On Dynamical Systems with Nabla Half Derivative on Time Scales
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On Dynamical Systems with Nabla Half Derivative on Time Scales Tom´aˇs Kisela Abstract. This paper is devoted to study of dynamical systems involving nabla half derivative on an arbitrary time scale. We prove existence and uniqueness of the solution of such system supplied with a suitable initial condition. Both Riemann–Liouville and Caputo approaches to noninteger-order derivatives are covered. Under special conditions we present an explicit form of the solution involving a time scales analogue of Mittag–Leffler function. Also an algorithm for solving of such problems on isolated time scales is established. Moreover, we show that half power functions are positive and decreasing with respect to t − s on an arbitrary time scale. Mathematics Subject Classification. Primary 26A33, 26E70; Secondary 39A12, 39A13. Keywords. Fractional calculus, time scales, Nabla half derivative, dynamical systems, Mittag–Leffler function, existence and uniqueness.
1. Introduction Fractional calculus is a discipline dealing with integrals and derivatives of noninteger orders while time scales were developed to unify and generalize theories of differential and difference equations (for more information we refer to [4,11], respectively). During the last decade, several attempts to combine these two disciplines to introduce time scales fractional calculus can be observed. Some authors managed to extend fractional operators to new discrete settings such as (q, h)-calculus (see [6]). Some authors employed the phenomenon of time scales Laplace transform (see [1]) and others developed axiomatic definitions enabling the classical approach through Cauchy formula for m-th integral (see [7,14]). There is also an effort to introduce local fractional operators (see [2,12]). So far, results addressing properties of time scales fractional operators are very rare. The beauty of the joint time scales notation might create an impression that with an existing definition, most of the properties can be 0123456789().: V,-vol
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directly applied to any time scale, although typically the proof is available only for special time scales such as reals, q-calculus, h-calculus or (q, h)calculus. The approach employed throughout this paper is based on [7], i.e. we consider the less established area of nabla fractional calculus. In particular, we study properties of nabla half derivatives and corresponding dynamical systems on an arbitrary time scale. Namely, we deal with 1
∇a2 x(t) = Ax(t),
a, t ∈ T, t > a,
(1.1)
1 2
where A ∈ Rd×d , T is an arbitrary time scale and ∇a is a nabla half derivative. We note that (1.1) includes as special cases various well-known models occurring in applications of fractional calculus. Among others, we point out Basset equation for forces acting on a spherical object sinking in an incompressible viscous fuid (see [8]) or Bagley–Torvik equation describing a motion of an immersed plate bounded in a Newtonian fluid (see [13]). The paper is organized as follows. After recalling the key known results of time sca
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