Variable-order fractional operators were conceived and mathematically formalized only in recent years. The possibility of formulating evolutionary governing equations has led to the successful application of these operators to the modelling of complex real-world problems ranging from mechanics, to transport processes, to control theory, to biology. Variable-order fractional calculus (VO-FC) is a relatively less known branch of calculus that offers remarkable opportunities to simulate interdisciplinary processes. Recognizing this untapped potential, the scientific community has been intensively exploring applications of VO-FC to the modelling of engineering and physical systems. This review is intended to serve as a starting point for the reader interested in approaching this fascinating field. We provide a concise and comprehensive summary of the progress made in the development of VO-FC analytical and computational methods with application to the simulation of complex physical systems. More specifically, following a short introduction of the fundamental mathematical concepts, we present the topic of VO-FC from the point of view of practical applications in the context of scientific modelling.

This study investigates the use of fractional order differential models to simulate the dynamic response of non-homogeneous discrete systems and to achieve efficient and accurate model order reduction. The traditional integer order approach to the simulation of non-homogeneous systems dictates the use of numerical solutions and often imposes stringent compromises between accuracy and computational performance. Fractional calculus provides an alternative approach where complex dynamical systems can be modeled with compact fractional equations that not only can still guarantee analytical solutions, but can also enable high levels of order reduction without compromising on accuracy. Different approaches are explored in order to transform the integer order model into a reduced order fractional model able to match the dynamic response of the initial system. Analytical and numerical results show that, under certain conditions, an exact match is possible and the resulting fractional differential models have both a complex and frequency-dependent order of the differential operator. The implications of this type of approach for both model order reduction and model synthesis are discussed.

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This study explores the use of fractional calculus as a possible tool to model wave propagation in complex, heterogeneous media. While the approach presented could be applied to predict field transport in a variety of inhomogeneous systems, we illustrate the methodology by focusing on elastic wave propagation in a one-dimensional periodic rod. The governing equations describing the wave propagation problem in inhomogeneous systems typically consist of partial differential equations with spatially varying coefficients. Even for very simple systems, these models require numerical solutions which are computationally expensive and do not provide the valuable insights associated with closed-form solutions. We will show that fractional calculus can provide a powerful approach to develop comprehensive mathematical models of inhomogeneous systems that can effectively be regarded as homogenized models. Although at first glance the mathematics might appear more complex, these fractional order models can allow the derivation of closed-form analytical solutions that provide excellent estimations of the systems' dynamic responses. Equally important, these solutions are valid in a frequency range that goes largely beyond the well-known homogenization limit of traditional integer order approaches, therefore providing a possible route to high-frequency homogenization. More specifically, this study focuses on the analyses of the dispersion and propagation properties of a periodic medium under single-tone harmonic excitation and illustrates the methodology to obtain a space-fractional wave equation capable of capturing the behavior of the physical system. The fractional wave equation and its analytical solution are compared with numerical results obtained via a traditional finite element method in order to assess their validity and evaluate their performance. It is found that the resulting fractional differential models are, in their most general form, of complex and frequency-dependent order. arXiv:1808.07422v1 [physics.class-ph]

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