a, Hamiltonian formulation of systems with Iinear velocities within Riemann-Liouville fractional derivatives, Journal of Mathematical Analysis and Applications, 304, 599–606.
Muslih S.I. and Baleanu D., 2005b, Quantization of classical fields with fractional derivatives, Nuovo Cimento B, 120, 507–512.
Muslih S.I. and Baleanu D., 2005c, Hamiltonian formulation of systems with linear velocities within Riemann-Liouville fractional derivatives, Journal of Mathematical Analysis and Applications, 304, 599–606.
Naber M., 2004, Time fractional Schrödinger equation, Journal of Mathematical Physics, 45, 3339–3352.
Nesterenko V.V., 1989, Singular Lagrangians with higher deratives, J. Phy. A-Math. Gen., 22, 1673–1687.
Nigmatullin R.R. and Mehaute A.L., 2005, Is there geometrical/physical meaning of the fractional integral with complex exponent? Journal of Non-Crystalline Solids, 351, 2888–2899.
Oldham K.B. and S’panier, J., 1974, The Fractional Calculus, Academic Press, New York.
Pais A. and Uhlenbeck G.E., 1950, On field theories with non-localized action, Physical Review, 79, 145–165.
Podlubny I., 1999, Fractional Differential Equations, Academic Press, San Diego CA.
Rabei E.M., Nawafleh K.I., Hijjawi R.S., Muslih S.I. and Baleanu D., 2007, The Hamilton formalism with fractional derivatives, Journal of Mathematical Analysis and Applications, 327, 891–897.
Rabei E.M., Altarazi I.M.A., Muslih S.I. and Baleanu D., 2009, Fractional WKB approximation, Nonlinear Dynamics, 57, 171–175.
Raspini A., 2001, Simple solutions of the fractional Dirac equation of order 2/3, Physica Scripta, 64, 20–22.
Riewe F., 1996, Nonconservative Lagrangian and Hamiltonian mechanics, Physical Review E, 53, 1890–1899.
Riewe F., 1997, Mechanics with fractional derivatives, Physical Review E, 55, 3581–3592
Samko S.G., Kilbas A.A. and Marichev O.I., 1993, Fractional Integrals and Derivatives — Theory and Applications, Gordon and Breach, Linghome.
Scalas E., Gorenflo R. and Mainardi F., 2004, Uncoupled continuous-time random walks: Solution and limiting behavior of the master equation, Physical Review E, 69, 011107–1.
Seiberg N., Susskind L. and Toumbas T., 2000, Strings in background electric field, space/time noncommutativity and a new noncritical string theory, Journal of High; Energy Physics, 6, Art. No. 021.
Silva M.F., Tenreiro-Machado J.A.T. and Barbosa R.S., 2008, Using fractional derivatives in joint control of hexapod robots, Journal of Vibration and Control, 14, 1473–1485.
Simon J.Z., 1990, Higher-derivative Lagrangians, nonlocality, problems and Solutions, Physical Review D, 41, 3720–3733.
Solomon T.H., Weeks E.R. and Swinney H.L., 1993, Observation of anomalous diffusion and Lévy flights in a two-dimensional rotating flow, Physical Review Letters, 71, 3975–3978.
Tarasov V.E., 2005, Continuous medium model for fractal media, Physics Letters A, 336, 167–174.
Tarasov V.E., 2006, Fractional statistical mechanics, Chaos, 16, 033108–033115.
Tarasov V.E. and Zaslavsky G.M., 2006, Nonholonomic constraints with fractional derivatives, Journal of Physics A:Mathematical and General, 39, 9797–9815.
Tenreiro Machado, J.A., 2001, Discrete-time fractional-order controllers, Fractional Calculus and Applied Analysis, 4, 47–66.
Tenreiro Machado J. A., 2003, A probabilistic interpretation of the fractional order differentiation, Fractional Calculus and Applied Analysis, 6, 73–81.
Trujillo J.J., 1999, On a Riemann-Liouville generalized Taylor’s formula, Journal of Mathematical Analysis and Applications, 231, 255–264.
Unser M. and Blu T., 1999, Construction of fractional spline wavelet bases, in proc. SPIE Wavelets Applications in Signal and Image Processing VII, Denver, CO, 3813, 422–431.
Unser M. and Blu T., 2000a, Fractional splines and wavelets, SIAM Review, 42, 43–67.
Unser M. and Blu T., 2000b, The fractional spline wavelet transform: definition and implementation, Proceedings of the Twenty-Fifth IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP’00), Istanbul, Turkey, June 5-9, vol. I, pp. 512–515.
Unser M. and Blu T., 2002, Wavelets, fractals, and radial basis functions, IEEE Transactions on Signal Processing, 50, 543–553.
Walczak B., 2000, Wavelets in Chemistry, Elsevier Press, Amsterdam.
West B.J., Bologna M. and Grigolini P., 2003, Physics of Fractal Operators, Springer, New York.
Zaslavsky G.M., 2002, Chaos, fractional kinetics, and anomalous transport, Physics Reports, 371, 461–580.
Zaslavsky G., 2005, Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, Oxford.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Baleanu, D. (2011). New Treatise in Fractional Dynamics. In: Luo, A.C.J., Sun, JQ. (eds) Complex Systems. Nonlinear Physical Science, vol 0. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17593-0_1
Download citation
DOI: https://doi.org/10.1007/978-3-642-17593-0_1
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-17592-3
Online ISBN: 978-3-642-17593-0
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)