It is well known that the complex step method is a tool that calculates derivatives by imposing a complex step in a strict sense. We extended the method by employing the fractional calculus differential operator in this paper. The fractional calculus can be taken in the sense of the Caputo operator, Riemann-Liouville operator, and so forth. Furthermore, we derived several approximations for com...

The theory of derivatives and integrals of non-integer order goes back to Leibniz, Liouville, Grunwald, Letnikov and Riemann. The history of fractional vector calculus (FVC) has only 10 years. The main approaches to formulate a FVC, which are used in the physics during the past few years, will be briefly described in this paper. We solve some problems of consistent formulations of FVC by using ...

Derivatives and integrals of non-integer order were introduced more than three centuries ago, but only recently gained more attention due to their application on nonlocal phenomena. In this context, the Caputo derivatives are the most popular approach to fractional calculus among physicists, since differential equations involving Caputo derivatives require regular boundary conditions. Motivated...

The main aim of this contribution is to introduce the fundamentals of the Fractional Order Calculus (FOC) and outline its possible application to analysis and synthesis of control systems. The basic theoretical concepts of FOC are followed by techniques for potential fractional order systems description and stability investigation. Moreover, the paper offers the overview of the existing fractio...

The problems formulated in the fractional calculus framework often require numerical fractional integration/differentiation of large data sets. Several existing fractional control toolboxes are capable of performing fractional calculus operations, however, none of them can efficiently perform numerical integration on multiple large data sequences. We developed a Fractional Integration Toolbox (...

The term fractional calculus is more than 300 years old. It is a generalization of the ordinary differentiation and integration to non-integer (arbitrary) order. The subject is as old as the calculus of differentiation and goes back to times when Leibniz, Gauss, and Newton invented this kind of calculation. In a letter to L’Hospital in 1695 Leibniz raised the following question (Miller and Ross...

The capability calculus is a framework for statically reasoning about program resources such as deallocatable memory regions. Fractional capabilities, originally proposed by Boyland for checking the determinism of parallel reads in multi-thread programs, extend the capability calculus by extending the capabilities to range over the rational numbers. Fractional capabilities have since found nume...

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 inte...