Article
Mathematics, Applied
Antonio M. Lopes, J. A. Tenreiro Machado
Summary: This paper explores the performance evaluation and visualization of generalized mean discrete-time fractional controllers using multidimensional scaling, comparing time and frequency responses of controlled systems with different parameter combinations. Numerical experiments with fractional PID and two linear plants demonstrate the feasibility of this method for comparing and visualizing multiple test cases.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2021)
Article
Chemistry, Analytical
Antonio M. Lopes, Jose A. Tenreiro Machado
Summary: This paper examines the use of multidimensional scaling to evaluate the performance of fractional-order variable structure controllers on a revolute planar robotic manipulator. The study demonstrates the feasibility and effectiveness of this approach through numerical experiments.
Article
Mathematics, Applied
Vasily E. Tarasov
Summary: This article discusses the scale invariance in nonlinear fractional dynamics in continuous and discrete time approaches. It uses non-integer-order integro-differential operators and considers nonlinear integro-differential equations with Hadamard type operators of non-integer orders and periodic sequence of kicks. Exact solutions are derived without using approximations, and mappings with non-local scaling in time are obtained from proposed equations for discrete time points.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2021)
Article
Engineering, Mechanical
Lin He, Chunqiu Wei, Jiang Sha, Delong Mao, Kangshuo Wang
Summary: This paper presents a general numerical scheme for the optimal control problem of fractional Birkhoffian systems. The scheme derives the fractional forced Birkhoff equations and directly discretizes the fractional Pfaff-Birkhoff-d'Alembert principle to convert the original problem into a nonlinear optimization problem. An illustrative example demonstrates the efficiency and simplicity of the proposed method.
NONLINEAR DYNAMICS
(2022)
Article
Mathematics, Interdisciplinary Applications
Bo Yu, Yifei Pu, Qiuyan He, Xiao Yuan
Summary: This study presents a high-resolution variable-order scaling fractal-ladder circuit for achieving variable-order fractional calculus using a high-resolution multiplying digital-to-analog converter. The experimental results demonstrate that the circuit exhibits variable-order characteristics and can operate at different orders and frequencies.
FRACTAL AND FRACTIONAL
(2022)
Article
Computer Science, Information Systems
Miguel Romero, Carolina Manoso
Summary: Control strategies based on Model-based Predictive Control (MPC) have been widely used due to their ability to handle constrains and predict and optimize future behavior. This paper extends the use of fractional calculus in a fractional-order Generalized Predictive Control (FGPC) strategy for constraints handling, and proposes a new method to soften constraints using fractional operators. Practical tuning of other control parameters is also included.
Article
Mathematics, Interdisciplinary Applications
Arran Fernandez, Hafiz Muhammad Fahad
Summary: We conducted a formal study on weighted fractional calculus and its extension, comparing it with classical Riemann-Liouville fractional calculus, proving fundamental properties and solving ordinary differential equations in specific cases.
FRACTAL AND FRACTIONAL
(2022)
Article
Mathematics
Vasily E. Tarasov
Summary: This paper proposes a generalization of fractional vector calculus to account for a general form of non-locality, using the general fractional calculus in the Luchko approach. It introduces self-consistent definitions of general fractional differential and integral vector operators.
Article
Computer Science, Artificial Intelligence
Lu Liu, Dingyu Xue, Shuo Zhang
Summary: This paper proposes a fuzzy fractional-order PID control algorithm for a general type industrial temperature control system. A fractional-order elementary system is used to describe the temperature control process, aiming to improve production quality and controlled model accuracy. The gain coefficients of the proposed fractional-order PID controller are updated online based on a set of fractional-order fuzzy rules defined by Mittag-Leffler functions. The effectiveness of the proposed controller is verified through examples of temperature control systems, demonstrating its superior dynamic performance and robustness to internal and external disturbances caused by environment changes.
COMPLEX & INTELLIGENT SYSTEMS
(2023)
Article
Automation & Control Systems
Aldo Jonathan Munoz-Vazquez, Guillermo Fernandez-Anaya, Juan Diego Sanchez-Torres
Summary: This article proposes a parallel controller based on the approximation capabilities of fractional sliding modes and the computation of weights for each control action online through an adaptation algorithm to compensate for various disturbances. The stability of the closed-loop system is demonstrated in the Lyapunov framework, and a simulation study highlights the reliability of the proposed controller.
INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING
(2022)
Review
Computer Science, Artificial Intelligence
Jinde Cao, K. Udhayakumar, R. Rakkiyappan, Xiaodi Li, Jianquan Lu
Summary: This study provides an exhaustive review of the dynamical studies of multidimensional FONNs in continuous/discontinuous time. It covers various neural network models and their applications in different mathematical fields. Theoretical findings from multidimensional FONNs with different types of delays are thoroughly evaluated, and stability and synchronization requirements for fractional-order NNs without delays are mentioned.
IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS
(2023)
Review
Mathematics
Vasily E. Tarasov
Summary: General fractional dynamics is an interdisciplinary science that studies the nonlocal properties of linear and nonlinear dynamical systems using general fractional calculus and operator kernels. The concept of general nonlocal mappings is introduced in this research, and exact solutions for equations with general fractional integrals and derivatives are obtained, showcasing the importance of nonlocality in general fractional differential and integral equations.
Article
Engineering, Mechanical
Timi Karner, Janez Gotlih
Summary: Successful control of a dielectric elastomer actuator (DEA) can be challenging, especially when trying to eliminate overshoot. This study introduced a (PID mu)-D-lambda controller to tackle this issue, using mathematical modeling and simulations to validate its effectiveness.
Article
Acoustics
Khalid Zguaid, Asmae Tajani, Amin Jajarmi, Dumitru Baleanu
Summary: The main goal of this manuscript is to investigate a fractional optimal control problem involving Hadamard fractional derivatives in a dynamical system. Necessary conditions for optimality are derived by solving the corresponding Euler-Lagrange equations. An iterative method is proposed to numerically solve the obtained equations. Two illustrative examples are simulated to demonstrate the applicability and efficiency of the proposed method. Numerical simulations show satisfactory results in terms of absolute error values.
JOURNAL OF VIBRATION AND CONTROL
(2022)
Review
Mathematics, Interdisciplinary Applications
Kishore Bingi, B. Rajanarayan Prusty, Abhaya Pal Singh
Summary: Robot manipulators are widely used in various fields and are crucial for future complex infrastructures in space. They are also valuable in places where it is unsafe for humans, such as deep-sea exploration and radioactive environments. However, the time-varying constraints and uncertainties of robotic manipulators pose challenges in their modeling and control.
FRACTAL AND FRACTIONAL
(2023)
Article
Computer Science, Interdisciplinary Applications
O. Nikan, A. Golbabai, J. A. Tenreiro Machado, T. Nikazad
Summary: The paper proposes a novel numerical method, the RBF-FD, to approximate the time-fractional cable model involving two fractional temporal derivatives. The method combines time discretization using the Grunwald-Letnikov expansion and spatial discretization using the RBF-FD. The proposed method is efficient and the numerical results confirm the theoretical formulation.
ENGINEERING WITH COMPUTERS
(2022)
Article
Acoustics
Fakhrodin Mohammadi, Leila Moradi, Jose Antonio Tenreiro Machado
Summary: This study develops an efficient numerical method for solving optimal control problems governed by fractional Volterra integro-differential equations, which has advantages in terms of computational cost and complexity.
JOURNAL OF VIBRATION AND CONTROL
(2022)
Article
Engineering, Multidisciplinary
Waleed M. Abd-Elhameed, Jose A. Tenreiro Machado, Youssri H. Youssri
Summary: This paper introduces an explicit formula for approximating the fractional derivatives of Chebyshev polynomials of the first-kind in the Caputo sense. It is applied to a spectral solution of a certain type of fractional delay differential equations using an explicit Chebyshev tau method. The efficiency and accuracy of the proposed algorithm are demonstrated through numerical results.
INTERNATIONAL JOURNAL OF NONLINEAR SCIENCES AND NUMERICAL SIMULATION
(2022)
Article
Engineering, Multidisciplinary
Zahra Sadat Aghayan, Alireza Alfi, J. A. Tenreiro Machado
Summary: In this article, the delay-dependent robust stability of uncertain fractional order neutral-type systems with distributed delays, nonlinear perturbations, and input saturation is addressed. Using the Lyapunov-Krasovskii functional, criteria on asymptotic robust stability of the systems, expressed in terms of linear matrix inequalities, are constructed to compute the state-feedback controller gains. The controller gains are determined through the cone complementarity linearization algorithm to maximize the domain of attraction. Numerical simulations are conducted to validate the theoretical results.
INTERNATIONAL JOURNAL OF NONLINEAR SCIENCES AND NUMERICAL SIMULATION
(2023)
Article
Acoustics
Ahmed S. Hendy, Mahmoud A. Zaky, Jose A. Tenreiro Machado
Summary: Dealing with fractional differential equations and fractional optimal control problems is more challenging than integer-order problems, and using traditional methods intended for integer-order problems may lead to erroneous results. Incorrectly applying the Cole-Hopf transformation to simplify equations in fractional-order problems can result in incorrect outcomes.
JOURNAL OF VIBRATION AND CONTROL
(2022)
Article
Mathematics, Interdisciplinary Applications
Seyed Mehdi Abedi Pahnehkolaei, Alireza Alfi, J. A. Tenreiro Machado
Summary: Mathematical modeling is crucial in describing the dynamics of infectious diseases, and fractional order calculus has been proposed as a tool for improving heuristic models. This paper studies the convergence of FO particle swarm optimization algorithm and confirms its stability and practical application through simulations.
CHAOS SOLITONS & FRACTALS
(2022)
Article
Computer Science, Interdisciplinary Applications
O. Nikan, Z. Avazzadeh, J. A. Tenreiro Machado, M. N. Rasoulizadeh
Summary: This paper presents an accurate localized meshfree collocation technique for solving the second-order two-dimensional telegraph model. The technique involves two steps: discretizing the time variable and discretizing the spatial variable. By decomposing the problem into multiple subdomains, the computational burden is reduced, resulting in a small condition number and limited computational cost.
ENGINEERING WITH COMPUTERS
(2023)
Article
Computer Science, Interdisciplinary Applications
H. Hassani, J. A. Tenreiro Machado, E. Naraghirad, Z. Avazzadeh
Summary: 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 differential equations. The solution of the nonlinear system of fractional partial differential equations is expanded in terms of the transcendental Bernstein series and the generalized shifted Chebyshev polynomials, as basis functions with unknown free coefficients and control parameters. The corresponding operational matrices of fractional derivatives are then derived for the basis functions. These basis functions, with their operational matrices of fractional order derivatives and the Lagrange multipliers, transform the problem into a nonlinear system of algebraic equations. By means of Darbo's fixed point theorem and Banach contraction principle, an existence result and a unique result for the solution of the nonlinear system of fractional partial differential equations are obtained, respectively. The convergence analysis is discussed and several illustrative experiments illustrate the efficiency and accuracy of the proposed method.
ENGINEERING WITH COMPUTERS
(2023)
Article
Mathematics
Jocemar Q. Chagas, Jose A. Tenreiro Machado, Antonio M. Lopes
Summary: The main contribution of this paper is the proposal of a closed expression for the Ramanujan constant of alternating series based on the Euler-Boole summation formula. It also highlights the unique choice for the parameter a in Hardy's formula for a series of positive terms to obtain a Ramanujan constant that agrees with other summation methods for divergent series. The paper further derives a closed-formula for the Ramanujan constant of a series with the chosen parameter, under a natural interpretation of the integral term in the Euler-Maclaurin summation formula. Several examples of the Ramanujan constant of divergent series are presented.
Article
Mathematics, Applied
Juan P. Ugarte, J. A. Tenreiro Machado, Catalina Tobon
Summary: This study characterizes rotors using a fractional generalization of the entropy concept and investigates the dynamics of atrial fibrillation propagation in computational models. The results demonstrate that the fractional entropy approach provides a better spatio-temporal characterization of rotor dynamics than conventional entropy analysis under various simulated fibrillation conditions.
APPLIED MATHEMATICS AND COMPUTATION
(2022)
Editorial Material
Mathematics
Antonio M. Lopes, J. A. Tenreiro Machado
Article
Mathematics, Applied
Zakieh Avazzadeh, Omid Nikan, Jose Tenreiro Machado, Mohammad Navaz Rasoulizadeh
Summary: This paper proposes a local meshless radial basis function method to solve the two-dimensional time-fractional Sobolev equation. The method approximates the spatial operator using RBF and uses a finite-difference algorithm for time stepping. The stability of the technique is examined using the matrix method, and numerical examples are provided to verify the method's performance and efficiency.
ADVANCES IN CONTINUOUS AND DISCRETE MODELS
(2022)
Article
Engineering, Electrical & Electronic
Manuel Duarte Ortigueira, J. A. Tenreiro Machado
Summary: Two different approaches for describing discrete-time fractional linear systems are presented, one based on discrete derivatives and the other on bilinear transformations. Algorithms for obtaining impulse, step, and frequency responses are provided, along with analysis of state-variable representation.
IEEE CIRCUITS AND SYSTEMS MAGAZINE
(2022)
Article
Mathematics, Interdisciplinary Applications
Liping Chen, Xiaobo Wu, Jose A. Tenreiro Machado, Antonio M. Lopes, Penghua Li, Xueping Dong
Summary: This paper presents a SOC estimation method based on fractional-order square-root unscented Kalman filter (FSR-UKF), and the effectiveness of the algorithm is proven through experiments.
FRACTAL AND FRACTIONAL
(2022)
Article
Biochemical Research Methods
Hossein Hassani, Zakieh Avazzadeh, J. A. Tenreiro Machado, Praveen Agarwal, Maryam Bakhtiar
Summary: This article presents a fractional mathematical model of the HIV/AIDS spread, with a focus on variables representing susceptible patients, HIV-infected patients, AIDS patients not receiving antiretroviral treatment, treated patients, and individuals immune to HIV infection. An optimization technique using generalized shifted Jacobi polynomials is employed to approximate the solution of the model. The article provides proofs for the existence, uniqueness, and convergence results of the method, and showcases its performance through illustrative examples.
JOURNAL OF COMPUTATIONAL BIOLOGY
(2022)
Article
Mathematics, Applied
Junfeng Cao, Ke Chen, Huan Han
Summary: This paper proposes a two-stage image segmentation model based on structure tensor and fractional-order regularization. In the first stage, fractional-order regularization is used to approximate the Hausdorff measure of the MS model. The solution is found using the ADI scheme. In the second stage, thresholding is used for target segmentation. The proposed model demonstrates superior performance compared to state-of-the-art methods.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Dylan J. Oliver, Ian W. Turner, Elliot J. Carr
Summary: This paper discusses a projection-based framework for numerical computation of advection-diffusion-reaction (ADR) equations in heterogeneous media with multiple layers or complex geometric structures. By obtaining approximate solutions on a coarse grid and reconstructing solutions on a fine grid, the computational cost is significantly reduced while accurately approximating complex solutions.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Nathan V. Roberts, Sean T. Miller, Stephen D. Bond, Eric C. Cyr
Summary: In this study, the time-marching discontinuous Petrov-Galerkin (DPG) method is applied to the Vlasov equation for the first time, using backward Euler for a Vlasov-Poisson discretization. Adaptive mesh refinement is demonstrated on two problems: the two-stream instability problem and a cold diode problem.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Yizhi Sun, Zhilin Sun
Summary: This work investigates the convexity of a specific class of positive definite probability measures and demonstrates the preservation of convexity under multiplication and intertwining product. The study reveals that any integrable function on an interval with a polynomial expansion of fast absolute convergence can be decomposed into a pair of positive convex interval probabilities, simplifying the study of interval distributions and discontinuous probabilistic Galerkin schemes.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Bhagwan Singh, Komal Jangid, Santwana Mukhopadhyay
Summary: This paper examines the prediction of bending characteristics of nanoscale materials using the Moore-Gibson-Thompson thermoelasticity theory in conjunction with the nonlocal strain gradient theory. The study finds that the stiffness of the materials can be affected by nonlocal and length-scale parameters, and the aspect ratios of the beam structure play a significant role in bending simulations.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Guoliang Wang, Bo Zheng, Yueqiang Shang
Summary: This paper presents and analyzes a parallel finite element post-processing algorithm for the simulation of Stokes equations with a nonlinear damping term, which integrates the algorithmic advantages of the two-level approach, the partition of unity method, and the post-processing technique. The algorithm generates a global continuous approximate solution using the partition of unity method and improves the smoothness of the solution by adding an extra coarse grid correction step. It has good parallel performance and is validated through theoretical error estimates and numerical test examples.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Hao Xu, Zeng-Qi Wang
Summary: Fluid flow control problems are crucial in industrial applications, and solving the optimal control of Navier-Stokes equations is challenging. By using Oseen's approximation and matrix splitting preconditioners, we can efficiently solve the linear systems and improve convergence.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
Article
Mathematics, Applied
Zhengya Yang, Xuejuan Chen, Yanping Chen, Jing Wang
Summary: This paper focuses on the high-order stable numerical solutions of the time-space fractional diffusion equation. The Fourier spectral method is used for spatial discretization and the Spectral Deferred Correction (SDC) method is used for numerical solutions in time. As a result, a high-precision numerical discretization scheme for solving the fractional diffusion equation is obtained, and the convergence and stability of the scheme are proved. Several numerical examples are presented to demonstrate the effectiveness and feasibility of the proposed numerical scheme.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2024)
