Introduction to the Concepts and Applications of Fractional and Variable Order Differential Calculus

Understanding the concepts of fractional and variable order differential calculus requires a willingness to depart from the traditional physical interpretations through which calculus is generally understood. Fractional calculus involves the computation of a derivative or integral of any real order, rather than just an integer. Several definitions for calculating a real order derivative or integral have been proposed throughout history. The pros and cons of each one of these definitions, as well as some of the most relevant applications of fractional calculus are here discussed. Fractional calculus has been used in modeling control systems, heat flux, temperature, entropy generation and diffusion, among others. Some applications of fractional calculus require that the integral or derivative order is not maintained constant, but be a function of a system parameter. This is achieved with what is called variable order differential equations and applies mostly to dynamical problems of changing regimes, such as variable viscoelasticity oscillators. A previously proposed model for viscoelastic oscillators is discussed here, as well as proposing a new and potentially more accurate approach which more closely represents the behavior of this kind of system. The purpose of this paper is to generate curiosity and encourage further investigation of the potential applications of this branch of mathematics.