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Loss functions are an essential part in modern data-driven approaches, such as bi-level training scheme and machine learnings. In this paper, we propose a loss function consisting of a r -order (an)-isotropic total variation semi-norms TV r , r ∈ ℝ + , defined via the Riemann–Liouville (RL) fractional derivative. We focus on studying key theoretical properties, such as the lower semi-continuity and compactness with respect to both the function and the order of derivative r , of such loss functions. [ABSTRACT FROM AUTHOR]

This paper explores the truncated M -fractional three-component coupled nonlinear Schrödinger (tc-CNLS) equation, that regulates the behavior of optical pulses in optical fibers. These equations are utilized in various scientific and engineering fields, including nonlinear fiber optics, electromagnetic field waves, and signal processing through optical fibers. The study of multi-component NLS equations has gained significant attention due to their ability to elucidate various complex physical phenomena and exhibit more dynamic structures of localized wave solutions. The freshly invented integration tools, known as the fractional modified Sardar subequation method (MSSEM) and fractional enhanced modified extended tanh-expansion method (eMETEM), are employed to ensure the solutions. The study focuses on extracting various types of optical solitons, including bright, dark, singular, bright-dark, complex, and combined solitons. Optical soliton propagation in optical fibers is currently a subject of great interest due to the multiple prospects for ultrafast signal routing systems and short light pulses in communications. In nonlinear dispersive media, optical solitons are stretched electromagnetic waves that maintain their intensity due to a balance between the effects of dispersion and nonlinearity. Furthermore, hyperbolic, periodic and exponential solutions are generated. The utilized methodology is effective in explaining fractional nonlinear partial differential equations (FNLPDEs) as it offers pre-existing solutions and additionally derives novel exact solutions by mixing outcomes from various procedures. Furthermore, we plot the visualizations of solutions by plotting 3D, 2D, and contour graphs with the corresponding parameter values. The findings of this paper can improve the understanding of the nonlinear dynamical behavior of a specific system and demonstrate the efficacy of the methodology used. We anticipate that our study will provide substantial benefits to a considerable group of engineering model experts. The findings demonstrate the efficacy, efficiency, and applicability of the computational method employed, particularly in dealing with intricate systems. [ABSTRACT FROM AUTHOR]

This paper concerns a new semi analytical method for approximate solution of the first type generalized Abel integral equations. An especial numerical fractional derivative which is L1 method will be used for solving this kind of equations. Also, existence, uniqueness, convergence and stability of the given scheme will be studied through some theorems and lemmas. Moreover, some numerical examples are presented and the results are compared with their exact solutions and some other numerical methods to illustrate the capability of this algorithm for solving these generalized Abel integral equations. [ABSTRACT FROM AUTHOR]

This paper employs the fractional derivative to investigate the effect of curves in Lorentz–Minkowski space, which is of crucial significance in geometry and physics. In the method of examining this effect, the conformable fractional derivative is chosen because it best fits the algebraic structure of differential geometry. Therefore, with the aid of conformable fractional derivatives, numerous special curves and the Frenet frame that were previously derived using classical derivatives have been reinterpreted in Lorentz–Minkowski three-space. [ABSTRACT FROM AUTHOR]

The polar bear hairs have special hierarchical structure with fractal dimensions of golden ratio, which endows the creature with remarkable cool prevention. Fractal calculus is adopted in this paper to reveal its thermal properties, and a fractal derivative model of one-dimensional heat conduction along the hair is established and solved, the results reveal that there is an optimal hair length for the cool prevention. This paper sheds a new light on bio-inspired fabrics for fire-protection clothing and clothing for extreme environment. [ABSTRACT FROM AUTHOR]

The stochastic transition behavior of tri-stable states in a fractional-order generalized Van der Pol (VDP) system under multiplicative Gaussian white noise (GWN) excitation is investigated. First, according to the minimal mean square error (MMSE) concept, the fractional derivative can be equivalent to a linear combination of damping and restoring forces, and the original system can be simplified into an equivalent integer-order system. Secondly, the stationary Probability Density Function (PDF) of system amplitude is obtained by stochastic averaging, and based on singularity theory, the critical parameters for stochastic P -bifurcation of the system are found. Finally, the properties of stationary PDF curves of the system amplitude are qualitatively analyzed by choosing the corresponding parameters in each sub-region divided by the transition set curves. The consistency between numerical results obtained by Monte-Carlo simulation and analytical solutions verified the accuracy of the theoretical analysis process and the method used in this paper has a direct guidance in the design of fractional-order controller to adjust the system behavior. [ABSTRACT FROM AUTHOR]

In this paper, stochastic bifurcations of a fractional-order smooth and discontinuous (SD) oscillator composed of different viscoelastic materials are studied. As a widely applicable algorithm for various fractional-orders cases, an extended fast algorithm is introduced to obtain the statistics of the response, where the fractional derivative is separated into a history part and a local part with a predetermined memory length. The local part is approximated by a highly accurate algorithm while the history part is computed by an efficient convolution algorithm. Through this accurate and fast method, effects of the system parameters on the dynamic behaviors, such as the fractional order, smoothness parameter, and frequency of harmonic force, are thus successfully investigated. Abundant stochastic P-bifurcation phenomena are discussed in detail. Further, it is found that only when the damping material shows nearly elastic behaviors, the probability density functions of the system exhibit the crater shape. Experiments show that the fast algorithm is accurate for different fractional orders. [ABSTRACT FROM AUTHOR]

This paper focuses on synchronization control of the fractional-order unified chaotic system. First, the stability of equilibria is analyzed by the Routh–Hurwitz criteria. Then, three different controllers are designed to achieve chaos synchronization of the drive-response unified system. Further, the impact of the fractional order on the synchronization performance is illustrated. Numerical simulations are carried out to verify the feasibility of theoretical analysis. [ABSTRACT FROM AUTHOR]

The primary focus of this paper is the investigation of the truncated M fractional Kuralay equation, which finds applicability in various domains such as engineering, nonlinear optics, ferromagnetic materials, signal processing, and optical fibers. As a result of its capacity to elucidate a vast array of complex physical phenomena and unveil more dynamic structures of localized wave solutions, the Kuralay equation has received considerable interest in the scientific community. To extract the solutions, the recently developed integration method, referred to as the modified generalized Riccati equation mapping (mGREM) approach, is utilized as the solving tool. Multiple types of optical solitons, including mixed, dark, singular, bright-dark, bright, complex, and combined solitons, are extracted. Furthermore, solutions that are periodic, hyperbolic, and exponential are produced. To acquire a valuable understanding of the solution dynamics, the research employs numerical simulations to examine and investigate the exact soliton solutions. Graphs in both two and three dimensions are presented. The graphical representations offer significant insights into the patterns of voltage propagation within the system. The aforementioned results make a valuable addition to the current body of knowledge and lay the groundwork for future inquiries in the domain of nonlinear sciences. The efficacy of the modified GREM method in generating a wide range of traveling wave solutions for the coupled Kuralay equation is illustrated in this study. [ABSTRACT FROM AUTHOR]

In this paper, many important formulas of the subtrigonometric, subhyperbolic, pretrigonometric, prehyperbolic, supertrigonometric, and superhyperbolic functions sin Wiman class are developed for the first time. The subsine, subcosine, subhyperbolic sine, and subhyperbolic cosine associated with Kohlrausch–Williams–Watts (KWW) function and their scaling-law ODEs are proposed. The supersine, supercosine, superhyperbolic sine, and superhyperbolic cosine functions associated with Mittag-Leffler (ML) function and their fractional ODEs are obtained. The conjectures for the supercosine functions containing ML function are presented in detail. [ABSTRACT FROM AUTHOR]

In this paper, a new fractional derivative model for the non-Darcian seepage within the exponential decay kernel is addressed for the first time. The new fractional derivative model is for high-speed non-Darcian and low-speed non-Darcian seepage, in which the applied zone is enlarged. [ABSTRACT FROM AUTHOR]

This paper is mainly concerning the study of truncated M-fractional Kuralay equations that have applications in numerous fields, including nonlinear optics, ferromagnetic materials, signal processing, engineering fields and optical fibers. Due to its ability to clarify a wide range of sophisticated physical phenomena and reveal more dynamic structures of localized wave solutions, the Kuralay equation has captured a lot of attention in the research field. The newly designed integration methods, known as the modified Sardar subequation method and enhanced modified extended tanh expansion method are used as solving tools to validate the solutions. The goal of this study is to extract several kinds of optical solitons, such as mixed, dark, singular, bright-dark, bright, complex and combined solitons. Due to the many potential applications for superfast signal routing techniques and shorter light pulses in communications, the optical propagation of soliton in optical fibers is now a topic of significant interest. In nonlinear dispersive media, optical solitons are stretched electromagnetic waves that maintain their intensity due to a balance between the effects of dispersion and nonlinearity. In addition, exponential, periodic, hyperbolic solutions are generated. The applied approaches are efficient in explaining fractional nonlinear partial differential equations by providing pre-existing solutions and also producing new solutions by combining results from multiple processes. Additionally, we plot the contour, 2D, and 3D graphs with the associated parameter values to visualize the solutions. The results of this study show the effectiveness of the approaches adopted and help enhance comprehension of the nonlinear dynamical behavior of specific systems. We expect that a substantial amount of engineering model specialists will greatly benefit from our work. The findings demonstrate the efficacy, efficiency, and applicability of the computational method employed, particularly in dealing with intricate systems. [ABSTRACT FROM AUTHOR]

In this paper, new results on the α -fractal function with variable parameters are presented. The Weyl–Marchaud variable order fractional derivative of an α -fractal function with variable parameters is examined by imposing certain conditions on the scaling factors. Following the investigation of fractional derivative, the definite integral of the α -fractal function with variable parameters is evaluated for various intervals in the prescribed domain. Finally, an explicit structure for the α -fractal function is provided using the base q representation of numbers. [ABSTRACT FROM AUTHOR]

On the basis of fractional calculus, we introduce an explicit formulation of the integral of controlled paths along Hölder rough paths in terms of Lebesgue integrals for fractional derivatives. The additivity with respect to the interval of integration, a fundamental property of the integral, is not apparent under the formulation because the fractional derivatives depend heavily on the endpoints of the interval of integration. In this paper, we provide a proof of the additivity of the integral under the formulation. Our proof seems to be simpler than those provided in previous studies and is suitable for utilizing the fractional calculus approach to rough path analysis. [ABSTRACT FROM AUTHOR]

In this paper, a conformable fractional time derivative of order α ∈ (0 , 1 ] is considered in view of the Lax-pair of nonlinear operators to derive a fractional nonlinear evolution system of partial differential equations, called the Fractional-Six-Wave-Interaction-Equations, which is derived in terms of one temporal plus one and two spatial dimensions. Further, an ansatz consisting of linear combinations of hyperbolic functions with complex coefficients is utilized to obtain an infinite set of exact soliton solutions for this system. Certain numerical examples are introduced to show the effectiveness of the ansatz method in obtaining exact solutions for similar systems of nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

By using the conformable fractional of the Nikiforov–Uvarov (CF–NU) method, the radial Schrödinger equation is analytically solved. The energy eigenvalues and corresponding functions are obtained, in which the dependent temperature potential is employed. The effect of fraction-order parameter is studied on the heavy-quarkonium masses such as charmonium and bottomonium in a hot QCD medium in the 3D and the higher-dimensional space. This paper discusses the flavor dependence of their binding energies and explores the nature of dissociation by employing the perturbative, nonperturbative, and the lattice-parametrized form of the Debye masses in the medium-modified potential. A comparison is studied with recent works. We conclude that the fractional-order plays an important role in a hot QCD medium in the 3D with consideration of a form of the Debye mass. [ABSTRACT FROM AUTHOR]

In this paper, we establish an averaging principle for neutral stochastic fractional differential equations with non-Lipschitz coefficients and with variable delays, driven by Lévy noise. Our result shows that the solutions of the equations concerned can be approximated by the solutions of averaged neutral stochastic fractional differential equations in the sense of convergence in mean square. As an application, we present an example with numerical simulations to explore the established averaging principle. [ABSTRACT FROM AUTHOR]

Riemann-Liouville fractional calculus of Coalescence Hidden-variable Fractal Interpolation Function (CHFIF) is studied in this paper. It is shown in this paper that fractional integral of order of a CHFIF defined on any interval is also a CHFIF albeit passing through different interpolation points. Further, conditions for fractional derivative of order of a CHFIF is derived in this paper. It is shown that under these conditions on free parameters, fractional derivative of order of a CHFIF defined on any interval is also a CHFIF. [ABSTRACT FROM AUTHOR]

In this paper, we study a new class of Hardy-type inequalities involving fractional calculus operators. We derive the Hardy-type inequalities for the variant of Riemann–Liouville fractional calculus operators and k -Hilfer fractional derivative operator. The obtained inequalities involving fractional operators are more general as compared to some existing results in the literature. [ABSTRACT FROM AUTHOR]

The basic idea of this paper is to investigate the approximate solution to a well-known Fornberg–Whitham equation of arbitrary order. We consider the stated problem under ABC fractional order derivative. The proposed derivative is non-local and contains non-singular kernel of Mittag-Leffler type. With the help of Modified Homotopy Perturbation Method (MHPM), we find approximate solution to the aforesaid equations. The required solution is computed in the form of infinite series. The method needs no discretization or collocation and easy to implement to compute the approximate solution that we intend. We also compare our results with that of the exact solution for the initial four terms approximate solution as well as with that computed by the Laplace decomposition method. We also plot the approximate solution of considered model through surface plots. For numerical illustration, we use Matlab throughout this work. [ABSTRACT FROM AUTHOR]

In this paper, we give a formula for computing the eigenvalues of the Dirichlet problem for a differential equation of second-order with fractional derivatives in the lower terms. We obtained this formula using the perturbation theory for linear operators. Using this formula we can write out the system of eigenvalues for the problem under consideration. [ABSTRACT FROM AUTHOR]

Intermittent bursting in the fractional difference logistic map of matrices is studied in this paper. Analytic relationships governing the dynamics of the fractional difference map of matrices are derived in an explicit form. Computational experiments are used to prove the existence of intermittent bursts located far away from the initial conditions. Such isolated waves of temporary divergence in the fractional difference logistic map of matrices are demonstrated for the first time. Intermittent bursts of temporary divergence offer new possibilities for designing advanced information hiding algorithms. [ABSTRACT FROM AUTHOR]

Vibration energy harvester, which can convert mechanical energy to electrical energy so as to achieve self-powered micro-electromechanical systems (MEMS), has received extensive attention. In order to improve the efficiency of vibration energy harvesters, many approaches, including the use of advanced materials and stochastic loading, have been adopted. As the viscoelastic property of advanced materials can be well described by fractional calculus, it is necessary to further discuss the dynamical behavior of the fractional-order vibration energy harvester. In this paper, the stochastic P-bifurcation of a fractional-order vibration energy harvester subjected to colored noise is investigated. Variable transformation is utilized to obtain the approximate equivalent system. Probability density function for the amplitude of the system response is derived via the stochastic averaging method. Numerical results are presented to verify the proposed method. Critical conditions for stochastic P-bifurcation are provided according to the change of the peak number for the probability density function. Then bifurcation diagrams in the parameter planes are analyzed. The influences of parameters in the system on the mean harvested power are discussed. It is found that the mean harvested power increases with the enhancement of the noise intensity, while it decreases with the increase of the fractional order and the correlation time. [ABSTRACT FROM AUTHOR]

Glycolysis, which occurs in the cytoplasm of both prokaryotic and eukaryotic cells, is regarded to be the primary step employed in the breakdown of glucose to extract energy. As it is utilized by all living things on the planet, it was likely one of the first metabolic routes to emerge. It is a cytoplasmic mechanism that converts glucose into carbon molecules while also producing energy. The enzyme hexokinase aids in the phosphorylation of glucose. Hexokinase is inhibited by this mechanism, which generates glucose-6-P from adenosine triphosphate (ATP). This paper's primary goal is to quantitatively examine the general reaction–diffusion Glycolysis system. Since the Glycolysis model shows a positive result as the unknown variables represent chemical substance concentration, therefore, to evaluate the behavior of the model for the non-integral order derivative of both independent variables, we expanded the concept of the conventional order Glycolysis model to the fractional Glycolysis model. The nonlinearity of the model is decomposed through an Adomian polynomial for evaluation. More precisely, we used the iterative Laplace Adomian decomposition method (LADM) to determine the numerical solution for the underlying model. The model's necessary analytical/numerical solution was found by adding the first few iterations. Finally, we have presented numerical examples and graphical representations to explain the dynamics of the considered model to ensure the scheme's validity. [ABSTRACT FROM AUTHOR]

The employment of nonlinear fractional partial differential equations (NLFPDEs) is not limited to branches of mathematics entirely but also applicable in other science fields such as biology, physics and engineering. This paper derives some solitary wave solutions for the space-time fractional symmetric regularized long wave (SRLW) equation by means of the improved P -expansion approach and the F ′ / F -expansion method. We use the definition of the Jumarie's modified Riemann–Liouville derivative to handle the fractional derivatives appearing in this equation. Diverse types of soliton solutions are successfully expressed on the form of rational, hyperbolic, trigonometric, and complex functions. We extract kink wave, internal solitary wave, and solitary wave solutions. The performances of the proposed methods are compared with each other. Moreover, we compare the constructed results with some published solutions. The long behaviors of the obtained solutions are plotted in 2D and 3D figures. The resulting outcomes point out that the used techniques promise to empower us to deal with more NLFPDEs arising in mathematical physics. [ABSTRACT FROM AUTHOR]

This paper demonstrates an effective and powerful technique, namely fractional He–Laplace method (FHe-LM), to study a nonlinear coupled system of equations with time fractional derivative. The FHe-LM is designed on the basis of Laplace transform to elucidate the solution of nonlinear fractional Hirota–Satsuma coupled KdV and coupled mKdV system but the series coefficients are evaluated in an iterative process with the help of homotopy perturbation method manipulating He's polynomials. The fractional derivatives are considered in the Caputo sense. The obtained results confirm the suggested approach is extremely convenient and applicable to provide the solution of nonlinear models in the form of a convergent series, without any restriction. Also, graphical representation and the error estimate when compared with the exact solution are presented. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to explore the influence of predation on the spread of a disease developed in the prey population where we assume that the prey has a social behavior. The memory of the prey and the predator measured by the time fractional derivative plays a crucial role in modeling the dynamical response in a predator–prey interaction. This memory can be modeled to articulate the involvement of interacting species by the presence of the time fractional derivative in the considered models. For the purpose of studying the complex dynamics generated by the presence of infection and the time-fractional-derivative we split our study into two cases. The first one is devoted to study the effect of a non-selective hunting on the spread of the disease, where the local stability of the equilibria is investigated. Further the backward bifurcation is obtained concerning basic reproduction rate of the infection. The second case is for explaining the impact of selecting the weakest infected prey on the edge of the herd by a predator on the prevalence of the infection, where the local behavior is scrutinized. Moreover, for the graphical representation part, a numerical simulation scheme has been achieved using the Caputo fractional derivative operator. [ABSTRACT FROM AUTHOR]

This paper studies the (random) uniform attractor for a class of non-autonomous stochastic evolution equations driven by a time-periodic forcing and multiplicative fractional noise with Hurst parameter bigger than 1/2. We first establish the existence and uniqueness results for the solution to the considered equation and show that the solution generates a jointly continuous non-autonomous random dynamical system (NRDS). Moreover, we prove the existence of the uniform attractor for this NRDS through stopping time technique. Particularly, a compact uniformly absorbing set is constructed under a smallness condition imposed on the fractional noise. [ABSTRACT FROM AUTHOR]

The paper discusses the fractional Rössler system and the dependence of its dynamics on some entry parameters. An explicit algorithm for a priori determination of fractional Hopf bifurcations is derived and scenarios documenting a route of the system from stability to chaos are performed with respect to a varying system’s fractional order as well as to a varying system’s coefficient. Contrary to the existing results, the searched values of the fractional Hopf bifurcations follow directly from a revealed analytical dependence between these two systems’ entries. Their various critical values are established and confirmed by numerical experiments demonstrating not only the loss of stability of an equilibrium point, but also other phenomena of transition to chaotic behavior. In addition, we suggest an active control method for synchronization of two chaotic fractional-order Rössler systems. Our theoretical analysis enables to synchronize them for any value of a free parameter under which the master system displays a chaotic behavior. [ABSTRACT FROM AUTHOR]

Superfast diffusion exists in various complex anisotropic systems. Its mean square displacement is an exponential function of time proved by several theoretical and experimental investigations. Previous studies have studied the superfast diffusion based on the time-space scaling local structural derivatives without considering the memory of dynamic behavior. This paper proposes a nonlocal time structural derivative model based on the Caputo fractional derivative to describe superfast diffusion in which the structural function is a power law function of time. The obtained concentration of the diffusive particles, i.e. the solution of the structural derivative model is a double-sided exponential distribution. The derived mean square displacement is a Mittag–Leffler function of time, which generalizes the exponential case. To verify the feasibility of the model, the charge and energy transfer at nanoscale interfaces in solar cells and the dynamics of the dripplons between two graphene sheets are employed. Compared with the existing models, the fitting results indicate that the proposed model is more accurate with higher credibility. The properties of the nonlocal structural derivative model with different structural functions are also discussed. [ABSTRACT FROM AUTHOR]

Fractional Dynamics and Control is emerging as a new hot topic of research which draws tremendous attention and great interest. Although the fractional calculus appeared almost in the same era when the classical (or integer-order) calculus was born, it has recently been found that it can better characterize long-memory processes and materials, anomalous diffusion, long-range interactions, long-term behaviors, power laws, allometric scaling laws, and so on. Complex dynamical evolutions of these fractional differential equation models, as well as their controls, are becoming more and more important due to their potential applications in the real world. This special issue includes one review article and twenty-three regular papers, covering fundamental theories of fractional calculus, dynamics and control of fractional differential systems, and numerical calculation of fractional differential equations. [ABSTRACT FROM AUTHOR]

We investigate the eigenvalue intervals of λ 1 , λ 2 , ... , λ n for which the iterative system of four-point fractional-order boundary value problem has at least one positive solution by utilizing Guo–Krasnosel'skii fixed point theorem under suitable conditions. The obtained results in the paper are illustrated with an example for their feasibility. [ABSTRACT FROM AUTHOR]

In this paper, we give some characterizations for the boundedness of weighted fractional composition operator D φ , u β from α -Bloch spaces into weighted type spaces by deriving the bounds of its norm. Also, estimates for essential norm are obtained which gives necessary and sufficient conditions for the compactness of the operator D φ , u β . [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of mild solutions of impulsive evolution fractional functional differential equation of order 0 < α < 1 involving a Lipschitz condition on term I k . We shall rely on a fixed point theorem for the sum of completely continuous and contraction operators due to Burton and Kirk. [ABSTRACT FROM AUTHOR]

Supersingular integrals arise in many areas of applied mathematics: Fluid dynamics and fracture mechanics are among the most important ones. This paper is devoted to investigating the relationship between Riesz, Riesz–Caputo, Hilfer fractional derivatives and the corresponding finite part integrals in Hadamard sense. We prove that Riesz, Riesz–Caputo, and Hilfer derivatives of a given function can be expressed by the finite part integrals of a supersingular integrals which do not exist. [ABSTRACT FROM AUTHOR]

Recently, experimental studies show that fractional calculus can depict the memory and hereditary attributes of neural networks more accurately. In this paper, we introduce temporal fractional derivatives into a six-neuron bidirectional associative memory (BAM) neural network with leakage delay. By selecting two different bifurcation parameters and analyzing corresponding characteristic equations, it is verified that the delayed fractional neural network generates Hopf bifurcation when the bifurcation parameters pass through some critical values. In order to measure how much is the impact of leakage delay on Hopf bifurcation, sensitivity analysis methods, such as scatter plots and partial rank correlation coefficients (PRCCs), are introduced to assess the sensitivity of bifurcation amplitudes to leakage delay. Numerical examples are carried out to illustrate the theoretical results and help us gain an insight into the effect of leakage delay. [ABSTRACT FROM AUTHOR]

This paper is devoted to the investigation of self-organization phenomena in time-fractional reaction-diffusion systems with multiple homogeneous states. It is shown that the fractional reaction-diffusion systems possess some new properties compared to the systems with derivatives of integer orders. In particular, some complex spatio-temporal solutions that cannot be found in the standard reaction-diffusion systems are identified. The simulation results are presented for the case of a incommensurate time-fractional reaction-diffusion system with a cubic nonlinearity. [ABSTRACT FROM AUTHOR]

This paper is divided into two parts. The first deals with some limit theorems to certain extensions of fractional Brownian motion like: bifractional Brownian motion, subfractional Brownian motion and weighted fractional Brownian motion. In the second part we give the similar results of their continuous additive functionals; more precisely, local time and its fractional derivatives involving slowly varying function. [ABSTRACT FROM AUTHOR]

A fractional-order model is proposed to describe the dynamic behaviors of the velocity of blood flow in cerebral aneurysm at the circle of Willis. The fractional-order derivative is used to model the blood flow damping term that features the viscoelasticity of the blood flow behaving between viscosity and elasticity, unlike the existing fractional models that use fractional-order derivatives to replace the integer-order derivatives as mathematical/logical generalization. A numerical analysis of the nonlinear dynamic behaviors of the model is carried out, and the influence of the damping term and the external power supply on the nonlinear dynamics of the model is investigated. It shows that not only chaos via period-doubling bifurcation is observed, but also two additional small period-doubling-bifurcation-like diagrams isolated from the big one are observed, a phenomenon that needs further investigation. [ABSTRACT FROM AUTHOR]

In this paper, we consider electric fields in media with power-law spatial dispersion (PLSD). Spatial dispersion means that the absolute permittivity of the media depends on the wave vector. Power-law type of this dispersion is described by derivatives and integrals of non-integer orders. We consider electric fields of point charge and dipole in media with PLSD, infinite charged wire, uniformly charged disk, capacitance of spherical capacitor and multipole expansion for PLSD-media. [ABSTRACT FROM AUTHOR]

Let (X , d) be a locally compact separable ultrametric space. Given a measure m on X and a function C (B) defined on the set of all non-singleton balls B of X , we consider the hierarchical Laplacian L = L C . The operator L acts in ℒ 2 (X , m) , is essentially self-adjoint and has a purely point spectrum. Choosing a sequence { 𝜀 (B) } of i.i.d. random variables, we consider the perturbed function C (B , ω) and the perturbed hierarchical Laplacian L ω = L C (ω). Under certain conditions, the density of states 𝔪 exists and it is a continuous function. We choose a point λ such that 𝔪 (λ) > 0 and build a sequence of point processes defined by the eigenvalues of L ω located in the vicinity of λ. We show that this sequence converges in distribution to the homogeneous Poisson point process with intensity 𝔪 (λ). [ABSTRACT FROM AUTHOR]

Active feedback control is commonly used to attenuate undesired vibrations in vibrating machineries and structures, such as bridges, highways and aircrafts. In this paper, we investigate the primary resonance and 1/3 subharmonic resonance of a harmonically forced Duffing oscillator under fractional nonlinear feedback control. By means of the first order averaging method, slow flow equations governing the modulations of amplitude and phase of the oscillator are obtained. An approximate solution for the steady state periodic response is derived and its stability is determined by the Routh-Hurwitz criterion. We demonstrate that appropriate choices on the control strategies and feedback gains can delay or eliminate the undesired bifurcations and reduce the amplitude peak both of the primary and subharmonic resonances. Analytical results are verified by comparisons with the numerical integration results. [ABSTRACT FROM AUTHOR]

In this paper, an arbitrary-dimensional quantum cellular neural network (QCNN) model and its fractional-order form are presented by using the polarization of quantum-dot cell and fractional derivatives. Two classes of fractional-order QCNN equations, either two-cell or three-cell models with different value of fractional order α are taken into consideration in detail. In particular, more complex and abundant fractional-order hyperchaotic behaviors can be observed by these two examples. Thus, the proposed fractional-order arbitrary-dimensional QCNN model can have an effective noninteger dimension and can generate rich hyperchaotic dynamics by nth cell. Numerical analysis and simulation results are provided to show the effectiveness of the proposed approach. This study provides valuable information about nth cell fractional-order QCNNs for further application in high-parallel signal processing and fractional quantum chaotic generators. [ABSTRACT FROM AUTHOR]

We prove that main properties represented by Eq. (4.2) for fractional derivative of power function and the non-fractional Leibniz rule in the form (4.3) of the considered paper, cannot hold together for derivatives of non-integer order. As a result, we prove that the usual Leibniz rule (4.3) cannot hold for fractional derivatives. [ABSTRACT FROM AUTHOR]

In this research, we examine the nonsingular memory effect when implementing the Atangana–Baleanu (AB) fractional derivative in the Caputo sense to the Hepatitis C virus (HCV) model. For this purpose, first, the sufficient conditions for the existence and uniqueness of the solutions under the AB fractional derivative of the model are expressed and proved. Therewithal, in order to show the response of the AB derivative on the system dynamics practically, a new version of the predictor–corrector method is used with the newly estimated model parameters from the literature. Finally, we provide some simulations of the results. [ABSTRACT FROM AUTHOR]

In this study, we examine a modified heat equation with memory and nonlinear source. The source function is considered under two different conditions, the global Lipschitz and the exponential growth functions. For the first condition, a special weighted Banach space is applied to deduce a desired result without any assumption on sufficiently small time and initial data. For the second condition of exponential growth, we apply the Moser–Trudinger inequality to cope with the source function, and a special time-space norm to deduce the unique existence of a global solution in regard to sufficiently small data. The main objective of this work is to prove the global existence and uniqueness of mild solutions. Besides the solution techniques, our main arguments are also based on the Banach fixed point theorem and linear estimates for the mild solution. The highlight of this study is that it is the first work on the global well-posedness for the mild solution of the fractional heat conduction with memory and nonlinear sources. [ABSTRACT FROM AUTHOR]

The article focuses on the numerical solution of the time-fractional partial integro differential equation with Caputo–Fabrizio fractional derivative. Topics include considered the problem is discretized by some finite difference schemes in the time direction and then the Sinc collocation method is applied to the resulting problems in the spatial direction.

Most biological systems have long-range temporal memory. Such systems can be modeled using fractional-order differential equations. The combination of fractional-order derivative and time delay provides the system more consistency with the reality of the interactions and higher degree of freedom. A fractional-order delayed prey–predator system with the fear effect has been proposed in this work. The time delay is considered in the cost of fear; therefore, there are no dynamical changes observed in the system due to time delay in the absence of fear. The existence and uniqueness of the solutions of the proposed system are studied along with non-negativity and boundedness. The existence of biologically relevant equilibria is discussed, and the conditions for local asymptotic stability are derived. Hopf bifurcation occurs in the system with respect to delay parameter. Further, a spatially extended system is proposed and analyzed. Hopf bifurcation also occurs in the extended system due to the delay parameter. Numerical examples are provided in support of analytical findings. Fractional-order derivative improves the stability and damps the oscillatory behaviors of the solutions of the system. Bistability behavior of the system admits stable dynamics by decreasing the fractional-order. Also, chaotic behavior is destroyed by decreasing fractional-order. [ABSTRACT FROM AUTHOR]

When a random environment with the Gaussian white noise function is considered, the Wick-type stochastic fractional quadratic-cubic nonlinear Schrödinger equation is used to govern the propagation of optical pulse in polarization-preserving fibers. Using a new strategy, namely combining the variable-coefficient fractional Riccati equation method with the fractional derivative, Mittag–Leffler function and Hermite transformation, some special fractional solutions with the Brownian motion function including fractional bright and dark solitons, and fractional combined soliton solutions are given. Under the influence of the stochastic effect from the stochastic Brownian motion function portrayed by using the Lorentz chaotic system, some wave packets randomly appear during the propagation, and thus make fractional bright soliton travel wriggled in the both periodic dispersion system and the exponential dispersion decreasing system. However, the stochastic Brownian motion function has a more significant impact on the propagation of fractional bright soliton in the periodic dispersion system than that in the exponential dispersion decreasing system. [ABSTRACT FROM AUTHOR]