José A. Tenreiro Machado

Polytechnic Institute of Porto | IPP · Department of Electrical Engineering

Fractal geometry plays an important role in the description of the characteristics of nature. Local fractional calculus, a new branch of mathematics, is used to handle the non-differentiable problems in mathematical physics and engineering sciences. The local fractional inequalities, local fractional ODEs and local fractional PDEs via local fractio...

This article presents a new strategy for solving the fractional epidemic model of COVID-19 (FEMC-19) in Italy. First a new set of basis functions is proposed to deal with the solutions in the system of FEMC-19. For that purpose the generalized Genocchi polynomials and their operational matrices are adopted. The problem is transformed into a system...

This paper adopts the transcendental Bernstein series (TBS), a set of basis functions based on the Bernstein polynomials (BP), for approximating analytical functions. The TBS is more accurate than the BP method, particularly in approximating functions including one or more transcendental terms. The numerical results reveal also the applicability an...

The treatment of fractional differential equations and fractional optimal control problems is more difficult to tackle than the standard integer-order counterpart and may pose problems to non-specialists. Due to this reason, the analytical and numerical methods proposed in the literature may be applied incorrectly. Often, such methods were establis...

This paper analyses several features of fundamental and composite particles using a computational approach. Different distances are used to unravel the connections among particles emerging from their characteristics. Two clustering and visualization techniques are adopted, namely hierarchical clustering (HC) and multidimensional scaling (MDS), for...

This paper proposes a local meshless radial basis function (RBF) method to obtain the solution of the two-dimensional time-fractional Sobolev equation. The model is formulated with the Caputo fractional derivative. The method uses the RBF to approximate the spatial operator, and a finite-difference algorithm as the time-stepping approach for the so...

This chapter presents a new nonlinear variable-order (VO) time fractional convection-diffusion equation (NV-TFCDE). The model generalizes the standard fixed-order nonlinear time fractional convection-diffusion equation. The VO time fractional derivative is described in the Caputo type leading to an optimization method for the NV-TFCDE. The proposed...

This chapter considers a class of variable-order fractional optimal control problems (V-FOCP). An optimization method based on the generalized polynomials (GP) for solving V-FOCP is proposed. The solution of the problem is expanded in terms of the GP with some free coefficients (FC) and control parameters (CP). The FC and CP are obtained optimally...

This paper presents the continuous-time fractional linear systems and their main properties. Two particular classes of models are introduced: the fractional autoregressive-moving average type and the tempered linear system. For both classes, the computations of the impulse response, transfer function, and frequency response are discussed. It is sho...

This paper proposes a nonlinear smoking model (SM) by means of a system of fractional-order differential equations. The SM is formulated in the sense of the fractional Caputo derivative. The method consists of an optimization based on a new class of basis functions, namely the generalized shifted Legendre polynomials (GSLP), to solve the fractional...

Time delay in actuators is mainly caused by electrical and mechanical components. The effect is visible in the system response particularly when changing in the input command. Therefore, input delay is a problem in the control system design that must be taken into account. Besides, ignoring uncertainty in the dynamic models may compromise the contr...

This paper proposes a quantitative approach to specify potentially hazardous asteroids using clustering tools to group a set of near-Earth asteroids (NEAs). The data pool adopted in the study contains a number of distinct indices characterizing ∼ 25 , 000 NEAs. The hierarchical clustering (HC) and multidimensional scaling (MDS) algorithmic techniqu...

Catastrophic events have been commonly referred to as phase transitions in complex systems (CS). This paper proposes an approach based on unsupervised machine learning to identify phases and phase transitions in the dynamics of CS. The testbed is a dataset of causalities and events associated with global large-scale accidents. Multidimensional time...

Atrial fibrillation (AF) underlies disordered spatiotemporal electrical activity, that increases in complexity with the persistence of the arrhythmia. It has been hypothesized that a specific arrhythmogenic mechanism, known as rotor, is the main driver sustaining the AF. Thus, the ablation of rotors has been suggested as a therapeutic strategy to t...

This paper introduces a general class of nonlinear system of fractional partial differential equations with initial and boundary conditions. A hybrid method based on the transcendental Bernstein series and the generalized shifted Chebyshev polynomials is proposed for finding the optimal solution of the nonlinear system of fractional partial differe...

This paper presents an accurate localized meshfree collocation technique for the approximate solution of the second-order two-dimensional telegraph model. This model is an useful description of the propagation of electrical signals in a transmission line as well as wave phenomena. The proposed algorithm approximates the unknown solution in two step...

The area of fractional calculus (FC) has been fast developing and is presently being applied in all scientific fields. Therefore, it is of key relevance to assess the present state of development and to foresee, if possible, the future evolution, or, at least, the challenges identified in the scope of advanced research works. This paper gives a vis...

This article presents a fractional mathematical model of the human immunodeficiency virus (HIV)/AIDS spread with a fractional derivative of the Caputo type. The model includes five compartments corresponding to the variables describing the susceptible patients, HIV-infected patients, people with AIDS but not receiving antiretroviral treatment, pati...

The microscale heat transport equation (MHTE) is an important model in the microtechnology. The MHTE differs from the classical model of heat diffusion since it includes temperatures derivatives of second- and third-order with respect to time, and space and time, respectively. This paper studies the application of the localized radial basis functio...

The cable equation is one useful description for modeling phenomena such as neuronal dynamics and electrophysiology. The time-fractional cable model (TFCM) generalizes the classical cable equation by considering the anomalous diffusion that occurs in the ionic motion present for example in the neuronal system. This paper proposes a novel meshless n...

Mathematical modeling plays an important role in biology for describing the dynamics of infectious diseases. A useful strategy for controlling infections and disorder conditions is to adopt computational algorithms for determining interactions among their processes. The use of fractional order (FO) calculus has been proposed as one relevant tool fo...

This paper presents a function approximation technique (FAT) including a fractional-order (FO) compound learning controller in the framework of backstepping algorithm. The controller is applied to uncertain fractional-order nonlinear systems with time-varying input delay in the presence of unknown external disturbances. An FAT is adopted in the lea...

The generalized Cattaneo model describes the heat conduction system in the perspective of time-nonlocality. This paper proposes an accurate and robust meshless technique for approximating the solution of the time fractional Cattaneo model applied to the heat flow in a porous medium. The fractional derivative is formulated in the Caputo sense with o...

It is a well-known fact that convex and non-convex fuzzy mappings play a critical role in the study of fuzzy optimization. Due to the behavior of its definition, the idea of convexity also plays a significant role in the subject of inequalities. The concepts of convexity and symmetry have a tight connection. We may use whatever we learn from both t...

This paper studies the bifurcation analysis of the discrete time Lorenz system considering its generalization for two control parameters. The one- and two-parameter bifurcations of the system, including pitchfork, period-doubling, Neimark–Sacker, 1:2, 1:3, and 1:4 resonances, are surveyed thoroughly. The critical coefficients are computed to analyz...

Journal of Vibration Testing and System Dynamics is an interdisciplinary journal as a platform for facilitating the synergy of dynamics, experimentation, design, and education. The journal publishes high-quality, original articles that explore the theory, modelling, and application of dynamical systems and data-driven dynamics for high-impact engin...

The variation in the option pricing of the fractal transmission system is modelled by the time fractional Black–Scholes equation (TFBSE). This paper proposes an efficient local meshless method for the numerical simulation of the TFBSE. At the first step, a difference formula of L1 type is employed to get a semi-discrete algorithm in the temporal va...

This work presents an overview of the summability of divergent series and fractional finite sums, including their connections. Several summation methods listed, including the smoothed sum, permit obtaining an algebraic constant related to a divergent series. The first goal is to revisit the discussion about the existence of an algebraic constant re...

This paper presents a third order iterative method for obtaining the Moore–Penrose and Drazin inverses with a computational cost of O(n3), where n∈N. The performance of the new approach is compared with other methods discussed in the literature. The results show that the algorithm is remarkably efficient and accurate. Furthermore, sufficient criter...

This paper studies a localized meshless algorithm for calculating the solution of a nonlinear biological population model (NBPM). This model describes the dynamics in the biological population and may provide valuable predictions under different scenarios. The solution of the NBPM is approximated by means of local radial basis function based on the...

Given a data-set of Ribonucleic acid (RNA) sequences we can infer the phylogenetics of the samples and tackle the information for scientific purposes. Based on current data and knowledge, the SARS-CoV-2 seemingly mutates much more slowly than the influenza virus that causes seasonal flu. However, very recent evolution poses some doubts about such c...

A R T I C L E I N F O Keywords: bifurcation numerical normal form critical normal form coefficient numerical continuation Neimark-Sacker period doubling transcritical strong resonances A B S T R A C T This paper studies the dynamic behavior of a discrete-time prey-predator model. It is shown that this model undergoes codimension one and codimension...

This study analyzes the problem of robust stability of fractional-order delay systems of neutral type under actuator saturation. A Lyapunov-Krasovskii (LK) function is constructed and conditions of the asymptotic robust stability of such systems are given, which are formulated by linear matrix inequalities (LMIs), using the Lyapunov direct method....

This paper addresses the area of particle swarm optimization (PSO) algorithms and, in particular, investigates the dynamics of the complex-order PSO (COPSO). The core of the COPSO adopts the concepts of complex derivative and conjugate order differential in the position and velocity adaption mechanisms to improve the algorithmic performance. The wo...

JVTSD is an interdisciplinary journal as a platform for facilitating the synergy of dynamics, experimentation, design, and education. The journal publishes high-quality, original articles that explore the theory, modelling, and application of dynamical systems and data-driven dynamics for high-impact engineering solutions. Manuscripts exploring dat...

In this article, we address the delay-dependent robust stability of uncertain fractional order neutral-type (FONT) systems with distributed delays, nonlinear perturbations, and input saturation. With the aid of the Lyapunov–Krasovskii functional, criteria on asymptotic robust stability of FONT, expressed in terms of linear matrix inequalities, are...

The area of fractional calculus (FC) has been fast developing and is presently being applied in all scientific fields. Therefore, it is of key relevance to assess the present state of development and to foresee, if possible, the future evolution, or, at least, the challenges identified in the scope of advanced research works. This paper gives a vis...

In this article, a fractional order breast cancer competition model (F-BCCM) under the Caputo fractional derivative is analyzed. A new set of basis functions, namely the generalized shifted Legendre polynomials, is proposed to deal with the solutions of F-BCCM. The F-BCCM describes the dynamics involving a variety of cancer factors, such as the ste...

This paper proposes a conceptual experiment embedding the model of a bouncing ball and the Grünwald-Letnikov (GL) formulation for derivative of fractional order. The impacts of the ball with the surface are modeled by means of a restitution coefficient related to the coefficients of the GL fractional derivative. The results are straightforward to i...

Dear colleagues, It is a matter of great pleasure to invite you to submit your high quality research papers in our special issue "Numerical Simulation and Control in Energy Systems". Deadline for manuscript submissions: 24 December 2021. https://www.mdpi.com/journal/mathematics/special_issues/Numerical_Simulation_Control_Energy_Systems

This paper addresses the solution of the Rayleigh–Stokes problem for an edge in a generalized Oldroyd-B fluid using fractional derivatives and the radial basis function-generated finite difference (RBF-FD) method. The time discretization is accomplished via the finite difference approach, while the spatial derivative terms are discretized using the...

This paper presents a comprehensive study of the Particle Swarm Optimization (PSO) algorithm, called complex-order PSO (CPSO). In the core of new set of algorithms, we employ the complex-order derivative and the conjugate order differential concepts in the position and velocity adaption mechanisms. To determine the influence of the control paramete...

The Bernstein operators (BO) are not orthogonal, but they have duals, which are obtained by a linear combination of BO. In recent years dual BO have been adopted in computer graphics, computer aided geometric design, and numerical analysis. This paper presents a numerical method based on the Bernstein operational matrices to solve the time-space fr...

In professional soccer, the choices made in forming a team lineup are crucial for achieving good results. Players are characterized by different skills and their relevance depends on the position that they occupy on the pitch. Experts can recognize similarities between players and their styles, but the procedures adopted are often subjective and pr...

This chapter introduces the fundamental concepts for the mathematical modeling of dynamical systems. In the study of dynamical systems, models for understanding, describing, predicting and controlling its behavior, are needed. In general, a mathematical model consists of a set of differential equations that try to represent the dynamics of the syst...

This chapter starts from the fundamental concepts regarding the bond graph technique that were introduced in Chapter 2. The material in this chapter applies bond graphs in the modeling and dynamical analysis of hydraulic circuits. Readers are guided through several worked examples that are discussed in detail. This set of solved examples allows rea...

An Introduction to Bond Graph Modeling with Applications presents a collection of exercises on dynamical systems, modeling and control for university students in the areas of engineering, physics and applied mathematics. We can find several books on bond graphs, but most merely a small set of exercises and, in a few cases, some commands for compute...

The bond graph technique was first presented in Chapter 2. Based on this general introductory material, this chapter applies bond graphs with special focus in the modeling and dynamical analysis of mechanical systems. Both translational and rotational systems are considered. Several examples are discussed in detail for readers to get experience wit...

This chapter introduces the fundamental ideas regarding the bond graph modeling. A bond graph is a graphical approach for the modeling of physical systems. A bond graph has some similarities to other system representations, such as block diagrams and signal-flow graphs. However, in the bond graphs the arcs/lines connecting elements stand for a bi-d...

The bond graph technique was first presented in Chapter 2. Stemming from this general introductory material, this chapter applies bond graphs with focus in the modeling and dynamical analysis of electrical circuits. Several examples are discussed step by step for readers to grasp the main ideas and to obtain some practical experience. Based on this...

This chapter offers a bond graph guided example on how to model and study the behavior of a physical system using the 20-sim® software application, available at www.20sim.com. The 20-sim software is a modeling and simulation tool used to examine dynamical systems from multiple physical domains, including systems that are a combination of several. T...

This chapter presents the modeling of thermal systems with bond graphs. The introductory material and concepts were presented in Chapter 2. Therefore, the main idea of this chapter is to guide readers through a set of solved problems that highlight the main issues for the application of this powerful graphical tool. The solved exercises are discuss...

This chapter addresses the modeling of multi-domain systems. In fact, the previous chapters of the book discussed the adoption of bond graphs in the scope of specific types of systems, often called as ‘domains’. Therefore, the book chapters included the modeling and dynamical analysis of electrical, mechanical, hydraulic and thermal systems. Noneth...

This article addresses the stability of uncertain fractional order systems of neutral type under actuator saturation. Some criteria regarding the asymptotic robust stability of such type of systems are constructed with the help of the Lyapunov–Krasovskii functional. Moreover, a state-feedback control law is formulated by means of linear matrix ineq...

This paper adopts a meshless method based on the finite difference scheme derived from the local radial basis function (RBF-FD). The algorithm is used for finding the approximate solution of nonlinear anomalous reaction–diffusion models. The time discretization procedure is carried out by means of a weighted discrete scheme covering second-order ap...

In this paper, we analyze the nontrivial zeros of the Riemann zeta-function using the multidimensional scaling (MDS) algorithm and computational visualization features. The nontrivial zeros of the Riemann zeta-function as well as the vectors with several neighboring zeros are interpreted as the basic elements (points or objects) of a data set. Then...

A double color image encryption method based on DNA (deoxyribonucleic acid) computation and chaos is proposed. Differently from the conventional algorithms, double color images are encrypted at the same time so that we can save information of each other, which makes the encryption more safe and reliable. In addition, a new chaotic fractional order...

In this editorial paper, we start by surveying of the main milestones in the organization, foundation, and development of the journal Fractional Calculus and Applied Analysis ( FCAA ). The main potential of FCAA is in its readers, authors, and editors who contribute to the scientific advance and promote the progress of the journal. Among the editor...

Uncertainty about the time of death is part of one’s life, and plays an important role in demographic and actuarial sciences. Entropy is a measure useful for characterizing complex systems. This paper analyses death uncertainty through the concept of entropy. For that purpose, the Shannon and the cumulative residual entropies are adopted. The first...

Time-series generated by complex systems (CS) are often characterized by phenomena such as chaoticity, fractality and memory effects, which pose difficulties in their analysis. The paper explores the dynamics of multidimensional data generated by a CS. The Dow Jones Industrial Average (DJIA) index is selected as a test-bed. The DJIA time-series is...

Nonlinear fractional differential equations reflect the true nature of physical and biological models with non-locality and memory effects. This paper considers nonlinear fractional differential equations with unknown analytical solutions. The Adomian decomposition and the fractional power series methods are adopted to approximate the solutions. Th...

In this paper, we analyze the nontrivial zeros of the Riemann zeta-function using the multidimensional scaling (MDS) algorithm and computational visualization features. The nontrivial zeros of the Riemann zeta-function as well as the vectors with several neighboring zeros are interpreted as the basic elements (points or objects) of a data set. Then...

The substantial, tempered, and shifted fractional derivatives, useful in medium range systems, are reviewed and highlighted in a unified framework. Their historical evolution is described and their properties studied. Moreover, they are characterized and assessed under the light of the strict sense criterion for the definition of derivatives. The r...

This paper adopts concepts of systems’ theory and multidimensional scaling to study the competitiveness in soccer leagues of four countries. A season for a given league is read as a dynamical system with states that are measured at discrete time samples corresponding to the rounds. The system state is inferred from the accumulated points won and lo...

This paper introduces the blossomed and grafted blossomed trees (BT and GBT, respectively) which are two new types of rooted trees. The trees consist of a finite number of solid and hollow vertices that represent buds and blossoms, respectively. Then the relation between them and derivative operators in a differential equation is analyzed. These co...

In the past decade, various types of wavelet-based algorithms were proposed, leading to a key tool in the solution of a number of numerical problems. This work adopts the Chebyshev wavelets for the numerical solution of several models. A Chebyshev operational matrix is developed, for selected collocation points, using the fundamental properties. Mo...

During the last years the modeling of dynamical phenomena has been advanced by including concepts borrowed from fractional order differential equations. The diffusion process plays an important role not only in heat transfer and fluid flow problems, but also in the modelling of pattern formation that arise in porous media. The modified time-fractio...

This paper presents an explicit formula that approximates the fractional derivatives of Chebyshev polynomials of the first-kind in the Caputo sense. The new expression is given in terms of a terminating hypergeometric function of the type 4 F 3 (1). The integer derivatives of Chebyshev polynomials of the first-kind are deduced as a special case of...

Journal of Vibration Testing and System Dynamics is an interdisciplinary journal as a platform for facilitating the synergy of dynamics, experimentation, design, and education. The journal publishes high-quality, original articles that explore the theory, modelling, and application of dynamical systems and data-driven dynamics for high-impact engin...

This article introduces a new class of basis functions, namely, the generalized Bernoulli polynomials (GBP). The GBP are adopted for solving nonlinear fractional optimal control problems (NFOCP) generated by nonlinear fractional dynamical systems (NFDS) and boundary conditions (BC). The corresponding operational matrices (OM) of fractional derivati...

This paper discusses the fractional order breast cancer competition model (F-BCCM), which considers population dynamics among cancer stem, tumor and healthy cells, as well as the effects of excess estrogen and the body’s natural immune response on the cell populations. Generalized shifted Legendre polynomials and their operational matrices are pres...

An effective harmony between the exploration and exploitation phases in meta-heuristics is an essential design consideration to provide reliable performance on a wide range of optimization problems. This paper proposes a novel approach to enhance the exploratory behavior of the Harris hawks optimizer (HHO) based on the fractional calculus (FOC) mem...

In practice the states of a system are not directly available that are difficult to measure by sensors. This article proposes an observer‐based controller for fractional‐order neutral‐type delay systems with actuator saturation. A feedback algorithm is constructed by applying the Lyapunov direct method. Moreover, based on the observer, the gains of...

This manuscript focuses on one of the most famous open problems in mathematics, namely the Collatz conjecture. The first part of the paper is devoted to describe the problem, providing a historical introduction to it, as well as giving some intuitive arguments of why is it hard from the mathematical point of view. The second part is dedicated to th...

This paper studies the numerical solution of the nonlinear time-fractional telegraph equation formulated in the Caputo sense‎. ‎This model is a useful description of the neutron transport process inside the core of a nuclear reactor‎. ‎The proposed method approximates the unknown solution with the help of two main stages‎. ‎At a first stage‎, ‎a se...

In this paper, we study the robust stability of uncertain fractional order (FO) nonlinear systems having neutral-type delay and input saturation. From the Lyapunov–Krasovskii functional, sufficient criteria on asymptotic robust stability of such FO systems with the help of linear matrix inequalities are specified to compute the gain of state-feedba...

The extended Jacobi elliptic function expansion (EJEFE) method is used to retrieve several types of optical solitons of two nonlinear Schrödinger equations, namely the Heisenberg ferromagnetic spin chains and Alfvén envelop equations. The obtained traveling wave solutions and the corresponding plots are analysed by means of the symbolic package Mat...