Implementation of fractional-order electromagnetic potential through a genetic algorithm

At present, the theory of fractional calculus is widely used in many fields of science for modeling various processes. Differential equations with fractional derivatives are used to model the migration of pollutants in porous inhomogeneous media and allow a more correct description of the behavior of pollutants at large distances from the source. The analytical solution of differential equations with fractional order derivatives is often very complicated or even impossible. There has been proposed a numerical method for solving fractional differential equations in partial derivatives with respect to time to describe the migration of pollutants in groundwater. An implicit difference scheme is developed for the numerical solution of a non-stationary fractional differential equation, which is an analogue of the well-known implicit Crank-Nicholson difference scheme. The system of difference equations is presented in matrix form. The solution of the problem is reduced to the multiple solution of a tridiagonal system of linear algebraic equations by the tridiagonal matrix algorithm. The results of evaluating the spread of pollutant in groundwater based on the numerical method for model examples are presented. The concentrations of the substance obtained on the basis of the analytical and numerical solutions of the unsteady one-dimensional fractional differential equation are compared. The results obtained using the proposed method and on the basis of the well-known analytical solution of the fractional differential equation are in fairly good agreement with each other. The relative error is on average 9%. In contrast to the well-known analytical solution, the developed numerical method can be used to model the spread of pollutants in groundwater, taking into account their biodegradation.

The advantages of using fractional calculus in designing control algorithms has been long discussed and demonstrated. In this paper, the theory of fractional calculus is used in combination with the Internal Model Control methodology to offer an alternative control solution for the concentration of biodegradable organic matter in the aeration tank of a waste water treatment facility. In contrast to the integer order Internal Model Controller, the fractional order controller has two tuning parameters, which allow for an increased flexibility in the tuning procedure. The fractional order Internal Model Controller is designed based on a simplified model of the process that occurs in the aeration tank. The dynamics of this controller is then analyzed using the nonlinear model of the process. Simulation results are provided including set point tracking, disturbance rejection and overall robustness.

The main purpose of this research is to explore the evolution of fractional calculus from its inception to its diverse applications in science and engineering. The concept of calculus dates back to the work of Sir Isaac Newton, an English mathematician, and Gottfried Leibniz, a French mathematician who presented similar ideas using different notations. Initially, the order of differentiation was understood as an integer. However, as time progressed, mathematicians encountered a paradox: what if the order of a derivative were not an integer but a fraction? This question marked the beginning of fractional calculus. Over the years, a lot of definitions were developed by renowned mathematicians such as Euler, Lagrange, and Laplace. In its early stages, fractional calculus was primarily theoretical with little application to practical problems. However, as the field matured, mathematicians began to apply this theory to practical situations. Today, fractional calculus has seen rapid development and application across various domains in science and engineering. Some applications are shortlisted as follows. In the field of computer and electrical engineering fractional calculus is used in noise filtering processes, particularly in echocardiographic imaging, to minimize noise interference. It also plays a crucial role in developing de-noising models in digital imaging. In mineral engineering, leaching column test is conducted by using new rate equation formed by fractional calculus. The Behaviour of Hydrological processes on Earth has been mathematically modelled using fractional calculus. In the field of quantum mechanics, the Schrödinger equation has been further improved using the fractional calculus theory in the formation of a new version of the fractional Schrödinger equation in the context of space-time.

The transient analysis of multi‐conductor transmission lines should consider the frequency‐dependent characteristics due to the skin effect. Fully considering the inherent fractional order characteristics of the frequency‐dependent effect, a general wide‐band modelling method is proposed. The fractional order vector fitting method is adopted to approximate the frequency‐dependent parameters and the corresponding fractional differential equations can be obtained by the inverse Laplace transformation. The backward difference is a practical method to solve the fractional differential equations; however, a linear convolution must be calculated, which will lead to a heavy computation complexity. To address this issue, a new recursive convolution method for the fractional differential equations is proposed and an efficient solution is achieved. Furthermore, considering the indispensability of the passivity verification of a system for the transient simulation, the passivity verification by extending the Hamiltonian matrix for the fractional order systems is studied and a practical criterion is proposed. Three examples are considered to validate the proposed method: (i) a single underground cable, (ii) three‐phase underground cable, and (iii) an experimental transformer under very fast transient voltage. The simulation results are compared with the results obtained by power systems computer‐aided design or measurements and good agreements are achieved.

Aiming at the existing problems that the image denoising algorithm based on integer-order partial differential equation could lost part of edge and texture information. This image denoising algorithm based on fractional variational theory was proposed by the theory of fractional calculus and partial differential equation. The denoising model proposed in this paper introduces and implements the numeric computation of the fractional variation partial differential equations by constructing the fractional differential mask operators along eight directions of image. The simulation data prove that the image denoising algorithm based on fractional variation theory compared with the traditional image denoising algorithm could better retain the edge and texture detail information, obtain visual effect, and properly improve the signal-to-noise ratio.

A spring’s nonlinear load-displacement function is described by three factors, the (i) shape function, (ii) load-range, and (iii) displacement-range. The shape function encompasses the nonlinear relationship between the load and displacement, and therefore, is the most difficult factor to match. In this paper, we present a general scheme for topology, size, and shape optimization of nonlinear springs for prescribed load–displacement shape functions, while simultaneously meeting manufacturing, space, and stress constraints. This paper presents the objective function and a novel, floating point parametric model used within a genetic algorithm optimization scheme. The nonlinear springs all undergo large deformations and are evaluated by nonlinear finite element analysis. Two examples are included to demonstrate the effectiveness of the methodology in synthesizing nonlinear springs that match a prescribed load-displacement shape function.

Nonlinear analysis has widespread and significant applications in many areas at the core of many branches of pure and applied mathematics and modern science, including nonlinear ordinary and partial differential equations, critical point theory, functional analysis, fixed point theory, nonlinear optimization, fractional calculus, variational analysis, convex analysis, dynamical system theory, mathematical economics, data mining, signal processing, control theory, and many more [...]

Fuzzy discrete fractional calculus and fuzzy fractional discrete equations

In the recent past, genetic algorithm (GA) and evolutionary optimization scheme have become increasingly popular for the design of electromagnetic (EM) devices. However, the conventional GA suffers from computational drawback and parameter dependency when applied to a computationally expensive problem, such as practical EM optimization design. To overcome these issues, a hybrid optimization scheme using GA in conjunction with Kriging is proposed. The algorithm is validated by using two mathematical problems and by optimizing rotor structure of interior permanent magnet synchronous machine.

Последние десятилетия количество работ, посвященных исследованию задач для дифференциальных уравнений дробного порядка, заметно растет. Интерес исследователей вызван тем, что количество областей науки, в которых используются уравнения, содержащие дробные производные, варьируется от биологии и медицины до теории управления, инженерии, финансов, а также оптики, физики и так далее. Включение запаздывания в уравнение дробного порядка существенно влияет на ход процесса, описываемого этим уравнением, так как неизвестная функция задается при различных значениях аргумента, что вносит эффект предыстории в уравнение. Поэтому, математические модели, содержащие дробный оператор и запаздывающий аргумент, более точны, чем модели, содержащие производные целого порядка. В данной работе исследуется задача Коши для линейного обыкновенного дифференциального уравнения с запаздывающим аргументом c оператором дробного дифференцирования Джрбашяна – Нерсесяна, обобщающим известные дробные операторы Римана – Лиувилля и Герасимова – Капуто. Результаты работы получены с использованием методов теории целого и дробного исчислений, методов теории дифференциальных уравнений с запаздывающим аргументом, метода специальных функций. В работе доказывается теорема о справедливости аналога формулы Лагранжа. Также доказано, что специальная функция Wγm​​(t), которая, в свою очередь, определяется через обобщенную функцию Миттаг – Леффлера (или функция Прабхакара), удовлетворяет уравнению и условиям, сопряженным исследуемому, и является фундаментальным решением рассматриваемого уравнения. Сформулирована и доказана теорема существования и единственности решения начальной задачи. Решение поставленной задачи выписано в терминах специальной функции Wν​(t). In recent, the number of works devoted to the study of problems for fractional order differential equations is growing noticeably. The interest of researchers is due to the fact that the number of areas of science in which equations containing fractional derivatives are used varies from biology and medicine to control theory, engineering, finance, as well as optics, physics, and so on. The inclusion of delay in the fractional order equation significantly affects the course of the process described by this equation, since the unknown function is given for different values of the argument, which includes a history effect into the equation. Therefore, mathematical models containing a fractional operator and a delay argument are more accurate than models containing integer derivatives. In this paper, we study the Cauchy problem for a linear ordinary delay differential equation with the Dzhrbashyan – Nersesyan fractional differentiation operator, which is generalizing the Riemann – Liouville and Gerasimov – Caputo fractional operators. The results of the work are obtained using the methods of the theory of integer and fractional calculus, methods of the theory of delay differential equations, the method of special functions. In this paper proves a theorem on the validity of an analogue of the Lagrange formula. It is also proved that the special function Wγm​​(t), which is defined in terms of the generalized Mittag-Leffler function (or the Prabhakar function), satisfies the equation and conditions associated with the one under study, and is the fundamental solution of the considered equation. The main result is that the existence and uniqueness theorem to the initial value problem is proved. The solution to the problem is written out in terms of the special function Wν​(t).

In the process of frozen food production, its refrigeration system is a typical large inertia, nonlinear, strongly coupled system, which is prone to large overshooting during operation, resulting in cost increase and quality control problems. This paper solves the problem of large overshoot in the digital PID control process by designing a fuzzy control, and designs a practical fuzzy controller so that the design method of the fuzzy PID controller can be realized in the simulation and actual control of the freezing system. A genetic algorithm optimization scheme is also designed for the fuzzy PID control system to optimize the fuzzy controller’s affiliation function, control rules, as well as proportionality and quantization factors, to enhance the model’s dynamic characteristics and robustness. Design of food quick-freezing simulation experiments and use the model in the refrigeration system control in practice, compare the performance of this model with the PID, fuzzy PID model, and found that the temperature difference set value of 4 ℃ and 3 ℃, respectively, the parameters of the traditional PID controller in the commissioning stage of the system after the end of the rectification to remain fixed, which makes the actual quality of the cooling water temperature difference control degradation. And the fuzzy genetic PID controller still has good control quality. In this paper, the model curve converges rapidly, the overshoot is only 3.9%, and the regulation time is only 450M seconds. The designed method of the study is feasible and has high industrial application value.

This research has proposed the iterative genetic algorithm (GA) optimization scheme to synthesize the radiation pattern of an aperiodic (nonuniform) linear array antenna. The aim of the iterative optimization is to achieve a radiation pattern with a side lobe level (SLL) of ≤−20 dB. In the optimization, the proposed scheme iteratively optimizes the array range (spacing) and the number of array elements, whereby the array element with the lowest absolute complex weight coefficient is first removed and then the second lowest and so on. The removal (the element reduction) is terminated once the SLL is greater than −20 dB (>−20 dB) and the elemental increment mechanism is triggered. The results indicate that the proposed iterative GA optimization scheme is applicable to the nonuniform linear array antenna and also is capable of synthesizing the radiation pattern with SLL ≤ −20 dB.

Many studies have proved that the fractional-order system is more accurate than the integer-order system. The theory of fractional calculus has received extensive attention in many fields. The research on flux-controlled memristor has been carried out for several years; however, fractional-order flux-controlled memristor has not received widespread attention. In light of the theory of fractional calculus, a generalized fractional-order dynamic model of flux-controlled memristor is proposed for the first time in this paper. The mathematical model is established and its characteristics of the pinched hysteresis loops are analyzed by changing the order and the frequency. The specific computational formulas of areas of pinched hysteresis loops are given. The equivalent circuit of the fractional-order flux-controlled memristor is constructed, and circuit simulations are given to illustrate the effectiveness of theoretical analysis and numerical calculation. In this study, the flux-controlled memristor model is extended to the fractional-order model that is closer to the real device, and the analysis method of areas of pinched hysteresis loops is further expanded.

Most physical systems in nature display inherently nonlinear and dynamical properties; hence, it would be difficult for nonlinear equations to be solved merely by analytical methods, which has given rise to the emerging of engrossing phenomena such as bifurcation and chaos. Conjointly, due to nonlinear systems’ exhibiting more exotic behavior than harmonic distortion, it becomes compelling to test, classify and interpret the results in an accurate way. For this reason, avoiding preconceived ideas of the way the system is likely to respond is of pivotal importance since this facet would have effect on the type of testing run and processing techniques used in nonlinear systems. Paradigms of nonlinear science may suggest that it is ‘the study of every single phenomenon’ due to its interdisciplinary nature, which is another challenge encountered and needs to be addressed by generating and designing a systematic mathematical framework where the complexity of natural phenomena hints the requirement of identifying their commonalties and classifying their various manifestations in different nonlinear systems. Studying such common properties, concepts or paradigms can enable one to gain insight into nonlinear problems, their essence and consequences in a broad range of disciplines all forthwith. Fractional differential equations associated with non-local phenomena in physics have arisen as a powerful mathematical tool within a multidisciplinary research framework. Fractional differential equations, as one extension of the fractional calculus theory, can yield the evolution of various systems properly, which reinforces its position in mathematics and science while setting stage for the description of dynamic, complicated and nonlinear events. Through the reflection of the systems’ actual properties, fractional calculus manifests unforeseeable and hidden variations, and thus, enables integration and differentiation, with the solutions to be approximated by numerical methods along with modeling and predicting the dynamics of multiphysics, multiscale and physical systems. Neural Networks (NNs), consisting of hidden layers with nonlinear functions that have vector inputs and outputs, are also considerably employed owing to their versatile and efficient characteristics in classification problems as well as their sophisticated neural network architectures, which make them capable of tackling complicated governing partial differential equation problems. Furthermore, partial differential equations are used to provide comprehensive and accurate models for many scientific phenomena owing to the advancements of data gathering and machine learning techniques which have raised opportunities for data-driven identification of governing equations derived from experimentally observed data. Given these considerations, while many problems are solvable and have been solved, efforts are still needed to be able to respond to the remaining open questions in the fields that have a broad range of spectrum ranging from mathematics, physics, biology, virology, epidemiology, chemistry, engineering, social sciences to applied sciences. With a view of different aspects of such questions, our special issue provides a collection of recent research focusing on the advances in the foundational theory, methodology and topical applications of fractals, fractional calculus, fractional differential equations, differential equations (PDEs, ODEs, to name some), delay differential equations (DDEs), chaos, bifurcation, stability, sensitivity, machine learning, quantum machine learning, and so forth in order to expound on advanced fractional calculus, differential equations and neural networks with detailed analyses, models, simulations, data-driven approaches as well as numerical computations.

An accurate state of charge (SOC) estimation is the basis of the Battery Management System (BMS). In this paper, a new estimation method which considers fractional calculus is proposed to estimate the lithium battery state of charge. Firstly, a modified second-order RC model based on fractional calculus theory is developed to model the lithium battery characteristics. After that, a pulse characterization test is implemented to obtain the battery terminal voltage and current, in which the parameter identification is completed based on least square method. Furthermore, the proposed method based on Fractional Unscented Kalman Filter (FUKF) algorithm is applied to estimate the battery state of charge value in both static and dynamic battery discharging experiment. The experimental results have demonstrated that the proposed method shows high accuracy and efficiency for state of charge estimation and the fractional calculus contributes to the battery state of charge estimation.