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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 279681, 8 pages
http://dx.doi.org/10.1155/2013/279681
Research Article

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

1Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, Bloemfontein 9300, South Africa
2Yildiz Technical University, Department of Mathematical Engineering, Davutpasa, 34210 İstanbul, Turkey

Received 10 January 2013; Accepted 1 March 2013

Academic Editor: Mustafa Bayram

Copyright © 2013 Abdon Atangana and Aydin Secer. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974. View at Zentralblatt MATH · View at MathSciNet
  2. I. Podlubny, Fractional Differential Equations, Academic Press, New York, NY, USA, 1999. View at Zentralblatt MATH · View at MathSciNet
  3. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. 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. View at Publisher · View at Google Scholar
  5. 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. View at Publisher · View at Google Scholar
  6. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, NY, USA, 1993.
  7. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, Switzerland, 1993. View at Zentralblatt MATH · View at MathSciNet
  8. G. M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, 2005. View at Zentralblatt MATH · View at MathSciNet
  9. A. Atangana and A. Secer, “Time-fractional coupled-the Korteweg-de Vries equations,” Abstract Applied Analysis, vol. 2013, Article ID 947986, 2013. View at Publisher · View at Google Scholar
  10. 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.
  11. 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. View at Zentralblatt MATH · View at MathSciNet
  12. A. Atangana and A. Kilicman, “Analytical solutions the Space-time-Fractional Derivative of advection dispersion equation,” Mathematical Problem in Engineering. In press.
  13. 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.
  14. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. M. Davison and C. Essex, “Fractional differential equations and initial value problems,” The Mathematical Scientist, vol. 23, no. 2, pp. 108–116, 1998. View at Zentralblatt MATH · View at MathSciNet
  17. C. F. M. Coimbra, “Mechanics with variable-order differential operators,” Annalen der Physik, vol. 12, no. 11-12, pp. 692–703, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. 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. View at Publisher · View at Google Scholar
  19. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. R. L. Magin, Fractional Calculus in Bioengineering, Begell House, Connecticut, UK, 2006.
  21. 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. View at Publisher · View at Google Scholar · View at Scopus
  22. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. 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. View at Publisher · View at Google Scholar
  24. 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. View at Publisher · View at Google Scholar
  25. 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. View at Publisher · View at Google Scholar
  26. 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. View at Publisher · View at Google Scholar · View at MathSciNet
  27. 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.
  28. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. B. Ross and S. Samko, “Fractional integration operator of variable order in the holder spaces Hλ(x),” International Journal of Mathematics and Mathematical Sciences, vol. 18, no. 4, pp. 777–788, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. 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. View at Publisher · View at Google Scholar
  32. 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. View at Publisher · View at Google Scholar · View at Scopus
  33. 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.
  34. L. Schwartz, Théorie des Distributions, Hermann, Paris, Farnce, 1978.
  35. R. Estrada and R. P. Kanwal, “Regularization and distributional derivatives of (x12+x22++xp2)-(1/2)n in p,” Proceedings of the Royal Society A, vol. 401, no. 1821, pp. 281–297, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  36. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  37. A. Sellier, “Hadamard's finite part concept in dimension n2, distributional definition, regularization forms and distributional derivatives,” Proceedings of the Royal Society A, vol. 445, no. 1923, pp. 69–98, 1994. View at Publisher · View at Google Scholar
  38. 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. View at Publisher · View at Google Scholar · View at MathSciNet