Abstract

Recently fractional calculus is under strong attention as useful tool in modelling. In many cases fractional derivatives and difference operators proved their usefulnes and effectiveness in describing many real-life processes and phenomenas. For a review of theory and applications of fractional calculus, we refer the reader to (Hilfer, 2000; Kaczorek, 2009; Mozyrska & Wyrwas, 2015; Ostalczyk, 2016; Podlubny, 1999). For variable-order applications the reader can see more in Mozyrska & Ostalczyk (2017). In the paper we investigate discrete time operators with variable orders. We define variable-, fractional order backward difference of the Grunwald-Letnikov-type which means that the order is a single-variable, positive-valued function. Our goal is to start investigations of fitting data for noised eigenvalue function for initial value problem for fractional difference with variable-order.