Initial value problems in discrete fractional calculus.

*(English)*Zbl 1166.39005Authors’ abstract: This paper is devoted to the study of discrete fractional calculus; the particular goal is to define and solve well-defined discrete fractional difference equations. For this purpose we first carefully develop the commutativity properties of the fractional sum and the fractional difference operators. Then a \(\nu\)-th (\(0<\nu \leq 1\)) order fractional difference equation is defined. A nonlinear problem with an initial condition is solved and the corresponding linear problem with constant coefficients is solved as an example. Further, the half-order linear problem with constant coefficients is solved with a method of undetermined coefficients and with a transform method.

Reviewer: Fozi Dannan (Damascus)

##### MSC:

39A12 | Discrete version of topics in analysis |

26A33 | Fractional derivatives and integrals |

39A20 | Multiplicative and other generalized difference equations |

39A10 | Additive difference equations |

##### Keywords:

discrete fractional calculus; fractional difference equations; commutativity; fractional sum; method of undetermined coefficients
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\textit{F. M. Atici} and \textit{P. W. Eloe}, Proc. Am. Math. Soc. 137, No. 3, 981--989 (2009; Zbl 1166.39005)

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##### References:

[1] | F. M. ATICI AND P. W. ELOE, A transform method in discrete fractional calculus, International Journal of Difference Equations, Vol. 2, 2 (2007), pp. 165-176. |

[2] | Ferhan M. Atici and Paul W. Eloe, Fractional \?-calculus on a time scale, J. Nonlinear Math. Phys. 14 (2007), no. 3, 333 – 344. · Zbl 1157.81315 |

[3] | Martin Bohner and Gusein Sh. Guseinov, The convolution on time scales, Abstr. Appl. Anal. , posted on (2007), Art. ID 58373, 24. · Zbl 1155.39010 |

[4] | Martin Bohner and Allan Peterson, Dynamic equations on time scales, Birkhäuser Boston, Inc., Boston, MA, 2001. An introduction with applications. · Zbl 0978.39001 |

[5] | Martin Bohner and Allan Peterson, Laplace transform and \?-transform: unification and extension, Methods Appl. Anal. 9 (2002), no. 1, 151 – 157. · Zbl 1031.44001 |

[6] | Walter G. Kelley and Allan C. Peterson, Difference equations, 2nd ed., Harcourt/Academic Press, San Diego, CA, 2001. An introduction with applications. · Zbl 0970.39001 |

[7] | Kenneth S. Miller and Bertram Ross, Fractional difference calculus, Univalent functions, fractional calculus, and their applications (Kōriyama, 1988) Ellis Horwood Ser. Math. Appl., Horwood, Chichester, 1989, pp. 139 – 152. · Zbl 0693.39002 |

[8] | Kenneth S. Miller and Bertram Ross, An introduction to the fractional calculus and fractional differential equations, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1993. · Zbl 0789.26002 |

[9] | Igor Podlubny, Fractional differential equations, Mathematics in Science and Engineering, vol. 198, Academic Press, Inc., San Diego, CA, 1999. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. · Zbl 0924.34008 |

[10] | Stefan G. Samko, Anatoly A. Kilbas, and Oleg I. Marichev, Fractional integrals and derivatives, Gordon and Breach Science Publishers, Yverdon, 1993. Theory and applications; Edited and with a foreword by S. M. Nikol\(^{\prime}\)skiĭ; Translated from the 1987 Russian original; Revised by the authors. |

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