Abstract and Applied Analysis

A Note on Fractional Order Derivatives and Table of Fractional Derivatives of Some Special Functions

Abdon Atangana and Aydin Secer

Full-text: Open access

Abstract

The purpose of this note is to present the different fractional order derivatives definition that are commonly used in the literature on one hand and to present a table of fractional order derivatives of some functions in Riemann-Liouville sense On other the hand. We present some advantages and disadvantages of these fractional derivatives. And finally we propose alternative fractional derivative definition.

Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 279681, 8 pages.

Dates
First available in Project Euclid: 26 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393448857

Digital Object Identifier
doi:10.1155/2013/279681

Mathematical Reviews number (MathSciNet)
MR3039169

Zentralblatt MATH identifier
1276.26010

Citation

Atangana, Abdon; Secer, Aydin. A Note on Fractional Order Derivatives and Table of Fractional Derivatives of Some Special Functions. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 279681, 8 pages. doi:10.1155/2013/279681. https://projecteuclid.org/euclid.aaa/1393448857


Export citation

References

  • K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974.
  • I. Podlubny, Fractional Differential Equations, Academic Press, New York, NY, USA, 1999.
  • A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006.
  • Abdon Atangana and J. F. Botha, “Generalized groundwater flow equation using the concept of variable order derivative,” Boundary Value Problems, vol. 2013, 53 pages, 2013.
  • M. Caputo, “Linear models of dissipation whose Q is almost frequency independent, part II,” Geophysical Journal International, vol. 13, no. 5, pp. 529–539, 1967.
  • K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, NY, USA, 1993.
  • S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, Switzerland, 1993.
  • G. M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, 2005.
  • A. Atangana and A. Secer, “Time-fractional coupled-the Korteweg-de Vries equations,” Abstract Applied Analysis, vol. 2013, Article ID 947986, 2013.
  • S. G. Samko, A. A. Kilbas, and O. I. Maritchev, Integrals and Derivatives of the Fractional Order and Some of Their Applications, in Russian, Nauka i Tekhnika, Minsk, Belarus, 1987.
  • I. Podlubny, “Geometric and physical interpretation of fractional integration and fractional differentiation,” Fractional Calculus and Applied Analysis, vol. 5, no. 4, pp. 367–386, 2002.
  • A. Atangana and A. Kilicman, “Analytical solutions the Space-time-Fractional Derivative of advection dispersion equation,” Mathematical Problem in Engineering. In press.
  • A. Atangana, “Numerical solution of space-time fractional derivative of groundwater flow equation,” in Proceedings of the International Conference of Algebra and Applied Analysis, p. 20, Istanbul, Turkey, June 2012.
  • G. Jumarie, “On the solution of the stochastic differential equation of exponential growth driven by fractional Brownian motion,” Applied Mathematics Letters, vol. 18, no. 7, pp. 817–826, 2005.
  • G. Jumarie, “Modified Riemann-Liouville derivative and fractional Taylor series of non-differentiable functions further results,” Computers & Mathematics with Applications, vol. 51, no. 9-10, pp. 1367–1376, 2006.
  • M. Davison and C. Essex, “Fractional differential equations and initial value problems,” The Mathematical Scientist, vol. 23, no. 2, pp. 108–116, 1998.
  • C. F. M. Coimbra, “Mechanics with variable-order differential operators,” Annalen der Physik, vol. 12, no. 11-12, pp. 692–703, 2003.
  • T. H. Solomon, E. R. Weeks, and H. L. Swinney, “Observation of anomalous diffusion and Lévy flights in a two-dimensional rotating flow,” Physical Review Letters, vol. 71, no. 24, pp. 3975–3978, 1993.
  • S. Bhalekar, V. Daftardar-Gejji, D. Baleanu, and R. Magin, “Fractional Bloch equation with delay,” Computers & Mathematics with Applications, vol. 61, no. 5, pp. 1355–1365, 2011.
  • R. L. Magin, Fractional Calculus in Bioengineering, Begell House, Connecticut, UK, 2006.
  • R. L. Magin, O. Abdullah, D. Baleanu, and X. J. Zhou, “Anomalous diffusion expressed through fractional order differential operators in the Bloch-Torrey equation,” Journal of Magnetic Resonance, vol. 190, no. 2, pp. 255–270, 2008.
  • A. V. Chechkin, R. Gorenflo, and I. M. Sokolov, “Fractional diffusion in inhomogeneous media,” Journal of Physics A, vol. 38, no. 42, pp. L679–L684, 2005.
  • F. Santamaria, S. Wils, E. de Schutter, and G. J. Augustine, “Anomalous diffusion in Purkinje cell dendrites caused by spines,” Neuron, vol. 52, no. 4, pp. 635–648, 2006.
  • H. G. Sun, W. Chen, and Y. Q. Chen, “Variable order fractional differential operators in anomalous diffusion modeling,” Physica A, vol. 388, no. 21, pp. 4586–4592, 2009.
  • H. G. Sun, Y. Q. Chen, and W. Chen, “Random order fractional differential equation models,” Signal Processing, vol. 91, no. 3, pp. 525–530, 2011.
  • Y. Q. Chen and K. L. Moore, “Discretization schemes for fractional-order differentiators and integrators,” IEEE Transactions on Circuits and Systems I, vol. 49, no. 3, pp. 363–367, 2002.
  • E. N. Azevedo, P. L. de Sousa, R. E. de Souza et al., “Concentration-dependent diffusivity and anomalous diffusion: a magnetic resonance imaging study of water ingress in porous zeolite,” Physical Review E, vol. 73, no. 1, part 1, Article ID 011204, 2006.
  • S. Umarov and S. Steinberg, “Variable order differential equations with piecewise constant order-function and diffusion with changing modes,” Zeitschrift für Analysis und ihre Anwendungen, vol. 28, no. 4, pp. 431–450, 2009.
  • B. Ross and S. Samko, “Fractional integration operator of variable order in the holder spaces H$\lambda $(x),” International Journal of Mathematics and Mathematical Sciences, vol. 18, no. 4, pp. 777–788, 1995.
  • H. T. C. Pedro, M. H. Kobayashi, J. M. C. Pereira, and C. F. M. Coimbra, “Variable order modeling of diffusive-convective effects on the oscillatory flow past a sphere,” Journal of Vibration and Control, vol. 14, no. 9-10, pp. 1659–1672, 2008.
  • D. Ingman and J. Suzdalnitsky, “Application of differential operator with servo-order function in model of viscoelastic deformation process,” Journal of Engineering Mechanics, vol. 131, no. 7, pp. 763–767, 2005.
  • Y. L. Kobelev, L. Y. Kobelev, and Y. L. Klimontovich, “Statistical physics of dynamic systems with variable memory,” Doklady Physics, vol. 48, no. 6, pp. 285–289, 2003.
  • A. H. Cloot and J. P. Botha, “A generalized groundwater flow equation using the concept of non-integer order,” Water SA, vol. 32, no. 1, pp. 1–7, 2006.
  • L. Schwartz, Théorie des Distributions, Hermann, Paris, Farnce, 1978.
  • R. Estrada and R. P. Kanwal, “Regularization and distributional derivatives of ${({x}_{1}^{2}+{x}_{2}^{2}+\cdots +{x}_{p}^{2})}^{-(1/2)n}$ in ${\mathbb{R}}^{p}$,” Proceedings of the Royal Society A, vol. 401, no. 1821, pp. 281–297, 1985.
  • R. Estrada and R. P. Kanwal, “Regularization, pseudofunction, and Hadamard finite part,” Journal of Mathematical Analysis and Applications, vol. 141, no. 1, pp. 195–207, 1989.
  • A. Sellier, “Hadamard's finite part concept in dimension $n\geq 2$, distributional definition, regularization forms and distributional derivatives,” Proceedings of the Royal Society A, vol. 445, no. 1923, pp. 69–98, 1994.
  • L. Bel, T. Damour, N. Deruelle, J. Ibañez, and J. Martin, “Poincaré-invariant gravitational field and equations of motion of two pointlike objects: the postlinear approximation of general relativity,” General Relativity and Gravitation, vol. 13, no. 10, pp. 963–1004, 1981.