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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References:

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