Non-integer Order Calculus Research Articles

Intelligent frequency stabilization of low-inertia islanded power grids-based redox battery

Renewable energy sources (RES)-generated electricity is increasing at a fast pace and is becoming more prevalent in microgrids (MGs). System inertia is much decreased due to the fact that MGs are mostly powered by RES rather than synchronous generators. As a result, system stability suffers, which is especially problematic with MGs. A suitable solution for this problem is an energy storage system and an appropriate inertial control technique. Redox flow batteries (RFBs) have an unusually long charge-discharge life cycle and a fast reaction time, which allows them to reduce oscillations in the power system when there is a sudden disturbance in the power system. This paper suggests a unique concept for VIC based on RFB in an isolated MG, which emulates the primary frequency control, virtual inertia, and damping all at once. In addition to the VIC concept, we suggest a novel nonlinear model predictive control (NMPC) method for MG secondary control. The NMPC utilizes non-integer order calculus and interval type-3 (IT3) fuzzy logic systems (FLSs). Finally, OPAL-RT-based real-time hardware-in-the-loop (HIL) simulation results are used to examine the validity and applicability of the suggested techniques in low inertia islanded MG.

Analysis of the spread of COVID-19 in Morocco based on a SEIR epidemic model

Fractional calculus has been widely used in mathematical modeling of evolutionary systems with memory effect on dynamics. In order to illustrate the efficiency of this non-integer order calculus, we employ SEIR models to model the dynamics, with and without memory, of the spread of Covid-19 in Morocco country.

The memory effect on fractional calculus: an application in the spread of COVID-19

Fractional calculus has been widely used in mathematical modeling of evolutionary systems with memory effect on dynamics. The main interest of this work is to attest, through a statistical approach, how the hysteresis phenomenon, which describes a type of memory effect present in biological systems, can be treated by fractional calculus. We also analyse the contribution of the historical values of a function in the evaluation of fractional operators according to their order. To illustrate the efficiency of this non-integer order calculus, we consider the SIR (susceptible–infected–recovered) compartmental model which is widely used in epidemiology. We employ this compartmental model to study the dynamics of the spread of COVID-19 in some countries, one version with memory and one without memory.

Fractional Mathematical Oncology: On the potential of non-integer order calculus applied to interdisciplinary models

Mathematical Oncology investigates cancer-related phenomena through mathematical models as comprehensive as possible. Accordingly, an interdisciplinary approach involving concepts from biology to materials science can provide a deeper understanding of biological systems pertaining the disease. In this context, fractional calculus (also referred to as non-integer order) is a branch in mathematical analysis whose tools can describe complex phenomena comprising different time and space scales. Fractional-order models may allow a better description and understanding of oncological particularities, potentially contributing to decision-making in areas of interest such as tumor evolution, early diagnosis techniques and personalized treatment therapies. By following a phenomenological (i.e. mechanistic) approach, the present study surveys and explores different aspects of Fractional Mathematical Oncology, reviewing and discussing recent developments in view of their prospective applications.

On multistep tumor growth models of fractional variable-order

Fractional mathematical oncology is a research topic that applies non-integer order calculus to tackle cancer problems such as tumor growth analysis or its optimal treatment. This work proposes a multistep exponential model with a fractional variable-order representing the evolution history of a tumor. Model parameters are tuned according to variable fractional order profiles while assessing their capability of fitting a clinical time series. The results point to the superiority of the proposed model in describing the experimental data, thus providing new perspectives for modeling tumor growth.

Transmission Dynamics of Fractional Order Brucellosis Model Using Caputo–Fabrizio Operator

In this paper, a noninteger order Brucellosis model is developed by employing the Caputo–Fabrizio noninteger order operator. The approach of noninteger order calculus is quite new for such a biological phenomenon. We establish the existence, uniqueness, and stability conditions for the model via the fixed-point theory. The initial approachable approximate solutions are derived for the proposed model by applying the iterative Laplace transform technique. Finally, numerical simulations are conducted for the analytical results to visualize the effect of various parameters that govern the dynamics of infection, and the results are presented using plots.

Exploring the impact of fractional-order capacitive behavior on the hysteresis effects of perovskite solar cells: A theoretical perspective

Perovskite-based solar cells are devices of increasing interest among new generation of photovoltaic technologies due to the outstanding power conversion efficiency (PCE) achieved in the last few years, and the favorable fabrication conditions. Among all these excellent photovoltaic properties, perovskite light-harvesting devices also show relevant issues such as the hysteresis mechanisms in the current density-voltage (J-V) characteristics, whose underlying origins have not yet been clarified from a single and universally accepted point of view. Here, we focus our attention on the ideal capacitive mechanisms which, on the other hand, are often modeled using fractional-order capacitors in the equivalent circuits used to fit impedance measurements. In particular, we provide the theoretical framework for the transient-photocurrent analysis described by fractional-order responses which involve the generalized Mittag-Leffler function. Importantly, the model introduces a connection between perovskite traps and defects, memory processes, fractional dynamics, and Cole-Cole behavior. Crucial dependences of non-ideal capacitive dynamics on fractional-order α and the selected scan rate were found in the J-V hysteresis behavior from the perspective of the non-integer order calculus. In this sense, it is necessary to point out that our numerical simulations of dark J-V curves by considering non-ideal capacitive effects reveal more prominent distortions than those of the ideal case, leading from free-hysteretic influences (α = 1) to significant hysteresis distortions (0<α<1) under certain voltage rates. Thus, this study can help to advance in the origin and the understanding of the experimental hysteresis mechanisms under pristine conditions or during degradation processes (α value decreases indicating a highly disordered morphology).

Local non-integer order dynamic problems on time scales revisited

This paper deals with a non-integer order calculus on time scales in a local sense. Taking advantage of a new field structure over real and complex numbers, the researchers have normalized the proposed derivative. Non-integer order mean value theorem on time scales is stated. In addition to a hypothesis on the existence of a H\( \ddot{o} \)lder continuous function h, a given time scale \( \mathbb {T} \) is explored. This consideration provided an extension to a recently proposed local fractional differentiation theory and it paved the ground to a local fractional integration on time scales. Also, making use of the presented non-integer order calculus, a class of dynamic initial value problems of fractional orders on an arbitrary time scale is studied. Eventually, an approximate solution to the potential-free Schrdinger equation of time-fractional order is obtained and illustrated.

Analytical solution of the local fractional Klein--Gordon equation for generalized Hulthen potential

The one-dimensional Klein-Gordon (KG) equation is investigated in the domain of conformable fractional calculus for one-dimensional scalar potential, namely generalized Hulthen potential. The conformable fractional calculus is based on conformable fractional derivative, which is the most natural definition in noninteger order calculus. Fractional order differential equations can be solved analytically by means of this derivative operator. We obtained exact eigenvalue and eigenfunction solutions of the local fractional KG equation and investigated the evolution of relativistic effects in correspondence with the fractional order.