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**Initial value problems in discrete fractional calculus.**
*(English)*
Zbl 1166.39005

Authors’ 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:

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