3,428 Results Found

Active Galactic Nuclei emit radiation over the whole electromagnetic spectrum up to TeV energies. Blazars are one subtype with their jets pointing towards the observer. One of their typical features is extreme variability on timescales, from minutes...

In this study, the variable order fractional calculus of the hidden variable fractal interpolation function is explored. It extends the constant order fractional calculus to the case of variable order. The Riemann–Liouville and the Weyl–M...

Malware software has become a pervasive threat in computer and mobile technology attacks. Attackers use this software to obtain information about users of the digital world to obtain benefits by hijacking their data. Antivirus software has been devel...

This article presents a method for the approximate calculation of fractional Caputo derivatives, including a crucial aspect of the ability to handle arbitrary—even variable—terminals and order. The proposed method involves rearranging the...

This article describes an approximation technique based on fractional order Bernstein wavelets for the numerical simulations of fractional oscillation equations under variable order, and the fractional order Bernstein wavelets are derived by means of...

In this paper, we investigate properties of solutions to a space-time fractional variable-order conformable nonlinear differential equation with a generalized tempered fractional Laplace operatorby using the maximum principle. We first establish some...

Given the recent advances regarding the studies of discrete fractional calculus, and the fact that the dynamics of discrete-time neural networks in fractional variable-order cases have not been sufficiently documented, herein, we consider a novel cla...

In this research paper, we solve the problem of synchronization and anti-synchronization of chaotic systems described by discrete and time-delayed variable fractional-order differential equations. To guarantee the synchronization and anti-synchroniza...

Modeling of wall-bounded turbulent flows is still an open problem in classical physics, with relatively slow progress in the last few decades beyond the log law, which only describes the intermediate region in wall-bounded turbulence, i.e., 30–50 y+...

The aim of this paper is to establish the existence and uniqueness results for differential equations of Hilfer-type fractional order with variable coefficient. Firstly, we establish the equivalent Volterra integral equation to an initial value probl...

Few papers have been published to date regarding the stability of neural networks described by fractional difference operators. This paper makes a contribution to the topic by presenting a variable-order fractional discrete neural network model and b...

We introduce a new numerical method, based on Bernoulli polynomials, for solving multiterm variable-order fractional differential equations. The variable-order fractional derivative was considered in the Caputo sense, while the Riemann–Liouvill...

This study investigates a variety of novel estimations involving the expectation, variance, and moment functions of continuous random variables by applying a generalized proportional fractional integral operator. Additionally, a continuous random var...

The fractional gradient method has garnered significant attention from researchers. The common view regarding fractional-order gradient methods is that they have a faster convergence rate compared to classical gradient methods. However, through condu...

The main goal of the paper is to present and study models of multi-agent systems for which the dynamics of the agents are described by a Caputo fractional derivative of variable order and a kernel that depends on an increasing function. Also, the ord...

A time-fractional substantial diffusion equation of variable order is investigated, in which the variable-order fractional substantial derivative accommodates the memory effects and the structure change of the surroundings of the physical processes w...

The current paper presents a novel numerical technique to handle variable-order multiterm fractional differential equations (VO-MTFDEs) supplemented with initial conditions (ICs) by introducing generalized fractional Jacobi functions (GFJFs). These G...

This paper presents a study of the application of the finite element method for solving a fractional differential filtration problem in heterogeneous fractured porous media with variable orders of fractional derivatives. A numerical method for the in...

We propose a fractional-order shifted Jacobi–Gauss collocation method for variable-order fractional integro-differential equations with weakly singular kernel (VO-FIDE-WSK) subject to initial conditions. Using the Riemann–Liouville fracti...

In the field of nonlinear mathematical physics, Ablowitz et al.’s algorithm has recently made significant progress in the inverse scattering transform (IST) of fractional-order nonlinear evolution equations (fNLEEs). However, the solved fNLEEs...

The paper deals with the problem of representing special functions by branched continued fractions, particularly multidimensional A- and J-fractions with independent variables, which are generalizations of associated continued fractions and Jacobi co...

An initial value problem for nonlinear fractional differential equations with a tempered Caputo fractional derivative of variable order with respect to another function is studied. The absence of semigroup properties of the considered variable order...

In this work, we study a general class of retarded linear systems with distributed delays and variable-order fractional derivatives of Caputo type. We propose an approach consisting of finding an associated one-parameter family of constant-order frac...

We use classical Galerkin approximations, the generalized Aubin–Lions Lemma as well as the Bellman–Gronwall Lemma to study the asymptotical behavior of a two-dimensional fractional Navier–Stokes equation with variable delay. By modi...

A five-dimensional hyperchaotic system is a dynamical system with five state variables that exhibits chaotic behavior in multiple directions. In this work, we incorporated a 5D hyperchaotic system with constant- and variable-order Caputo and the Capu...

An initial value problem for a scalar nonlinear differential equation with a variable order for the generalized proportional Caputo fractional derivative is studied. We consider the case of a piecewise constant variable order of the fractional deriva...

Variable-order fractional discrete calculus is a new and unexplored part of calculus that provides extraordinary capabilities for simulating multidisciplinary processes. Recognizing this incredible potential, the scientific community has been researc...

Variability in reclaimed asphalt pavement (RAP) properties, such as aggregate gradation, asphalt content, and moisture content, poses a significant challenge to producing consistent and reliable recycled asphalt mixtures. This study systematically ev...

This paper is concerned with the adaptive backstepping synchronization control for a class of variable-order (VO) fractional uncertain nonlinear system with external disturbances and a dead zone. A kind of VO fractional command filter is employed to...

In the theory of differential equations, the study of existence and the uniqueness of the solutions are important. In the last few decades, many researchers have had a keen interest in finding the existence–uniqueness solution of constant fract...

In this manuscript we introduce a quadratic integral equation of the Urysohn type of fractional variable order. The existence and uniqueness of solutions of the proposed fractional model are studied by transforming it into an integral equation of fra...

In this paper, we present a more general approach based on a Picard integral scheme for nonlinear partial differential equations with a variable time-fractional derivative of order $\alpha (\mathbf{x},t)\in (1,2)$ and space-fractional order $s\in (0,1)$, wher...

Macroeconomic mathematical models are practical instruments structured to carry out theoretical analyses of macroeconomic developments. In this manuscript, the Caputo-like fractional operator of variable order is used to introduce and investigate the...

This paper presents a new procedure for designing a fractional order unknown input observer (FOUIO) for nonlinear systems represented by a fractional-order Takagi–Sugeno (FOTS) model with unmeasurable premise variables (UPV). Most of the curren...

In this work, analytic solutions of initial value problems for fractional Langevin equations involving Hilfer fractional derivatives and variable coefficients are studied. Firstly, the equivalence of an initial value problem to an integral equation i...

We propose two efficient numerical approaches for solving variable-order fractional optimal control-affine problems. The variable-order fractional derivative is considered in the Caputo sense, which together with the Riemann–Liouville integral...

The variable-order (VO) fractional calculus can be seen as a natural extension of the constant-order, which can be utilized in physical and biological applications. In this study, we derive a new numerical approximation for the VO fractional Riemann&...

Using Mellin-Barnes contour integrals, we aim at suggesting a q-analogue (q-extension) of the several variable Aleph-function. Then we present Riemann Liouville fractional q-integral and q-differential formulae for the q-extended several variable Ale...

In this paper, we propose the extended Boussinesq–Whitham–Broer–Kaup (BWBK)-type equations with variable coefficients and fractional order. We consider the fractional BWBK equations, the fractional Whitham–Broer–Kaup (WBK) equations and the fractiona...

In recent years, various complex systems and real-world phenomena have been shown to include memory and hereditary properties that change with respect to time, space, or other variables. Consequently, fractional partial differential equations contain...

The dynamics of the Caputo-fractional variable-order difference form of the Tinkerbell map are studied. The phase portraits, bifurcation, and largest Lyapunov exponent (LLE) were employed to demonstrate the presence of chaos over a different fraction...

Significant progress has been made in incorporating fractional calculus into the projection and lag synchronization of complex networks. However, real-world networks are highly complex, making the fractional derivative used in complex dynamics more s...

A variable-order time-fractional Kelvin peridynamics model is proposed, where the variable order is utilized to reflect the changes of viscosity in viscoelastic materials to effectively capture the damage and deformation of rock steady creep. The cor...

The thermistor model captures the complex interaction between heat dissipation, electrical current conduction, and Joule heat generation. Our research examines the diverse properties and implications of employing fractional calculus in the analysis w...

This study establishes the existence and stability of solutions for a general class of Riemann–Liouville (RL) fractional differential equations (FDEs) with a variable order and finite delay. Our findings are confirmed by the fixed-point theorem...

A class of boundary value problems for fractional differential systems involving variable-order derivatives is considered. Such problems can be transformed into some boundary value problems for nonlinear Caputo fractional differential systems. Here,...

Fractional calculus generalizes well-known differentiation and integration to noninteger orders, allowing a more accurate framework for modeling complex dynamical behaviors. The application of fractional-order systems is quite wide in engineering, bi...

Building an accurate constitutive model for soft materials is essential for better understanding its rate-dependent deformation characteristics and improving the design of soft material devices. To establish a concise constitutive model with few para...

This paper is devoted to solving boundary value problems for differential equations with fractional derivatives by the Fourier method. The necessary information is given (in particular, theorems on the completeness of the eigenfunctions and associate...

Viscoelasticity and variable mass are common phenomena in Micro-Electro-Mechanical Systems (MEMS), and could be described by a fractional derivative damping and a stochastic process, respectively. To study the dynamic influence cased by the viscoelas...