Energy Consideration of a Capacitor Modelled Using Conformal Fractional-Order Derivative .

Fractional order circuit elements have become important parts of electronic circuits to model systems including supercapacitors, filters, and many more. The conformal fractional derivative (CFD), which is a new basic fractional derivative, has been recently used to model supercapacitors successfully. It is essential to know how electronic components behave under excitation with different types of voltage and current sources. A CFD capacitor is not a well-known element and its usage in circuits is barely examined in the literature. In this research, it is examined how to calculate the stored energy of a CFD capacitor with a series resistor supplied from a DC voltage source. The solutions given in this study may be used in circuits where supercapacitors are used

Controllability of conformable differential systems

This paper deals with complete controllability of systems governed by linear and semilinear conformable differential equations. By establishing conformable Gram criterion and rank criterion, we give sufficient and necessary conditions to examine that a linear conformable system is null completely controllable. Further, we apply Krasnoselskii’s fixed point theorem to derive a completely controllability result for a semilinear conformable system. Finally, three numerical examples are given to illustrate our theoretical results. &nbsp

Modelo de operador fraccional para describir la dinámica de los supercondensadores

Este artículo propone un nuevo circuito equivalente para modelar supercondensadores. El circuito propuesto es un arreglo de circuitos RC serie descritos por ecuaciones diferenciales fraccionarias conformables. Se implementa un algoritmo de identificación de parámetros del circuito equivalente, que utiliza como entrada datos experimentales. Los resultados de validación obtenidos muestran que un circuito equivalente que emplea el operador conformable puede ser utilizado para modelar el comportamiento real del supercondensador.This paper proposes a new equivalent circuit to model supercapacitors. The proposed circuit is an array of RC branches described by differential conformable fractional derivative equations. A parameter identification algorithm based on experimental data is implemented. Validation results using experimental data show that the proposed equivalent circuit can be used to model the real behavior of the supercapacitors

Fractional Calculus and Special Functions with Applications

The study of fractional integrals and fractional derivatives has a long history, and they have many real-world applications because of their properties of interpolation between integer-order operators. This field includes classical fractional operators such as Riemann–Liouville, Weyl, Caputo, and Grunwald–Letnikov; nevertheless, especially in the last two decades, many new operators have also appeared that often define using integrals with special functions in the kernel, such as Atangana–Baleanu, Prabhakar, Marichev–Saigo–Maeda, and the tempered fractional equation, as well as their extended or multivariable forms. These have been intensively studied because they can also be useful in modelling and analysing real-world processes, due to their different properties and behaviours from those of the classical cases.Special functions, such as Mittag–Leffler functions, hypergeometric functions, Fox's H-functions, Wright functions, and Bessel and hyper-Bessel functions, also have important connections with fractional calculus. Some of them, such as the Mittag–Leffler function and its generalisations, appear naturally as solutions of fractional differential equations. Furthermore, many interesting relationships between different special functions are found by using the operators of fractional calculus. Certain special functions have also been applied to analyse the qualitative properties of fractional differential equations, e.g., the concept of Mittag–Leffler stability.The aim of this reprint is to explore and highlight the diverse connections between fractional calculus and special functions, and their associated applications