Search Results (18)

Cosmic rays exhibit anomalous diffusion behaviors in the heliospheric environment that cannot be adequately described by classical diffusion models. In this paper, we develop a theoretical framework employing a fractional Fokker–Planck equation to model the anomalous transport of cosmic rays. This approach accounts for the observed non-Gaussian distributions, long-range correlations and memory effects in cosmic ray fluxes. We derive analytical solutions using the Adomian Decomposition Method and express them in terms of Mittag-Leffler functions and Lévy stable distributions. The model parameters, including the fractional orders $\alpha$ and $\mu$ and the entropic index q, are estimated by a short comparison between theoretical predictions and observational data from cosmic ray experiments. Our findings suggest that the integration of fractional calculus and non-extensive statistics can be employed for describing the cosmic ray propagation and the anomalous diffusion observed in the heliosphere. Full article

For fractional-order systems, observer design is remarkable for the estimation of unavailable states from measurable outputs. In addition, the nonlinear dynamics and the presence of parameters that can vary over different operating conditions or time, such as load or temperature, increase the complexity of the observer design. In view of the aforementioned factors, this paper investigates the observer design problem for a class of Fractional-Order Polynomial Fuzzy Systems (FORPSs) depending on a parameter. The Caputo–Hadamard derivative is considered in this study. First, we prove the practical Mittag-Leffler stability, using the Lyapunov methods, for the general case of Caputo–Hadamard Fractional-Order Systems (CHFOSs) depending on a parameter. Secondly, based on this stability theory, we design an observer for the considered class of FORPSs. The state estimation error is ensured to be practically generalized Mittag-Leffler stable by solving Sum Of Squares (SOSs) conditions using the developed SOSTOOLS. Full article

We aim to investigate the stability property for the certain linear and nonlinear fractional $\mathtt{q}$-difference equations in the Ulam–Hyers and Ulam–Hyers–Rassias sense. To achieve this goal, we prove that three types of the linear $\mathtt{q}$-difference equations of the $\mathtt{q}$-Caputo-like type are Ulam–Hyers stable by using the quantum Laplace transform and quantum Mittag–Leffler function. Moreover, after proving the existence property for a nonlinear Cauchy $\mathtt{q}$-difference initial value problem, we use the same quantum Laplace transform and the $\mathtt{q}$-Gronwall inequality to show that it is generalized Ulam–Hyers–Rassias stable. Full article

Hyperbolic complete monotonicity property ($\mathrm{HCM}$) is a way to check if a distribution is a generalized gamma ($\mathrm{GGC}$), hence is infinitely divisible. In this work, we illustrate to which extent the Mittag-Leffler functions $E_{\alpha },\alpha \in (0,2]$, enjoy the $\mathrm{HCM}$ property, and then intervene deeply in the probabilistic context. We prove that for suitable $\alpha$ and complex numbers z, the real and imaginary part of the functions $x\mapsto E_{\alpha }\left(zx\right)$, are tightly linked to the stable distributions and to the generalized Cauchy kernel. Full article

The present study is concerned with studying the dynamical behavior of two space-dimensional nonlinear time-fractional models governing the unsteady-flow of polytropic-gas (in brief, pGas) that occurred in cosmology and astronomy. For this purpose, two efficient hybrid methods so-called optimal homotopy analysis $\mathbb{J}$-transform method (${}_O$HA$\mathbb{J}$TM) and $\mathbb{J}$-variational iteration transform method ($\mathbb{J}$-VITM) have been adopted. The _OHA$\mathbb{J}$TM is the hybrid method, where optimal-homotopy analysis method (${}_O$HAM) is utilized after implementing the properties of $\mathbb{J}$-transform ($\mathbb{J}$T), and in $\mathbb{J}$-VITM is the $\mathbb{J}$-transform-based variational iteration method. Banach’s fixed point approach is adopted to analyze the convergence of these methods. It is demonstrated that $\mathbb{J}$-VITM is $\mathrm{T}$-stable, and the evaluated dynamics of pGas are described in terms of Mittag–Leffler functions. The proposed evaluation confirms that the implemented methods perform better for the referred model equation of pGas. In addition, for a given iteration, the proposed behavior via ${}_O$HA$\mathbb{J}$TM performs better in producing more accurate behavior in comparison to $\mathbb{J}$-VITM and the methods introduced recently. Full article

This paper discusses the robust stability and stabilization of polynomial fractional differential (PFD) systems with a Caputo derivative using the sum of squares. In addition, it presents a novel method of stability and stabilization for PFD systems. It demonstrates the feasibility of designing problems that cannot be represented in LMIs (linear matrix inequalities). First, sufficient conditions of stability are expressed for the PFD equation system. Based on the results, the fractional differential system is Mittag–Leffler stable when there is a polynomial function to satisfy the inequality conditions. These functions are obtained from the sum of the square (SOS) approach. The result presents a valuable method to select the Lyapunov function for the stability of PFD systems. Then, robust Mittag–Leffler stability conditions were able to demonstrate better convergence performance compared to asymptotic stabilization and a robust controller design for a PFD equation system with unknown system parameters, and design performance based on a polynomial state feedback controller for PFD-controlled systems. Finally, simulation results indicate the effectiveness of the proposed theorems. Full article

This paper proposes a new fractional Poisson process through a recursive fractional differential governing equation. Unlike the homogeneous Poison process, the Caputo derivative on the probability distribution of k jumps with respect to time is linked to all probability distribution functions of j jumps, where j is a non-negative integer less than or equal to k. The distribution functions of arrival times are derived, while the inter-arrival times are no longer independent and identically distributed. Further, this new fractional Poisson process can be interpreted as a homogeneous Poisson process whose natural time flow has been randomized, and the underlying time randomizing process has been studied. Finally, the conditional distribution of the kth order statistic from random number samples, counted by this fractional Poisson process, is also discussed. Full article

The article considers the fractional Poisson process as a mathematical model of deformation activity in a seismically active region. The dislocation approach is used to describe five modes of the deformation process. The change in modes is determined by the change in the intensity of the event stream, the regrouping of dislocations, and the change in and the appearance of stable connections between dislocations. Modeling of the change of deformation modes is carried out by changing three parameters of the proposed model. The background mode with independent events is described by a standard Poisson process. To describe variations from the background mode of seismic activity, when connections are formed between dislocations, the fractional Poisson process and the Mittag–Leffler function characterizing it are used. An approximation of the empirical cumulative distribution function of waiting time of the foreshocks obtained as a result of processing the seismic catalog data was carried out on the basis of the proposed model. It is shown that the model curves, with an appropriate choice of the Mittag–Leffler function’s parameters, gives results close to the experimental ones and can be allowed to characterize the deformation process in the seismically active region under consideration. Full article

This study investigates the problem of finite-time boundedness of a class of neural networks of Caputo fractional order with time delay and uncertain terms. New sufficient conditions are established by constructing suitable Lyapunov functionals to ensure that the addressed fractional-order uncertain neural networks are finite-time stable. Criteria for finite-time boundedness of the considered fractional-order uncertain models are also achieved. The obtained results are based on a newly developed property of Caputo fractional derivatives, properties of Mittag–Leffler functions and Laplace transforms. In addition, examples are developed to manifest the usefulness of our theoretical results. Full article

In this paper, we present a new result that allows for studying the global stability of the disease-free equilibrium point when the basic reproduction number is less than 1, in the fractional calculus context. The method only involves basic linear algebra and can be easily applied to study global asymptotic stability. After proving some auxiliary lemmas involving the Mittag–Leffler function, we present the main result of the paper. Under some assumptions, we prove that the disease-free equilibrium point of a fractional differential system is globally asymptotically stable. We then exemplify the procedure with some epidemiological models: a fractional-order SEIR model with classical incidence function, a fractional-order SIRS model with a general incidence function, and a fractional-order model for HIV/AIDS. Full article

A generalized multivariate problem due to V. M. Zolotarev is considered. Some related results on geometric random sums and (multivariate) geometric stable distributions are extended to a more general case of “anisotropic” random summation where sums of independent random vectors with multivariate random index having a special multivariate geometric distribution are considered. Anisotropic-geometric stable distributions are introduced. It is demonstrated that these distributions are coordinate-wise scale mixtures of elliptically contoured stable distributions with the Marshall–Olkin mixing distributions. The corresponding “anisotropic” analogs of multivariate Laplace, Linnik and Mittag–Leffler distributions are introduced. Some relations between these distributions are presented. Full article

In this paper, we consider a fractional-order eco-epidemic model based on the Rosenzweig–MacArthur predator–prey model. The model is derived by assuming that the prey may be infected by a disease. In order to take the memory effect into account, we apply two fractional differential operators, namely the Caputo fractional derivative (operator with power-law kernel) and the Atangana–Baleanu fractional derivative in the Caputo (ABC) sense (operator with Mittag–Leffler kernel). We take the same order of the fractional derivative in all equations for both senses to maintain the symmetry aspect. The existence and uniqueness of solutions of both eco-epidemic models (i.e., in the Caputo sense and in ABC sense) are established. Both models have the same equilibrium points, namely the trivial (origin) equilibrium point, the extinction of infected prey and predator point, the infected prey free point, the predator-free point and the co-existence point. For a model in the Caputo sense, we also show the non-negativity and boundedness of solution, perform the local and global stability analysis and establish the conditions for the existence of Hopf bifurcation. It is found that the trivial equilibrium point is a saddle point while other equilibrium points are conditionally asymptotically stable. The numerical simulations show that the solutions of the model in the Caputo sense strongly agree with analytical results. Furthermore, it is indicated numerically that the model in the ABC sense has quite similar dynamics as the model in the Caputo sense. The essential difference between the two models is the convergence rate to reach the stable equilibrium point. When a Hopf bifurcation occurs, the bifurcation points and the diameter of the limit cycles of both models are different. Moreover, we also observe a bistability phenomenon which disappears via Hopf bifurcation. Full article

We characterize the complete monotonicity of the Kilbas-Saigo function on the negative half-line. We also provide the exact asymptotics at $-\infty$, and uniform hyperbolic bounds are derived. The same questions are addressed for the classical Le Roy function. The main ingredient for the proof is a probabilistic representation of these functions in terms of the stable subordinator. Full article

In the paper, multivariate probability distributions are considered that are representable as scale mixtures of multivariate stable distributions. Multivariate analogs of the Mittag–Leffler distribution are introduced. Some properties of these distributions are discussed. The main focus is on the representations of the corresponding random vectors as products of independent random variables and vectors. In these products, relations are traced of the distributions of the involved terms with popular probability distributions. As examples of distributions of the class of scale mixtures of multivariate stable distributions, multivariate generalized Linnik distributions and multivariate generalized Mittag–Leffler distributions are considered in detail. Their relations with multivariate ‘ordinary’ Linnik distributions, multivariate normal, stable and Laplace laws as well as with univariate Mittag–Leffler and generalized Mittag–Leffler distributions are discussed. Limit theorems are proved presenting necessary and sufficient conditions for the convergence of the distributions of random sequences with independent random indices (including sums of a random number of random vectors and multivariate statistics constructed from samples with random sizes) to scale mixtures of multivariate elliptically contoured stable distributions. The property of scale-mixed multivariate elliptically contoured stable distributions to be both scale mixtures of a non-trivial multivariate stable distribution and a normal scale mixture is used to obtain necessary and sufficient conditions for the convergence of the distributions of random sums of random vectors with covariance matrices to the multivariate generalized Linnik distribution. Full article

We look at estimates for the Green’s function of time-fractional evolution equations of the form $D_{0+\ast }^{\nu }u=Lu$ , where $D_{0+\ast }^{\nu }$ is a Caputo-type time-fractional derivative, depending on a Lévy kernel $\nu$ with variable coefficients, which is comparable to $y^{-1-\beta }$ for $\beta \in (0,1)$ , and L is an operator acting on the spatial variable. First, we obtain global two-sided estimates for the Green’s function of $D_0^{\beta }u=Lu$ in the case that L is a second order elliptic operator in divergence form. Secondly, we obtain global upper bounds for the Green’s function of $D_0^{\beta }u=\mathrm{\Psi }(-i\nabla )u$ where $\mathrm{\Psi }$ is a pseudo-differential operator with constant coefficients that is homogeneous of order $\alpha$ . Thirdly, we obtain local two-sided estimates for the Green’s function of $D_0^{\beta }u=Lu$ where L is a more general non-degenerate second order elliptic operator. Finally we look at the case of stable-like operator, extending the second result from a constant coefficient to variable coefficients. In each case, we also estimate the spatial derivatives of the Green’s functions. To obtain these bounds we use a particular form of the Mittag-Leffler functions, which allow us to use directly known estimates for the Green’s functions associated with L and $\mathrm{\Psi }$ , as well as estimates for stable densities. These estimates then allow us to estimate the solutions to a wide class of problems of the form $D_0^{(\nu ,t)}u=Lu$ , where $D^{(\nu ,t)}$ is a Caputo-type operator with variable coefficients. Full article