Article
Mathematics, Interdisciplinary Applications
Kamal Kamran, Aisha Subhan, Kamal Shah, Suhad Subhi Aiadi, Nabil Mlaiki, Fahad M. Alotaibi
Summary: In this paper, a class of integrodifferential equations with the Caputo fractal-fractional derivative is studied. The exact and numerical solutions of the problem are obtained using the fractal-fractional differential operator, which is widely used in mathematical modeling of physical and biological problems. The paper presents a scheme to convert the integrodifferential equation into an equivalent equation involving the Riemann-Liouville and Caputo differential operators, and then applies Laplace transform to solve the equation in the Laplace space before converting the solution back to the real domain.
Article
Engineering, Civil
Magdy A. Ezzat
Summary: A new mathematical model was constructed to study the thermal and plasma transfer in organic semiconductors, utilizing numerical techniques for inversion and comparison to investigate the influence of fractional-order parameters on various fields.
SMART STRUCTURES AND SYSTEMS
(2021)
Article
Mathematics, Applied
Christian Maxime Steve Oumarou, Hafiz Muhammad Fahad, Jean-Daniel Djida, Arran Fernandez
Summary: This article discusses various types of fractional calculus and their general classes, emphasizing the importance of proving results in the most general setting. It highlights the significance of fractional integrals and derivatives with general analytic kernels in research and their applications in analyzing the general class that covers both.
COMPUTATIONAL & APPLIED MATHEMATICS
(2021)
Article
Thermodynamics
Magdy A. Ezzat, Roland W. Lewis
Summary: The study uses the system of equations for fractional thermo-viscoelasticity to investigate bioheat transfer and heat-induced mechanical response in human skin tissue. The results show that volume relaxation parameters and fractional order parameters play a significant role in all considered fields, impacting various distributions.
INTERNATIONAL JOURNAL OF NUMERICAL METHODS FOR HEAT & FLUID FLOW
(2022)
Article
Mathematics, Interdisciplinary Applications
Sunday Simon Isah, Arran Fernandez, Mehmet Ali Ozarslan
Summary: We present a general bivariate fractional calculus method using a kernel based on an arbitrary univariate analytic function. Various properties of the general operators are established, including a series formula, function space mappings, and Fourier and Laplace transforms. Notably, we derive a fractional Leibniz rule for the new operators and correct a minor error in a classic textbook on fractional calculus. Additionally, we solve fractional differential equations using transform methods and uncover an interesting connection between bivariate Mittag-Leffler functions.
CHAOS SOLITONS & FRACTALS
(2023)
Article
Mathematics, Applied
Luis L. Ferras, M. Luisa Morgado, Magda Rebelo
Summary: In this work, a generalised viscoelastic model using distributed-order derivatives is presented, allowing for a more accurate description of complex fluids. By choosing a proper weighting function of the order of the derivatives, this model generalises the fractional viscoelastic model and establishes connections between classical, fractional and distributed-order viscoelastic models.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics, Interdisciplinary Applications
Vasily E. Tarasov
Summary: This article proposes a general fractional calculus (GFC) of operators based on the Mellin convolution instead of the Laplace convolution. This Mellin convolution calculus can be considered as an analogue of the Luchko GFC for the Laplace convolution operators. The proposed general fractional differential operators are generalizations of scaling (dilation) differential operators for nonlocality cases. The properties of semi-group and scale-invariance of these operators are proven. The Hadamard and Hadamard-type fractional operators are special cases of the proposed operators. The fundamental theorems for the scale-invariant general fractional operators are also proven. The proposed GFC can be applied in the study of dynamics characterized by nonlocality and scale invariance.
FRACTAL AND FRACTIONAL
(2023)
Article
Mathematics, Interdisciplinary Applications
Ricardo Almeida
Summary: In this work, we investigate variational problems involving the use of generalized proportional fractional derivatives in place of ordinary derivatives. These fractional derivatives depend on a fixed parameter, which acts as a weight on the state function and its first-order derivative. We examine the problems with and without boundary conditions, and explore additional constraints such as isoperimetric and holonomic conditions. We also consider Herglotz's variational problem and the presence of time delays.
FRACTAL AND FRACTIONAL
(2022)
Article
Nuclear Science & Technology
Ahmed E. Aboanber, Abdallah A. Nahla, S. M. Aljawazneh
Summary: Nuclear reactor dynamics involves studying the time-dependent behavior of neutron density in a reactor, which is crucial for ensuring safe and economical operation. The use of fractional order differential operators allows for the development of a good representation of neutron density in reactors, with analytical solutions based on Laplace transformation and eigenvalues of coefficient matrices. The performance of these methods has been tested for both fast and thermal neutron density in the presence of time-dependent external neutron sources, discussing anomalous diffusion processes for various fractional orders.
ANNALS OF NUCLEAR ENERGY
(2021)
Article
Mathematics, Interdisciplinary Applications
Jun-Sheng Duan, Di-Chen Hu, Ming Li
Summary: The impulse response of the fractional oscillation equation was investigated using two different methods to obtain two different analytical forms, with detailed results in terms of analytic approximation and infinite integrals.
FRACTAL AND FRACTIONAL
(2021)
Article
Thermodynamics
Ibrahim Abbas, Aatef Hobiny, Alaa El-Bary
Summary: This article utilizes the fractional bioheat model in spherical coordinates to explain the transfer of heat in living tissues during magnetic hyperthermia treatment for tumors. It proposes a hybrid numerical approach to accurately estimate power dissipation and investigates the impact of fractional parameter and differences in thermophysical properties between diseased and healthy tissue on temperature.
JOURNAL OF NON-EQUILIBRIUM THERMODYNAMICS
(2023)
Article
Mathematics, Interdisciplinary Applications
Jun-Sheng Duan, Yu-Jie Lan, Ming Li
Summary: This passage mainly compares the applications of the Weyl fractional derivative and the Caputo fractional derivative in the fractional oscillator equation. Under the two perspectives, the equation has different solutions and response modes. One of the characteristics of the fractional case is the presence of a monotone recovery term in negative power law.
FRACTAL AND FRACTIONAL
(2022)
Article
Mathematics, Applied
Yasemin Basci, Adil Misir, Sueleyman Ogrekci
Summary: In this work, the most generalized derivatives and integrals in (k, psi)-Hilfer form are discussed, with their features explored. Additionally, a new generalized Laplace transform is defined for the generalized derivatives and integrals in (k, psi)-Hilfer form, and the new Laplace transforms of certain expressions are obtained. These findings encompass previous studies and emphasize the importance of parameters such as k, rho, and psi in the (k, psi)-generalized Laplace transform.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics
Duarte Valerio, Manuel D. Ortigueira, Antonio M. Lopes
Summary: In this paper, a unified fractional derivative is introduced, which can generate various interesting derivative forms. The results of this study are expected to reduce derivative differences and prevent the ambiguous use of fractional derivatives.
Article
Mathematics, Applied
Fahad Alsidrani, Adem Kilicman, Norazak Senu
Summary: This paper provides both analytical and numerical solutions for partial differential equations involving time-fractional derivatives. It implements three powerful techniques and uses the Laplace transformation to enhance the accuracy of the proposed numerical methods. The obtained results are shown through tables and graphs.
Editorial Material
Physics, Mathematical
Delfim F. M. Torres
Summary: The validity of Noether's theorem and the conclusions of Anerot et al. in the Journal of Mathematical Physics (2020, 61(11), 113502) are discussed.
JOURNAL OF MATHEMATICAL PHYSICS
(2023)
Article
Chemistry, Multidisciplinary
Manuel Duarte Ortigueira, Gabriel Bengochea
Summary: This paper discusses the fractionalization and solution of the Ambartsumian equation. It describes a general approach to fractional calculus suitable for applications in physics and engineering. Liouville-type derivatives are shown to be necessary as they preserve backward compatibility with classical results. These derivatives are used to define and solve the fractional Ambartsumian equation. The paper presents both a solution in terms of a slowly convergent fractional Taylor series and a simple solution expressed as an infinite linear combination of Mittag-Leffler functions. It also introduces a fast algorithm using a bilinear transformation and the fast Fourier transform for numerical approximation.
APPLIED SCIENCES-BASEL
(2023)
Article
Mathematics, Applied
Jose Vanterler da C. Sousa, Daniela S. Oliveira, Gastao S. F. Frederico, Delfim F. M. Torres
Summary: We present a new version of ?-Hilfer fractional derivative on arbitrary time scales and investigate its fundamental properties. We derive an integration by parts formula and propose a ?-Riemann-Liouville fractional integral using Laplace transform. We demonstrate the applicability of these new operators by studying a fractional initial value problem and prove the existence, uniqueness, and controllability of solutions in a suitable Banach space. The obtained results are interesting and nontrivial, suggesting new directions for further research. The article concludes with comments and future work.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics
Ashish Rayal, Bhagawati Prasad Joshi, Mukesh Pandey, Delfim F. M. Torres
Summary: This article presents an approximation technique using fractional order Bernstein wavelets for numerical simulations of fractional oscillation equations with variable order. The equations describe electrical circuits exhibiting various nonlinear dynamical behaviors. The proposed variable order model has current interest in engineering and applied sciences. To analyze the behavior of the equations under variable-order fractional operator, the proposed model is converted into nonlinear algebraic equations using collocation nodes. Different cases of the model are examined to demonstrate the precision and performance of the method. The results confirm the simplicity and efficiency of the scheme for studying nonlinear random order fractional models in engineering and science.
Article
Mathematics
Silverio Rosa, Delfim F. M. Torres
Summary: In this article, a simple mathematical code is developed using GNU Octave/MATLAB for simulating mathematical models governed by fractional-order differential equations and resolving fractional-order optimal control problems. The code is applied to a fractional-order model for respiratory syncytial virus (RSV) infection. Both the initial value problem and the fractional optimal control problem are numerically solved using the implemented algorithms. The code is available on GitHub and compatible with MATLAB.
Article
Mathematics
Jaouad Danane, Delfim F. M. Torres
Summary: Our study focuses on the behavior analysis of a stochastic predator-prey model with a time delay and logistic growth of prey under the influence of Levy noise. We establish the existence, uniqueness, and boundedness of a positive solution that spans globally. We explore the conditions for extinction and identify criteria for persistence. Numerical simulations validate our theoretical findings and illustrate the dynamics of the stochastic delayed predator-prey model based on different criteria.
Article
Mathematics, Interdisciplinary Applications
Manuel D. Ortigueira, Gary W. Bohannan
Summary: A general fractional scale derivative is introduced and its relation with Hadamard derivatives is established. A new derivative similar to the Grunwald-Letnikov's is deduced. Tempered versions are also introduced. Scale-invariant systems are described and a new logarithmic Mittag-Leffler series is proposed for solving the corresponding differential equations.
FRACTAL AND FRACTIONAL
(2023)
Article
Operations Research & Management Science
Sakine Esmaili, M. R. Eslahchi, Delfim F. M. Torres
Summary: This study investigates the optimal control problem for a stochastic model of tumour growth with drug application. The model consists of three stochastic hyperbolic equations for the tumour cells' evolution, and two stochastic parabolic equations for the diffusions of nutrient and drug concentrations. Stochastic terms are added to account for uncertainties, and control variables are added to control the drug and nutrient concentrations. The study proves the existence of unique optimal controls, derives necessary conditions using stochastic adjoint equations, and transforms the stochastic model and adjoint equations into deterministic ones to prove the existence and uniqueness of the optimal control.
Article
Multidisciplinary Sciences
Amal S. Alali, Shahbaz Ali, Muhammad Adnan, Delfim F. M. Torres
Summary: The metric dimension of a graph refers to the smallest set of vertices needed to differentiate or categorize every other vertex. This concept has applications in various domains, and the paper proposes two specific types of graphs and discusses their metric dimension upper bounds.
Article
Mathematics
Manuel Duarte Ortigueira, Gabriel Bengochea
Summary: This article introduces the duality of Laplace and Fourier transforms associated with integer-order derivatives and generalizes it to fractional derivatives. The results are further extended to other transforms such as Mellin, Z, and discrete-time Fourier transforms. The use of scale and nabla derivatives and some consequences are also described.
Article
Mathematics
Faical Ndairou, Delfim F. M. Torres
Summary: This paper introduces a new optimal control problem involving a controlled dynamical system with multi-order fractional differential equations. The continuity and differentiability of the state solutions are established with respect to perturbed trajectories. A Pontryagin maximum principle for incommensurate Caputo fractional optimal control problems is stated and proven. An example is provided to illustrate the applicability of the Pontryagin maximum principle.
Article
Mathematics, Interdisciplinary Applications
Gabriel Bengochea, Manuel Duarte Ortigueira
Summary: The paper introduces and studies fractional scale-invariant systems using an operational formalism. It is shown that the impulse and step responses of these systems belong to the vector space generated by special functions introduced in the paper. The method's effectiveness and accuracy are demonstrated through various numerical simulations.
FRACTAL AND FRACTIONAL
(2023)
Article
Mathematics, Applied
Mohamed A. Zaitri, Cristiana J. Silva, Delfim F. M. Torres
Summary: In this study, an analytical solution for the time-optimal control problem in the induction phase of anesthesia is obtained and compared with the conventional shooting method. The results show that the proposed analytical method aligns numerically with the shooting method. This method has advantages in solving the minimum-time problem in the induction phase of anesthesia.
Article
Mathematics, Interdisciplinary Applications
Manuel Duarte Ortigueira, Valeriy Martynyuk, Volodymyr Kosenkov, Arnaldo Guimaraes Batista
Summary: This article reviews the mathematical description of the charging process of time-varying capacitors and proposes a new formulation. Suitable fractional derivatives are described for it. The case of fractional capacitors that follow the Curie-von Schweidler law is considered. A similar scheme for fractional inductors is obtained through suitable substitutions. Formulae for voltage/current input/output are also presented. The backward coherence with classic results is established and generalized to the variable order case. The concept of a tempered fractor and its relation to the Davidson-Cole model are introduced.
FRACTAL AND FRACTIONAL
(2023)
Article
Mathematics
Duarte Valerio, Manuel D. Ortigueira
Summary: This paper introduces and investigates a new type of general variable-order fractional scale derivatives, considering both the stretching and shrinking cases for the definitions of the GL type and the Hadamard type derivatives. The properties of these derivatives are deduced and discussed. Additionally, fractional variable-order systems of autoregressive-moving-average type are introduced and exemplified, with the corresponding transfer functions obtained and used to find the corresponding impulse responses.
Article
Mathematics, Applied
Hao Liu, Yuzhe Li
Summary: This paper investigates the finite-time stealthy covert attack on reference tracking systems with unknown-but-bounded noises. It proposes a novel finite-time covert attack method that can steer the system state into a target set within a finite time interval while being undetectable.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
Article
Mathematics, Applied
Nikolay A. Kudryashov, Aleksandr A. Kutukov, Sofia F. Lavrova
Summary: The Chavy-Waddy-Kolokolnikov model with dispersion is analyzed, and new properties of the model are studied. It is shown that dispersion can be used as a control mechanism for bacterial colonies.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
Article
Mathematics, Applied
Qiang Ma, Jianxin Lv, Lin Bi
Summary: This paper introduces a linear stability equation based on the Boltzmann equation and establishes the relationship between small perturbations and macroscopic variables. The numerical solutions of the linear stability equations based on the Boltzmann equation and the Navier-Stokes equations are the same under the continuum assumption, providing a theoretical foundation for stability research.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
Article
Mathematics, Applied
Samuel W. Akingbade, Marian Gidea, Matteo Manzi, Vahid Nateghi
Summary: This paper presents a heuristic argument for the capacity of Topological Data Analysis (TDA) to detect critical transitions in financial time series. The argument is based on the Log-Periodic Power Law Singularity (LPPLS) model, which characterizes financial bubbles as super-exponential growth (or decay) with increasing oscillations approaching a tipping point. The study shows that whenever the LPPLS model fits the data, TDA generates early warning signals. As an application, the approach is illustrated using positive and negative bubbles in the Bitcoin historical price.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
Article
Mathematics, Applied
Xavier Antoine, Jeremie Gaidamour, Emmanuel Lorin
Summary: This paper is interested in computing the ground state of nonlinear Schrodinger/Gross-Pitaevskii equations using gradient flow type methods. The authors derived and analyzed Fractional Normalized Gradient Flow methods, which involve fractional derivatives and generalize the well-known Normalized Gradient Flow method proposed by Bao and Du in 2004. Several experiments are proposed to illustrate the convergence properties of the developed algorithms.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
Article
Mathematics, Applied
Lianwen Wang, Xingyu Wang, Zhijun Liu, Yating Wang
Summary: This contribution presents a delayed diffusive SEIVS epidemic model that can predict and quantify the transmission dynamics of slowly progressive diseases. The model is applied to fit pulmonary tuberculosis case data in China and provides predictions of its spread trend and effectiveness of interventions.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
Article
Mathematics, Applied
Shuangxi Huang, Feng-Fei Jin
Summary: This paper investigates the error feedback regulator problem for a 1-D wave equation with velocity recirculation. By introducing an invertible transformation and an adaptive error-based observer, an observer-based error feedback controller is constructed to regulate the tracking error to zero asymptotically and ensure bounded internal signals.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
Article
Mathematics, Applied
Weimin Liu, Shiqi Gao, Feng Xu, Yandong Zhao, Yuanqing Xia, Jinkun Liu
Summary: This paper studies the modeling and consensus control of flexible wings with bending and torsion deformation, considering the vibration suppression as well. Unlike most existing multi-agent control theories, the agent system in this study is a distributed parameter system. By considering the mutual coupling between the wing's deformation and rotation angle, the dynamics model of each agent is expressed using sets of partial differential equations (PDEs) and ordinary differential equations (ODEs). Boundary control algorithms are designed to achieve control objectives, and it is proven that the closed-loop system is asymptotically stable. Numerical simulation is conducted to demonstrate the effectiveness of the proposed control scheme.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
Article
Mathematics, Applied
Gourav Mandal, Lakshmi Narayan Guin, Santabrata Chakravarty
Summary: The ecological framework investigates the dynamical complexity of a system influenced by prey refuge and alternative food sources for predators. This study provides a thorough investigation of the stability-instability phenomena, system parameters sensitivity, and the occurrence of bifurcations. The bubbling phenomenon, which indicates a change in the amplitudes of successive cycles, is observed in the current two-dimensional continuous system. The controlling system parameter for the bubbling phenomena is found to be the most sensitive. The prediction and identification of bifurcations in the dynamical system are crucial for theoretical and field researchers.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
Article
Mathematics, Applied
Damian Trofimowicz, Tomasz P. Stefanski, Jacek Gulgowski, Tomasz Talaska
Summary: This paper presents the application of control engineering methods in modeling and simulating signal propagation in time-fractional electrodynamics. By simulating signal propagation in electromagnetic media using Maxwell's equations with fractional-order constitutive relations in the time domain, the equations in time-fractional electrodynamics can be considered as a continuous-time system of state-space equations in control engineering. Analytical solutions are derived for electromagnetic-wave propagation in the time-fractional media based on state-transition matrices, and discrete time zero-order-hold equivalent models are developed and their analytical solutions are derived. The proposed models yield the same results as other reference methods, but are more flexible in terms of the number of simulation scenarios that can be tackled due to the application of the finite-difference scheme.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
Article
Mathematics, Applied
Yuhao Zhao, Fanhao Guo, Deshui Xu
Summary: This study develops a vibration analysis model of a nonlinear coupling-layered soft-core beam system and finds that nonlinear coupling layers are responsible for the nonlinear phenomena in the system. By using reasonable parameters for the nonlinear coupling layers, vibrations in the resonance regions can be reduced and effective control of the vibration energy of the soft-core beam system can be achieved.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
Article
Mathematics, Applied
S. Kumar, H. Roy, A. Mitra, K. Ganguly
Summary: This study investigates the nonlinear dynamic behavior of bidirectional functionally graded plates (BFG) and unidirectional functionally graded plates (UFG). Two different methods, namely the whole domain method and the finite element method, are used to formulate the dynamic problem. The results show that all three plates exhibit hardening type nonlinearity, with the effect of material gradation parameters being more pronounced in simply supported plates.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
Article
Mathematics, Applied
Isaac A. Garcia, Susanna Maza
Summary: This paper analyzes the role of non-autonomous inverse Jacobi multipliers in the problem of nonexistence, existence, localization, and hyperbolic nature of periodic orbits of planar vector fields. It extends and generalizes previous results that focused only on the autonomous or periodic case, providing novel applications of inverse Jacobi multipliers.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
Article
Mathematics, Applied
Yongjian Liu, Yasi Lu, Calogero Vetro
Summary: This paper introduces a new double phase elliptic inclusion problem (DPEI) involving a nonlinear and nonhomogeneous partial differential operator. It establishes the existence and extremality results to the elliptic inclusion problem and provides definitions for weak solutions, subsolutions, and supersolutions.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
Article
Mathematics, Applied
Shangshuai Li, Da-jun Zhang
Summary: In this paper, the Cauchy matrix structure of the spin-1 Gross-Pitaevskii equations is investigated. A 2 x 2 matrix nonlinear Schrodinger equation is derived using the Cauchy matrix approach, serving as an unreduced model for the spin-1 BEC system with explicit solutions. Suitable constraints are provided to obtain reductions for the classical and nonlocal spin-1 GP equations and their solutions, including one-soliton solution, two-soliton solution, and double-pole solution.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
