publication . Preprint . 2009

Series expansion in fractional calculus and fractional differential equations

Li, Ming-Fan; Ren, Ji-Rong; Zhu, Tao;
Open Access English
  • Published: 26 Oct 2009
Fractional calculus is the calculus of differentiation and integration of non-integer orders. In a recently paper (Annals of Physics 323 (2008) 2756-2778), the Fundamental Theorem of Fractional Calculus is highlighted. Based on this theorem, in this paper we introduce fractional series expansion method to fractional calculus. We define a kind of fractional Taylor series of an infinitely fractionally-differentiable function. Further, based on our definition we generalize hypergeometric functions and derive corresponding differential equations. For finitely fractionally-differentiable functions, we observe that the non-infinitely fractionally-differentiability is ...
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21 references, page 1 of 2

[1] K. B. Oldham, J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, Academic Press, New York, 1974.

[2] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives Theory and Applications, Gordon and Breach, New York, 1993.

[3] K. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.

[4] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.

[5] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Application of Fractional Differential Equations, Elsevier, Amsterdam, 2006.

[6] A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, Springer, Wien, 1997.

[7] R. Hilfer (Ed.), Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.

[8] B. West, M. Bologna, P. Grigolini, Physics of Fractal Operators, Springer, New York, 2003.

[9] G. M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, Oxford, 2005.

[10] R. Metzler, J. Klafter, Phys. Rep. 339 (2000) 1-77; R. Metzler, J. Klafter, J. Phys. A 37 (2004) R161-R208.

[11] R. Herrmann, Phys. Lett. A 372 (2008) 5515.

[12] O. P. Agrawal, J. Math. Anal. Appl. 272 (2002) 368.

[13] Marcelo R. Ubriaco, Phys. Lett. A 373 (2009) 2516-2519.

[14] Ricardo Almeida, Delfim F. M. Torres, Appl. Math. Lett. 22 (2009) 1816-1820.

[15] Dumitru Baleanu, Phys. Scr. T136 (2009) 014006.

21 references, page 1 of 2
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