Abstract
Fractional calculus presents a new perspective for solution of scientific and engineering problems. There are still many fields where fractional calculus promise fresh understanding of real world problems. In this study, we present an investigation on fractional order motion, where we extend our comprehension from velocity and acceleration concepts to a fuzzy velocity and acceleration domains. Here, fundamentally, we suggest a debate on interpretation of fractional-order derivative system on the bases of integer-order derivative knowledge. We observed that the continuous fractional-order motion equation set can be considered to cover the discrete integer-order motion equation set and physical meaning of fractional-order motion systems can be better understood by using a fuzzy conceptualization of integer-order derivative systems according to the dissimilarity metric.
Highlights
THERE has been renewed interest to fractional-order system due to its practical outcomes in engineering and applied science [1,2,3,4,5,6,7,8,9,10,11,12,13,14]
There were several works aiming physical interpretation of fractional derivatives: Machado presented a probabilistic interpretation of the fractional-order differentiation using Grünwald-Letnikov definition [16]
Giannantoni present extensive [20] work on physical meaning in linear differential equations of fractional order. He stated that “a time differential problem described by one fractional differential equation generates new ‘special’ functions which can be interpreted as being the mathematical description of the evolution of a unique system, made up of a prefixed number of parts, which are in turn so strictly related to each other that they form one sole entity [20]
Summary
THERE has been renewed interest to fractional-order system due to its practical outcomes in engineering and applied science [1,2,3,4,5,6,7,8,9,10,11,12,13,14]. Observable states of system models such as displacement, velocity, acceleration are considered as related states (parts) of FOD modeling space Interaction between these concepts is classified by defining a distance metric, which forms fuzzy domains of observable states of physical systems. This paper presents a discussion on fuzzy conceptual domains in order to enhance our understanding of FOD modeling This can expand our view on the physical interpretation of FOD models. A fuzzy position, a fuzzy velocity and a fuzzy acceleration modeling domains are formed and we illustrate that any FOD model derived from the position data can be classified one of these fuzzy domains according to similarity of their characteristics This allows identifying a FOD model of motion by associating them to a most likely physical concept. A short-time average of absolute difference distance metric (STADD) to indicate the dissimilarities between elements of a fractional-derivative modeling set of f is defined as follows, Ewu
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