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Modeling with fractional difference equations. (English) Zbl 1204.39004

A fractional sum of a function f is introduced as

Δ a -α f(t)=1 Γ(α) s=a t-α (t-s-1) (α-1) f(s),

where aR, α>0, x (α) =Γ(x+1)/Γ(x-α+1), f is defined for s=a(mod1), and Δ a -α f is defined for t=a+α(mod1)· Besides some previously known properties of the fractional sum, additional properties such as a Leibniz type formula and a summation by parts formula are derived. A simple fractional calculus of a variation problem is defined and its Euler-Lagrange equation is derived. As an application, a so called Gompertz fractional difference equation is introduced and solved in terms of a series.

39A12Discrete version of topics in analysis
39A05General theory of difference equations
26A33Fractional derivatives and integrals (real functions)
34A08Fractional differential equations
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