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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.

MSC:

39A12 Discrete version of topics in analysis
39A05 General theory of difference equations
26A33 Fractional derivatives and integrals
34A08 Fractional ordinary differential equations
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