Abstract
In this paper, we introduce the analogue of Caputo type fractional derivatives on a $(q,h)$ -discrete time scale which can be reduced to Caputo type fractional differences studied by Abdeljawad (Comput. Math. Appl. 62:1602-1611, 2011) and Caputo type fractional q-differences studied by Atici and Eloe via the choice $q=h=1$ and $h=0$ , respectively. Then, we solve linear fractional difference equations involving Caputo type $(q,h)$ -derivatives and give the general solutions in terms of discrete Mittag-Leffler functions introduced by Cermak et al. In addition, we also apply the $(q,h)$ -Laplace transform method to solve these linear fractional order difference equations.
Highlights
Fractional calculus deals with the study of fractional order integrals and derivatives and their applications [ – ]
Differential equations with fractional derivative provided a natural framework for the discussion of various kinds of real problems modeled by the aid of fractional derivative, such as viscoelastic system, signal processing, diffusion processes, control processing, fractional stochastic system, allometry in biology and ecology
Among other significant papers dealing with these problems, we can mention, e.g., [, – ] or [ ], where discrete analogues of some topics of continuous fractional calculus have been developed
Summary
Fractional calculus deals with the study of fractional order integrals and derivatives and their applications [ – ]. We solve some linear fractional difference equations involving Caputo type (q, h)-derivatives and provide the general solutions in terms of discrete Mittag-Leffler functions introduced in [ ]. If f ∇ (t) is defined for all t ∈ Tκ , the function obtained is called the ∇-derivative of f . Let λ ∈ cRν(T) and s ∈ T, the generalized exponential function eλ(·, s) on time scale T is denoted by the unique solution of the initial value problem x∇ (t) = λx(t), t ∈ Tκ ; x(s) = .
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