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

A Conformal Fractional Derivative-based Leaky Integrate-and-Fire Neuron Model

Neuron model have been extensively studied and different models have been proposed. Nobel laureate Hodgkin-Huxley model is physiologically relevant and can demonstrate different neural behaviors, but it is mathematically complex. For this reason, simplified neuron models such as integrate-and-fire model and its derivatives are more popular in the literature to study neural populations. Lapicque’s integrate-and-fire model is proposed in 1907 and its leaky integrate-and-fire version is very popular due to its simplicity. In order to improve this simple model and capture different aspects of neurons, a variety of it have been proposed. Fractional order derivative-based neuron models are one of those varieties, which can show adaptation without necessitating additional differential equations. However, fractional-order derivatives could be computationally costly. Recently, a conformal fractional derivative (CFD) is suggested in literature. It is easy to understand and implement compared to the other methods. In this study, a CFD-based leaky integrate-and-fire neuron model is proposed. The model captures the adaptation in firing rate under sustained current injection. Results suggest that it could be used to easily and efficiently implement network models as well as to model different sensory afferents

Numerical Simulation of Parallel RLC Model Using Different Fractional Derivative Operators

In the current study, the theory of fractional calculus is applied to the electric parallel RLC circuit. The aim of this article is to alter the concept of a parallel RLC circuit by applying various fractional derivative operators. A fractional RLC circuit was investigated via Caputo, Caputo-Fabrizio, and Atangana-Baleanu derivatives. The Laplace transform technique was applied to resolve the system of governing differential equations. The results for the various orders are compared to each other. When the fractional order derivative tends to be one, the system\u27s performance is found to be very slow due to a decrease in damping capacity. The results for the various orders are compared to each other. When the fractional order derivative tends to be one, the system\u27s performance is found to be very slow due to a decrease in damping capacity. The results for the various orders are compared to each other. When the fractional order derivative tends to be one, the system\u27s performance is found to be very slow due to a decrease in damping capacity

Conformable Derivative Operator in Modelling Neuronal Dynamics

This study presents two new numerical techniques for solving time-fractional one-dimensional cable differential equation (FCE) modeling neuronal dynamics. We have introduced new formulations for the approximate-analytical solution of the FCE by using modified homotopy perturbation method defined with conformable operator (MHPMC) and reduced differential transform method defined with conformable operator (RDTMC), which are derived the solutions for linear-nonlinear fractional PDEs. In order to show the efficiencies of these methods, we have compared the numerical and exact solutions of fractional neuronal dynamics problem. Moreover, we have declared that the proposed models are very accurate and illustrative techniques in determining to approximate-analytical solutions for the PDEs of fractional order in conformable sense

Solution of Conformable Fractional Ordinary Differential Equations via Differential Transform Method

Recently, a new fractional derivative called the conformable fractional derivative is given which is based on the basic limit definition of the derivative in [1]. Then, the fractional versions of chain rules, exponential functions, Gronwall's inequality, integration by parts, Taylor power series expansions is developed in [2]. In this paper, we give conformable fractional differential transform method and its application to conformable fractional differential equations

Generalized squared remainder minimization method for solving multi-term fractional differential equations

In this paper, we introduce a generalization of the squared remainder minimization method for solving multi-term fractional differential equations. We restrict our attention to linear equations. Approximate solutions of these equations are considered in terms of linearly independent functions. We change our problem into a minimization problem. Finally, the Lagrange-multiplier method is used to minimize the resultant problem. The convergence of this approach is discussed and theoretically investigated. Some relevant examples are investigated to illustrate the accuracy of the method, and obtained results are compared with other methods to show the power of applied method

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

A THREE FRACTIONAL ORDER JERK EQUATION WITH ANTI PERIODIC CONDITIONS

We study a new Jerk equation involving three fractional derivatives and anti periodic conditions. By Banach contraction principle, we present an existence and uniqueness result for the considered problem. Utilizing Krasnoselskii fixed point theorem we prove another existence result governing at least one solution. We provide an illustrative example to claim our established results. At the end, an approximation for Caputo derivative is proposed and some chaotic behaviours are discussed by means of the Runge Kutta 4th order method

Uniqueness criteria for initial value problem of conformable fractional differential equation

This paper presents four uniqueness criteria for the initial value problem of a differential equation which depends on conformable fractional derivative. Among them is the generalization of Nagumo-type uniqueness theory and Lipschitz conditional theory, and advances its development in proving fractional differential equations. Finally, we verify the main conclusions of this paper by providing four concrete examples