United lattice fractional integro-differentiation

Abstract

Abstract New fractional operators of non-integer and integer orders for N-dimensional lattice ℤ N are suggested. The proposed lattice fractional integro-differentiation of positive order can be considered as a lattice analog of partial derivatives of non-integer orders. For negative values of order, it can be considered as a lattice fractional integration. For integer orders, the continuum limit of the united lattice derivatives gives the usual partial derivatives of integer orders. In contrast to the usual lattice derivatives, which are proposed in our recent paper (“Lattice fractional calculus”, Appl. Math. Comp. 257 (2015), 12–33), the suggested lattice operators give the usual partial derivatives for all integer orders. The united lattice integro-differentiation is represented by finite differences with infinite series that describe long-range lattice interactions. The suggested lattice operators of positive orders can be considered as an exact discretization of derivatives of corresponding orders. The basic mathematical feature of these united lattice operators is an implementation of the same algebraic properties that have the differentiation operator.