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A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: closed (15 October 2022) | Viewed by 26525

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Dr. Duarte Valério

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Instituto de Engenharia Mecânica (IDMEC), Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal
Interests: fractional calculus; fractional control; wave energy conversion; data mining
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Dear Colleagues,

Dynamic models that are linear and only involve integer-order derivatives are relatively simple to study and rather effective as tools for modeling systems in many branches of science and engineering. Their properties are consequently very well known, and they are widely applied.

More complex models may be nonlinear (and integer), may have fractional-order derivatives (and be linear), or may involve both nonlinearities and fractional calculus. This Special Issue is dedicated to the latter case, which has received increased attention in recent years, where fractional derivatives and nonlinearities interact.

Some of the related subjects that deserve to be studied and deepened are as follows:

Study of properties of fractional-order nonlinear systems
Identification techniques for fractional order nonlinear systems
Applications of fractional-order nonlinear systems in science and engineering
Fractional impulsive systems
Stochastic fractional order nonlinear systems
Approximations of nonlinearities using fractional derivatives
Approximations of fractional derivatives using nonlinearities
Design of fractional order controllers for nonlinear plants
Design of nonlinear controllers for fractional order plants
Fractional reset control
Fractional sliding mode control

Prof. Dr. Duarte Valério
Guest Editor

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Keywords

Fractional order nonlinear systems
Fractional calculus
Fractional order systems
Non-linear systems
Fractional control
Non-linear control
Impulsive systems
Reset control

Published Papers (14 papers)

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Research

18 pages, 3284 KiB

Open AccessArticle

Generalization of Reset Controllers to Fractional Orders

by Henrique Paz and Duarte Valério

Mathematics 2022, 10(24), 4630; https://0-doi-org.brum.beds.ac.uk/10.3390/math10244630 - 07 Dec 2022

Cited by 1 | Viewed by 798

Abstract

Reset control is a simple non-linear control technique that can help overcome the structural limitations of linear control. Fractional control uses the concept of fractional derivatives to expand the range of possibilities when modeling a controller, making it more robust. Fractional reset control merges the advantages of both areas and is the object of this paper. Fractional-order versions of different reset controllers were implemented, namely a fractional Clegg integrator, a fractional generalized first-order reset element, a fractional generalized second-order reset element, and fractional “constant in gain lead in phase” controllers with first- and second-order reset elements. These were computed directly from a numerical implementation of the Grünwald–Letnikov definition of fractional derivatives, and their performances were analyzed. Full article

(This article belongs to the Special Issue Fractional Calculus and Nonlinear Systems)

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18 pages, 354 KiB

Open AccessArticle

Existence of Hilfer Fractional Stochastic Differential Equations with Nonlocal Conditions and Delay via Almost Sectorial Operators

by Sivajiganesan Sivasankar, Ramalingam Udhayakumar, Velmurugan Subramanian, Ghada AlNemer and Ahmed M. Elshenhab

Mathematics 2022, 10(22), 4392; https://0-doi-org.brum.beds.ac.uk/10.3390/math10224392 - 21 Nov 2022

Cited by 6 | Viewed by 1313

Abstract

In this article, we examine the existence of Hilfer fractional

(HF)

stochastic differential systems with nonlocal conditions and delay via almost sectorial operators. The major methods depend on the semigroup of operators method and the

M \ddot{o} n c h

[...] Read more.

In this article, we examine the existence of Hilfer fractional

(HF)

stochastic differential systems with nonlocal conditions and delay via almost sectorial operators. The major methods depend on the semigroup of operators method and the

M \ddot{o} n c h

fixed-point technique via the measure of noncompactness, and the fundamental theory of fractional calculus. Finally, to clarify our key points, we provide an application. Full article

(This article belongs to the Special Issue Fractional Calculus and Nonlinear Systems)

30 pages, 988 KiB

Open AccessArticle

Dynamical Analysis of Discrete-Time Two-Predators One-Prey Lotka–Volterra Model

by Abdul Khaliq, Tarek F. Ibrahim, Abeer M. Alotaibi, Muhammad Shoaib and Mohammed Abd El-Moneam

Mathematics 2022, 10(21), 4015; https://0-doi-org.brum.beds.ac.uk/10.3390/math10214015 - 28 Oct 2022

Cited by 8 | Viewed by 1376

Abstract

This research manifesto has a comprehensive discussion of the global dynamics of an achievable discrete-time two predators and one prey Lotka–Volterra model in three dimensions, i.e., in the space

R^{3}

. In some assertive parametric circumstances, the discrete-time model has eight equilibrium [...] Read more.

This research manifesto has a comprehensive discussion of the global dynamics of an achievable discrete-time two predators and one prey Lotka–Volterra model in three dimensions, i.e., in the space

R^{3}

. In some assertive parametric circumstances, the discrete-time model has eight equilibrium points among which one is a special or unique positive equilibrium point. We have also investigated the local and global behavior of equilibrium points of an achievable three-dimensional discrete-time two predators and one prey Lotka–Volterra model. The conversion of a continuous-type model into its discrete counterpart model has been completed by adopting a dynamically consistent nonstandard difference scheme with the end goal that the equilibrium points are conserved in twin cases. The difficulty lies in how to find all fixed points

O, P, Q, R, S, T, U, V

and the Jacobian matrix and its characteristic polynomial at the unique positive fixed point. For that purpose, we use Mathematica software to find the equilibrium points and all of the Jacobian matrices at those equilibrium points. Moreover, we discuss boundedness conditions for every solution and prove the existence of a unique positive equilibrium point. We discuss the local stability of the obtained system about all of its equilibrium points. The discrete Lotka–Volterra model in three dimensions is given by system (3), where parameters

α, β, γ, δ, ζ, η, μ, ε, υ, ρ, σ, ω \in R^{+}

and initial conditions

x_{0}, y_{0}, z_{0}

are positive real numbers. Additionally, the rate of convergence of a solution that converges to a unique positive equilibrium point is discussed. To represent theoretical perceptions, some numerical debates are introduced, including phase portraits. Full article

(This article belongs to the Special Issue Fractional Calculus and Nonlinear Systems)

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14 pages, 1624 KiB

Open AccessArticle

Hierarchical Quasi-Fractional Gradient Descent Method for Parameter Estimation of Nonlinear ARX Systems Using Key Term Separation Principle

by Naveed Ishtiaq Chaudhary, Muhammad Asif Zahoor Raja, Zeshan Aslam Khan, Khalid Mehmood Cheema and Ahmad H. Milyani

Mathematics 2021, 9(24), 3302; https://0-doi-org.brum.beds.ac.uk/10.3390/math9243302 - 18 Dec 2021

Cited by 23 | Viewed by 2170

Abstract

Recently, a quasi-fractional order gradient descent (QFGD) algorithm was proposed and successfully applied to solve system identification problem. The QFGD suffers from the overparameterization problem and results in estimating the redundant parameters instead of identifying only the actual parameters of the system. This study develops a novel hierarchical QFDS (HQFGD) algorithm by introducing the concepts of hierarchical identification principle and key term separation idea. The proposed HQFGD is effectively applied to solve the parameter estimation problem of input nonlinear autoregressive with exogeneous noise (INARX) system. A detailed investigation about the performance of HQFGD is conducted under different disturbance conditions considering different fractional orders and learning rate variations. The simulation results validate the better performance of the HQFGD over the standard counterpart in terms of estimation accuracy, convergence speed and robustness. Full article

(This article belongs to the Special Issue Fractional Calculus and Nonlinear Systems)

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29 pages, 921 KiB

Open AccessArticle

Analysis of Fractional-Order Nonlinear Dynamic Systems with General Analytic Kernels: Lyapunov Stability and Inequalities

by Oscar Martínez-Fuentes, Fidel Meléndez-Vázquez, Guillermo Fernández-Anaya and José Francisco Gómez-Aguilar

Mathematics 2021, 9(17), 2084; https://0-doi-org.brum.beds.ac.uk/10.3390/math9172084 - 28 Aug 2021

Cited by 21 | Viewed by 2469

Abstract

In this paper, we study the recently proposed fractional-order operators with general analytic kernels. The kernel of these operators is a locally uniformly convergent power series that can be chosen adequately to obtain a family of fractional operators and, in particular, the main existing fractional derivatives. Based on the conditions for the Laplace transform of these operators, in this paper, some new results are obtained—for example, relationships between Riemann–Liouville and Caputo derivatives and inverse operators. Later, employing a representation for the product of two functions, we determine a form of calculating its fractional derivative; this result is essential due to its connection to the fractional derivative of Lyapunov functions. In addition, some other new results are developed, leading to Lyapunov-like theorems and a Lyapunov direct method that serves to prove asymptotic stability in the sense of the operators with general analytic kernels. The FOB-stability concept is introduced, which generalizes the classical Mittag–Leffler stability for a wide class of systems. Some inequalities are established for operators with general analytic kernels, which generalize others in the literature. Finally, some new stability results via convex Lyapunov functions are presented, whose importance lies in avoiding the calculation of fractional derivatives for the stability analysis of dynamical systems. Some illustrative examples are given. Full article

(This article belongs to the Special Issue Fractional Calculus and Nonlinear Systems)

16 pages, 350 KiB

Open AccessArticle

Approximate Iterative Method for Initial Value Problem of Impulsive Fractional Differential Equations with Generalized Proportional Fractional Derivatives

by Ravi P. Agarwal, Snezhana Hristova, Donal O’Regan and Ricardo Almeida

Mathematics 2021, 9(16), 1979; https://0-doi-org.brum.beds.ac.uk/10.3390/math9161979 - 19 Aug 2021

Cited by 2 | Viewed by 1357

Abstract

The main aim of the paper is to present an algorithm to solve approximately initial value problems for a scalar non-linear fractional differential equation with generalized proportional fractional derivative on a finite interval. The main condition is connected with the one sided Lipschitz condition of the right hand side part of the given equation. An iterative scheme, based on appropriately defined mild lower and mild upper solutions, is provided. Two monotone sequences, increasing and decreasing ones, are constructed and their convergence to mild solutions of the given problem is established. In the case of uniqueness, both limits coincide with the unique solution of the given problem. The approximate method is based on the application of the method of lower and upper solutions combined with the monotone-iterative technique. Full article

(This article belongs to the Special Issue Fractional Calculus and Nonlinear Systems)

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11 pages, 281 KiB

Open AccessArticle

Fractional Line Integral

by Gabriel Bengochea and Manuel Ortigueira

Mathematics 2021, 9(10), 1150; https://0-doi-org.brum.beds.ac.uk/10.3390/math9101150 - 20 May 2021

Viewed by 1812

Abstract

This paper proposed a definition of the fractional line integral, generalising the concept of the fractional definite integral. The proposal replicated the properties of the classic definite integral, namely the fundamental theorem of integral calculus. It was based on the concept of the fractional anti-derivative used to generalise the Barrow formula. To define the fractional line integral, the Grünwald–Letnikov and Liouville directional derivatives were introduced and their properties described. The integral was defined for a piecewise linear path first and, from it, for any regular curve. Full article

(This article belongs to the Special Issue Fractional Calculus and Nonlinear Systems)

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19 pages, 879 KiB

Open AccessArticle

Successive Approximation Technique in the Study of a Nonlinear Fractional Boundary Value Problem

by Kateryna Marynets

Mathematics 2021, 9(7), 724; https://0-doi-org.brum.beds.ac.uk/10.3390/math9070724 - 27 Mar 2021

Cited by 2 | Viewed by 1888

Abstract

We studied one essentially nonlinear two–point boundary value problem for a system of fractional differential equations. An original parametrization technique and a dichotomy-type approach led to investigation of solutions of two “model”-type fractional boundary value problems, containing some artificially introduced parameters. The approximate solutions of these problems were constructed analytically, while the numerical values of the parameters were determined as solutions of the so-called “bifurcation” equations. Full article

(This article belongs to the Special Issue Fractional Calculus and Nonlinear Systems)

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17 pages, 287 KiB

Open AccessArticle

Controllability for Retarded Semilinear Neutral Control Systems of Fractional Order in Hilbert Spaces

by Daewook Kim and Jin-Mun Jeong

Mathematics 2021, 9(6), 671; https://0-doi-org.brum.beds.ac.uk/10.3390/math9060671 - 21 Mar 2021

Viewed by 1364

Abstract

In this paper, we discuss the approximate controllability for a class of retarded semilinear neutral control systems of fractional order by investigating the relations between the reachable set of the semilinear retarded neutral system of fractional order and that of its corresponding linear system. The research direction used here is to find the conditions for nonlinear terms so that controllability is maintained even in perturbations. Finally, we will show a simple example to which the main result can be applied. Full article

(This article belongs to the Special Issue Fractional Calculus and Nonlinear Systems)

15 pages, 27758 KiB

Open AccessArticle

Fractional-Order Colour Image Processing

by Manuel Henriques, Duarte Valério, Paulo Gordo and Rui Melicio

Mathematics 2021, 9(5), 457; https://0-doi-org.brum.beds.ac.uk/10.3390/math9050457 - 24 Feb 2021

Cited by 13 | Viewed by 2442

Abstract

Many image processing algorithms make use of derivatives. In such cases, fractional derivatives allow an extra degree of freedom, which can be used to obtain better results in applications such as edge detection. Published literature concentrates on grey-scale images; in this paper, algorithms of six fractional detectors for colour images are implemented, and their performance is illustrated. The algorithms are: Canny, Sobel, Roberts, Laplacian of Gaussian, CRONE, and fractional derivative. Full article

(This article belongs to the Special Issue Fractional Calculus and Nonlinear Systems)

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14 pages, 441 KiB

Open AccessArticle

Fractional Vertical Infiltration

by Carlos Fuentes, Fernando Alcántara-López, Antonio Quevedo and Carlos Chávez

Mathematics 2021, 9(4), 383; https://0-doi-org.brum.beds.ac.uk/10.3390/math9040383 - 14 Feb 2021

Cited by 1 | Viewed by 1766

Abstract

The infiltration phenomena has been studied by several authors for decades, and numerical and approximate results have been shown through the asymptotic solution in short and long times. In particular, it is worth highlighting the works of Philip and Parlange, who used time [...] Read more.

t^{1 / 2}

that is valid for short times. However, several studies show that these models are not applicable to anomalous flows, in which case the application of fractional calculus is needed. In this work, a fractional time derivative of a Caputo type is applied to model anomalous infiltration phenomena. Fractional horizontal infiltration phenomena are studied, and the fractional Boltzmann transform is defined. To study fractional vertical infiltration phenomena, the asymptotic behavior is described for short and long times considering an arbitrary diffusivity and hydraulic conductivity. Finally, considering a constant flux-dependent relation and a relation between diffusivity and hydraulic conductivity, a fractional cumulative infiltration model applicable to various types of soil is built; its solution is expressed as a power series in

t^{ν / 2}

, where

ν \in (0, 2)

is the order of the fractional derivative. The results show the effect of superdiffusive and subdiffusive flows in different types of soil. Full article

(This article belongs to the Special Issue Fractional Calculus and Nonlinear Systems)

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10 pages, 290 KiB

Open AccessArticle

Initial Value Problems of Semilinear Supdiffusion Equations

by Yabing Gao and Yongxiang Li

Mathematics 2021, 9(1), 57; https://0-doi-org.brum.beds.ac.uk/10.3390/math9010057 - 29 Dec 2020

Cited by 1 | Viewed by 1198

Abstract

We consider the existence of a mild solution of supdiffusion equations and obtain some results under some growth and noncompactness conditions of nonlinearity without coefficient restriction; and some new results for Ulam-Hyers-Rassias stability are obtained. Full article

(This article belongs to the Special Issue Fractional Calculus and Nonlinear Systems)

13 pages, 328 KiB

Open AccessArticle

Delay-Dependent and Order-Dependent Guaranteed Cost Control for Uncertain Fractional-Order Delayed Linear Systems

by Fei Qi, Yi Chai, Liping Chen and José A. Tenreiro Machado

Mathematics 2021, 9(1), 41; https://0-doi-org.brum.beds.ac.uk/10.3390/math9010041 - 27 Dec 2020

Cited by 7 | Viewed by 1594

Abstract

This paper addresses the guaranteed cost control problem of a class of uncertain fractional-order (FO) delayed linear systems with norm-bounded time-varying parametric uncertainty. The study is focused on the design of state feedback controllers with delay such that the resulting closed-loop system is asymptotically stable and an adequate level of performance is also guaranteed. Stemming from the linear matrix inequality (LMI) approach and the FO Razumikhin theorem, a delay- and order-dependent design method is proposed with guaranteed closed-loop stability and cost for admissible uncertainties. Examples illustrate the effectiveness of the proposed method. Full article

(This article belongs to the Special Issue Fractional Calculus and Nonlinear Systems)

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18 pages, 6462 KiB

Open AccessArticle

Design of Fractional Order Controllers Using the PM Diagram

by Santiago Garrido, Concepción A. Monje, Fernando Martín and Luis Moreno

Mathematics 2020, 8(11), 2022; https://0-doi-org.brum.beds.ac.uk/10.3390/math8112022 - 13 Nov 2020

Cited by 4 | Viewed by 1539

Abstract

This work presents a modeling and controller tuning method for non-rational systems. First, a graphical tool is proposed where transfer functions are represented in a four-dimensional space. The magnitude is represented in decibels as the third dimension and a color code is applied to represent the phase in a fourth dimension. This tool, which is called Phase Magnitude (PM) diagram, allows the user to visually obtain the phase and the magnitude that have to be added to a system to meet some control design specifications. The application of the PM diagram to systems with non-rational transfer functions is discussed in this paper. A fractional order Proportional Integral Derivative (PID) controller is computed to control different non-rational systems. The tuning method, based on evolutionary computation concepts, relies on a cost function that defines the behavior in the frequency domain. The cost value is read in the PM diagram to estimate the optimum controller. To validate the contribution of this research, four different non-rational reference systems have been considered. The method proposed here contributes first to a simpler and graphical modeling of these complex systems, and second to provide an effective tool to face the unsolved control problem of these systems. Full article

(This article belongs to the Special Issue Fractional Calculus and Nonlinear Systems)

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