Fractional Variable-Order Derivative and Difference Operators and Their Applications to Dynamical Systems Modelling

Abstract

AbstractThe chapter presents an overview of some particular derivative and difference operators of fractional variable order, their properties, equivalent forms, and applications. When fundamental properties of a system or its structure are changing in time, a variation of the system’s order may be observed. In such a case, time-dependent variable order operators are taken into consideration. Recently, cases where order is time-varying, have began to be studied extensively. In order to give a deeper insight into fractional variable order calculus, alternative, intuitive descriptions of some particular variable order operators, in the form of equivalent switching schemes, is provided. According to such a schematic interpretation of variable order operators analysis of variable order systems can be simpler and more effective than on the basis of purely analytical definitions. Based on those switching schemes it is possible to categorize fractional order derivatives according to their behaviour and intrinsic properties. Thanks to this schematic description and duality property between chosen variable order operators, analytical solutions of variable order linear differential equations can be effectively derived. Examples of applications of these operators to automatic control and modelling of the heat transfer process in specific grid-holes and two-dimensional fractal-like structure media, of which the geometry is changing in time, are presented.