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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.

MSC:
39A12 Discrete version of topics in analysis
26A33 Fractional derivatives and integrals
39A20 Multiplicative and other generalized difference equations
39A10 Additive difference equations
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