From Fractional Differential Equations with Hilfer Derivatives: To Discrete Maps with Memory

Abstract: In this article, we proposed new discrete maps with memory (DMM). These maps are derived from fractional differential equations (FDE) with the Hilfer fractional derivatives of non-integer orders and periodic sequence of kicks. The suggested DMM are obtained from these equations without any approximation, and they are a discrete form of the exact solutions. DMM are proposed for arbitrary positive orders of equations with the Hilfer fractional derivatives. As an example, the suggested maps are described for the orders lying in the intervals (0,1) and (1,2). The maps, which are derived from the equations with the Hilfer operators, allow us to consider a whole range of the maps derived from the FDE with the Caputo and Riemann-Liouville fractional derivatives.