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Viral infections pose significant threats to public health globally. Understanding the behavior, transmission, and epidemiology of viruses is essential for developing strategies to prevent, control, and manage outbreaks. Mathematical models help in identifying emerging viral pathogens, assessing their risks, and implementing effective public health measures to mitigate their impact. In this work, we formulate the dynamics of Covid-19 viral infection with the effect of vaccination in fractional framework. Our study is mainly concerned with the dynamical behavior and qualitative analysis of Covid-19 dynamics. The model is investigated for basic properties and the threshold of the system is determined. To scrutinize the solution of the recommend system, we use the fixed point theorems of Schaefer and Banach to evaluate the existence and uniqueness of solutions. Moreover, sufficient conditions for the Ulam–Hyers stability of system is established through mathematical skills. We examine the solution pathways of our model through a numerical scheme to show the importance different input parameters of the system. Our findings emphasize the pivotal role of asymptomatic carriers and losing rate of immunity as critical determinants that can heighten the challenge of controlling Covid-19. Vaccination rate and the fractional parameters are attractive parameters while asymptotic fraction poses a significant risk since it has the ability to spread the illness to uninfected areas. We suggest that effective control of the transmission rate can effectively regulate the intensity of Covid-19 transmission. [ABSTRACT FROM AUTHOR]

• Obtaining mathematical discrete-time approach for fractional order models. • Rendering solution for utilizing actuators with fractional models. • Universal solution applicable to broad class of nonlinear systems. • Designing robust algorithm for discrete-time fractional control. This paper attempts to lay the theoretical foundations of the efficient employment of actuation systems with fractional order models. A discussion on the superiority of the discrete-time approaches, relative to continuous-time, is held and it is shown that discrete-time approaches demonstrate further compatibility to real world problems. Afterwards, a standard procedure is introduced that enables researchers and engineers to implement such actuation systems in practice. A general case of a nonlinear plant operated via an actuation system represented by Torvik's fractional three parameter model is delineated, analyzed and extensively discussed. The actuation system in discussion emerges frequently in numerous practical engineering problems and can be considered as a paragon of fractional actuation systems. To render the merits of the paper, the actuation system is simulated and it is illustrated that owing to the standard procedure offered in this paper, fractional actuation systems can be implemented both in simulation and practice hereafter to enhance the performance of numerous processes. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a new fractional derivative of the Caputo type is proposed and some basic properties are studied. The form of the definition shows that the new derivative is the natural extension of the Caputo one, and that it yields the Caputo derivative with designated memory length. By adaptively changing the memory length, the new definition is capable of capturing local memory effect in a distinct way, which is critical in modelling complex systems where the short memory properties has to be considered. Another attractive property of the new derivative is that it is naturally associated with the Riemann–Liouville definition and as a result, the well established Grünwald–Letnikov approach for numerically solving the fractional differential equation can be readily embedded to approximate the solution of differential equation that involves the new derivatives. Numerical simulations demonstrate the changeable memory effect of the new definition. Highlights • The paper presents a new fractional derivative of the Caputo type. • Some important properties and integral transform of the new derivative is derived. • The new derivative offers particular value for capturing the local memory effect. [ABSTRACT FROM AUTHOR]

This article discusses macroeconomic models, which take into account effects of power-law fading memory. The power-law long memory is described by using the mathematical tool of fractional calculus that includes the fractional derivatives and integrals of non-integer orders. We obtain solutions of the fractional differential equations of these macroeconomic models. Examples of dependence of macroeconomic dynamics on the memory effects are suggested. Asymptotic behaviors of the solutions, which characterize the rate of technological growth with memory, are described. We formulate principles of economic dynamics with one-parametric and multi-parametric memory. It has been shown that the effects of fading long memory can change the economic growth rate and change dominant parameters, which determine growth rates. [ABSTRACT FROM AUTHOR]

The complexity of fluid flows remains an intriguing problem and many scientists are still struggling to gain new and reliable insight into the dynamics of fluids. Transition from laminar to turbulent flows is even more complex and many of its features remain surprising and unexplained. To describe transition to turbulence we introduce some fractional models and use numerical approximations to reveal the existence of attractor points. Two different cases are studied; the classical situation corresponding to the integer dimension one and the pure fractional case. The observed simulations show, in both cases, the presence of attractors near which iterations converge faster than usual. The behavior observed in the conventional case is in concordance with the well-known results that exist in the literature for relatively low order ordinary differential equations. The results observed in the fractional case are innovative since they reveal, not only the persistence of attractors, but also a possible better description of the transition to turbulent flows due to the variation of the fractional parameter that allows the control of the dynamics. [ABSTRACT FROM AUTHOR]

For a disturbed dynamical system that can be transformed into fractional Birkhoffian representation with all dynamical information, under a more general fractional infinitesimal transformation of Lie group with high time extension and fractional extension, a new kind of fractional Mei symmetrical perturbation method which is most universal significance is presented and it is found that, using the new method, we can find a new kind of non-Noether adiabatic invariant; as the special cases of new method, we, respectively, reveal an autonomous disturbed fractional Birkhoffian system possesses more adiabatic invariants, a new kind of non-Noether exact invariant directly led by fractional Mei symmetry and a new kind of non-Noether exact and adiabatic invariant of integer Birkhoffian systems. Also, as the new method’s applications to nonlinear dynamical problems, we, respectively, explore the symmetrical perturbation and adiabatic invariant of a disturbed fractional general relativistic Buchdahl model and a disturbed fractional Emden model. It is worth pointing out that, for a disturbed dynamical system, this work reveals intrinsic relation between the fractional symmetrical perturbation and the adiabatic invariant, and provides a general method for finding adiabatic invariants of an actual disturbed fractional dynamical system that is related to science and engineering. [ABSTRACT FROM AUTHOR]

The nonlinear free vibration of a fractional dynamic model for the viscoelastic pipe conveying fluid is studied in this paper. The dynamic equations of coupled planar motion for the pipe are derived by employing the Euler beam theory and the generalized Hamilton principle when we consider both the fractional material model and the geometric non-linearity. Then the equations are simplified into a new nonlinear, fractional order dynamic model governing transverse vibration of the pipe in small but limited stretching issues. The method of multiple scales is directly applied for the analysis and simulation of the nonlinear vibration. Numerical results show the influence of the factional order, the mass ratio, the fluid velocity and the nonlinear coefficient on the nonlinear amplitudes and frequencies of the viscoelastic pipe. It is noticeable that the amplitudes of the fluid-conveying pipe constituted by the fractional viscoelastic material model display much higher than those predicted by the previous models. [ABSTRACT FROM AUTHOR]

For a dynamical system that can be transformed into fractional Birkhoffian representation, under a more general kind of fractional infinitesimal transformation of Lie group, we present the fractional conformal invariance method and it is found that, using the new method, we can find a new kind of non-Noether conserved quantity; and we find that, as a special case, an autonomous fractional Birkhoffian system possesses more conserved quantities. Also, as the fractional conformal invariance method’s applications, we, respectively, explore the conformal invariance and conserved quantities of a fractional Lotka biochemical oscillator and a fractional Hojman–Urrutia model. This work constructs a basic theoretical framework of fractional conformal invariance method, and provides a general method for finding conserved quantities of an actual fractional dynamical system that is related to science and engineering. [ABSTRACT FROM AUTHOR]

In this paper, a new three-dimensional autonomous fractional-order chaotic system is presented. There are four different nonlinear terms in the governed equations. By means of theoretical analysis and numerical simulation, some basic dynamical properties, such as phase diagrams, bifurcation diagram, Lyapunov exponents spectrum and chaos diagram of the new fractional-order chaotic system are investigated. Then, we completed the circuit design of the fractional-order chaotic system, and realized the analog hardware circuit. Both the hardware experiment results and circuit simulation results are agreed well with those of numerical simulation. Finally, based on the theory of sliding-mode control and the fractional stability theory, a suitable sliding mode controller is designed for the novel three-dimensional fractional-order chaotic system. Numerical simulations are provided to verify the effectiveness of the proposed synchronization scheme. [ABSTRACT FROM AUTHOR]

In the paper, the dynamical behaviors of a new fractional order hyperchaotic Rabinovich system are investigated, which include its local stability, hyperchaos, chaotic control and synchronization. Firstly, a new fractional order hyperchaotic Rabinovich system with Caputo derivative is proposed. Then, the hyperchaotic attractors of the commensurate and incommensurate fractional order hyperchaotic Rabinovich system are found. After that, four linear feedback controllers are designed to stabilize this fractional order system Finally, by using the active control method the synchronization is studied between the fractional order hyperchaotic and chaos controlled Rabinovich system In addition, the theoretical predictions are confirmed by numerical simulations. [ABSTRACT FROM AUTHOR]

Nonlocal generalization of classical statistical mechanics is proposed by using the general fractional calculus in the Luchko form. Some basic concepts of nonlocal statistical mechanics in Liouville picture are suggested. Nonlocal analogs of the standard law of probability conservation are suggested in integral and differential forms. Nonlocality is described by the pairs of Sonin kernels that belong to the Luchko set. Examples of solutions of general fractional Liouville equations are proposed for power-law nonlocality in phase space. The introduction gives a brief overview of some approaches to the description of nonlocal statistical mechanics. • Basic concepts of general nonlocal statistical mechanics are proposed. • Nonlocal law of conservation of probability is derived. • Nonlocal generalization of Liouville equations are obtained. • Solutions of general fractional Liouville equations are derived. • Correct standard physical dimensions are taken into account. [ABSTRACT FROM AUTHOR]

In this paper, we present a new method, i.e. fractional Birkhoffian method, for stability of equilibrium positions of dynamical systems, in terms of Riesz derivatives, and study its applications. For an actual dynamical system, the fractional Birkhoffian method of constructing a fractional dynamical model is given, and then the seven criterions for fractional Birkhoffian method of equilibrium stability are established. As applications, by using the fractional Birkhoffian method, we construct four kinds of actual fractional dynamical models, which include a fractional Duffing oscillator model, a fractional Whittaker model, a fractional Emden model and a fractional Hojman–Urrutia model, and we explore the equilibrium stability of these models respectively. This work provides a general method for studying the equilibrium stability of an actual fractional dynamical system that is related to science and engineering. [ABSTRACT FROM AUTHOR]

In this paper we use the conformable fractional derivative and integral to discuss the fractional Newtonian mechanics. The fractional version of the calculus of variations is introduced and the fractional Euler–Lagrange equation is constructed. Some mechanical problems such as fractional harmonic oscillator problem, the fractional damped oscillator problem and the forced oscillator problem are discussed in the one-dimensional fractional dynamics. [ABSTRACT FROM AUTHOR]

In the paper, we present fractional generalized Hamiltonian method of dynamical stability, in terms of Riesz–Riemann–Liouville derivative, and study its applications. For an actual dynamical system, the fractional generalized Hamiltonian method of constructing a fractional dynamical model is given, and then the six criterions for fractional generalized Hamiltonian method of dynamical stability are presented. As applications, by using the fractional generalized Hamiltonian method, we construct five kinds of actual fractional dynamical models, which include a fractional Euler–Poinsot model of rigid body that rotates with respect to a fixed-point, a fractional Hojman–Urrutia model, a fractional Lorentz–Dirac model, a fractional Whittaker model and a fractional Robbins–Lorenz model, and we explore the dynamical stability of these models, respectively. This work provides a general method for studying the dynamical stability of an actual fractional dynamical system that is related to science and engineering. [ABSTRACT FROM AUTHOR]

Integration and differentiation of non-integer orders for N-dimensional physical lattices with long-range particle interactions are suggested. The proposed lattice fractional derivatives and integrals are represented by kernels of lattice long-range interactions, such that their Fourier series transformations have a power-law form with respect to components of wave vector. Continuous limits for these lattice fractional derivatives and integrals give the continuum derivatives and integrals of non-integer orders with respect to coordinates. Lattice analogs of fractional differential equations that include suggested lattice differential and integral operators can serve as an important element of microscopic approach to nonlocal continuum models in mechanics and physics. [ABSTRACT FROM AUTHOR]

Abstract: Lattice models with long-range interactions of power-law type are suggested as a new type of microscopic model for fractional non-local elasticity. Using the transform operation, we map the lattice equations into continuum equation with Riesz derivatives of non-integer orders. The continuum equations that are obtained from the lattice model describe fractional generalization of non-local elasticity models. Particular solutions and correspondent asymptotic of the fractional differential equations for displacement fields are suggested for the static case. [Copyright &y& Elsevier]

• Analysis of the connection between PNP model and displacement current. • Electrical conductivity and evidence of the anomalous diffusion. • Implications of the fractional extension to the total current density. • Extensions of the PNP model and connection with CPE elements. We establish, in general terms, the conditions to be satisfied by a time-fractional approach formulation of the Poisson-Nernst-Planck model in order to guarantee that the total current across the sample be solenoidal, as required by the Maxwell equations. Only in this case the electric impedance of a cell can be determined as the ratio between the applied difference of potential and the current across the cell. We show that in the case of anomalous diffusion, the model predicts for the electric impedance of the cell a constant phase element behaviour in the low frequency region. In the parametric curve of the reactance versus the resistance, the slope coincides with the order of the fractional time derivative. [ABSTRACT FROM AUTHOR]

Abstract: We demonstrate that continuous-time FARIMA processes with -stable noise provide a new stochastic tool for studying the solar flare phenomenon in the framework of fractional Langevin equation. Simple computer tests to check the origins of -stability and self-similarity are implemented for empirical time series describing the energy of solar flares. Based on observed physical time series we solve the challenging problem of how to detect long-range dependence from real data and how to model it via fractional dynamics (Langevin or Fokker–Planck). We employ here codifference as a proper measure for long-range dependence. It is applicable to empirical data from the distribution lacking the second moment. [Copyright &y& Elsevier]

The analysis of the low-frequency conductivity spectra of the clay–water mixtures is presented. The frequency dependence of the conductivity is shown to follow the power-law with the exponent n =0.67 before reaching the frequency-independent part. When scaled with the value of the frequency-independent part of the spectrum the conductivity spectra for samples at different water content values are shown to fit to a single master curve. It is argued that the observed conductivity dispersion is a consequence of the anomalously diffusing ions in the clay–water system. The fractional Langevin equation is then used to describe the stochastic dynamics of the single ion. The results indicate that the experimentally observed dielectric properties originate in anomalous ion transport in clay–water system characterized with time-dependent diffusion coefficient. [Copyright &y& Elsevier]

Abstract: The nonlinear dielectric response due to the application of a strong dc bias electric field superimposed on a weak ac electric field is considered in the context of the anomalous diffusion (subdiffusion). A perturbation procedure is used to derive analytical expressions for the first three harmonic components of the electric polarization of an assembly of both polar and anisotropically polarizable symmetric-top molecules. To accomplish that, an infinite hierarchy of multiterm (21) differential-recurrence equations of noninteger order for the moments is established and solved for the stationary regime. The results so obtained are illustrated in the form of Argand diagrams and three-dimensional relaxation spectra for the complex nonlinear dielectric increment extracted from the first harmonic component of the electric susceptibility. These plots show the role and importance played by the fractional exponent α and the parameter P measuring the influence of the dipole moment over the permanent one. [Copyright &y& Elsevier]

The linear dielectric response of an assembly of symmetric top dipole molecules (each of which is free to rotate in space) is evaluated in the context of fractional dynamics. The infinite hierarchy of differential-recurrence relations for the relaxation functions appropriate to the dielectric response is derived by using the underlying inertial fractional Fokker–Planck (fractional Klein–Kramers) equation. On solving this hierarchy in terms of continued fractions (as in normal rotational diffusion), the complex dynamic susceptibility is calculated for typical values of the model parameters for the particular case of linear molecules. It is shown that the model can reproduce nonexponential anomalous dielectric relaxation behaviour at low frequencies (ωτ≤1, where τ is the Debye relaxation time) and the inclusion of inertial effects ensures that optical transparency is regained at very high frequencies (in the far infrared region) so that Gordon''s sum rule for integral dipolar absorption is satisfied. [Copyright &y& Elsevier]

The fractional Klein–Kramers equation describing the fractal time dynamics of an assembly of fixed axis dipoles rotating in a double-well cosine potential representing the internal field due to neighboring molecules is solved using matrix continued fractions. The solution includes both inertial and strong internal field effects which in combination produce a strong resonance peak (Poley absorption) at high frequencies due to librations of the dipoles in the potential, an anomalous relaxation band at low-frequencies mainly arising from overbarrier relaxation and a weaker relaxation band at mid frequencies due to the fast intrawell modes. The high frequency behavior is controlled by the inertia of a dipole, so that the Gordon sum rule for dipolar absorption is satisfied ensuring a return to optical transparency at very high frequencies. Application of the model to the broad-band (0-THz) dielectric loss spectrum of a dilute solution of the probe dipolar molecule CH2Cl2 in glassy decalin is demonstrated. [Copyright &y& Elsevier]

In this paper, we generalize simple model of quantum mechanics and statistics to take into account power-law nonlocality by using fractional differential equations, and fractal distribution of states. These equations with derivatives of non-integer orders are widely used in describing such complex properties of systems as power-law spatial nonlocality and spatial dispersion, long-range interactions, as well as the openness of quantum systems (interactions with the environment). We suggest a generalization of quantum statistical model of an ideal gas of particles for the case of spatial nonlocality, long-range interactions and fractal distribution of permitted states. To take into account the power-law nonlocality and long-range interactions, we use the fractional Laplacian in the Riesz–Trujillo form in the Schrodinger equation. To take into account the fractal distribution of states, we use non-integer dimensional space approach. Expression for the fractal density of states for fractal distribution of permitted quantum states is proposed. The equation to calculate energy and total number of particles for quantum systems with power-law non-locality and fractal distribution of quantum states are derived. Examples of these calculations for degenerate quantum system are suggested in the framework of the proposed quantum statistical model. • Nonlocal quantum system with fractal density of states is described. • Total number of particles and energy for nonlocal quantum system are proposed. • Fractional Schrodinger equation for power-law nonlocality is suggested. • Fractional Laplacian in Riesz–Trujillo form to describe nonlocality. • Non-integer dimensional space approach is used to described density of states. [ABSTRACT FROM AUTHOR]

In recent years, a new branch of the econophysics has appeared and began to actively develop, which can be called fractional econophysics. We can define fractional econophysics as a new direction of research applying methods developed in physical sciences, to describe processes in economics and finance, basically those including power-law memory and spatial nonlocality. The mathematical tool of this branch of econophysics is the fractional calculus. The birth of the fractional econophysics can be dated 2000 and it can be primarily associated with the works of a group, which includes E. Scalas, F. Mainardi, R. Gorenflo, M. Raberto, in the field of the continuous-time finance. The fractional econophysics was born on the border of the centuries: the first paper was submitted to Physica A on 10 December 1999. Then a lot of work was done on the adaptation and application of fractional dynamics methods, physical model and equations, previously used in the physical sciences, to the description of processes in economics and finance. In fact, at the end of 2019 and at the beginning of 2020 there will be a twenty-year anniversary of fractional econophysics. In this paper, using the fractional econophysics approach, we consider dynamics of market prices, in which power-law memory effects are taken into account. We propose new economic model of the price dynamics in the market for a single product. In this model we assume that economic entities (merchants, buyers, suppliers) can remember how stocks of goods and their prices have changed over time. • Fractional econophysics approach to price dynamics in market is proposed. • Fractional generalization of the Evans models of price dynamics in market is described. • Fractional differential equation of price dynamics with memory is solved. • Asymptotic behavior of proposed solutions is described. • Principle of pricing with memory is suggested. [ABSTRACT FROM AUTHOR]

• Fractional Dynamics of Dirac particle in external electromagnetic field is suggested. • Proper time non-locality of the Dirac particle is described by fractional-order derivatives. • Fock-Schwinger proper time method is used to have covariance of the model. • Fractional Heisenberg equations for Dirac particle are solved. • Asymptotic behavior of the proposed solutions is described. A generalization of the standard model of Dirac particle in external electromagnetic field is proposed. In the generalization we take into account interactions of this particle with environment, which is described by the memory function. This function takes into account that the behavior of the particle at proper time can depend not only at the present time, but also on the history of changes on finite time interval. In this case the Dirac particle can be considered an open quantum system with non-Markovian dynamics. The violation of the semigroup property of dynamic maps is a characteristic property of dynamics with memory. We use the Fock-Schwinger proper time method and derivatives of non-integer orders with respect to proper time. The fractional differential equation, which describes the Dirac particle with memory, and the expression of its exact solution are suggested. The asymptotic behavior of the proposed solutions is described. [ABSTRACT FROM AUTHOR]

The formulae for linear electric polarization induced by harmonic electric field in liquids composed of rigid noninteracting dipolar and asymmetric-top molecules in spherical solvents are derived. The model of noninertial, anomalous rotational Brownian motion is applied. The fractional rotational diffusion equations are solved and time evolution of the electric polarizability is investigated for the case when a dc-electric field is turned off and for the stationary state case when only an ac-field is present. Numerical analysis of the dispersion and absorption parts of electric polarizability is performed as well as the influence of molecular parameters on so-called Cole-Cole plots is investigated. [ABSTRACT FROM AUTHOR]