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Modeling with fractional difference equations. (English) Zbl 1204.39004
A fractional sum of a function $f$ is introduced as $$\Delta _{a}^{-\alpha}f(t)=\frac{1}{\Gamma (\alpha )}\sum_{s=a}^{t-\alpha }(t-s-1)^{(\alpha -1)}f(s),$$ where $a\in R,$ $\alpha >0$, $x^{(\alpha )}=\Gamma (x+1)/\Gamma (x-\alpha +1),$ $f$ is defined for $s=a\ (\text{mod }1),$ and $\Delta _{a}^{-\alpha }f$ is defined for $t=a+\alpha \ (\text{mod }1).$ 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
Full Text: DOI
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