14,200 Results Found

General variable-order fractional scale derivatives are introduced and studied. Both the stretching and the shrinking cases are considered for definitions of the derivatives of the GL type and of the Hadamard type. Their properties are deduced and di...

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...

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...

This work explores the growing field of fractional calculus, with particular emphasis on the complexities and opportunities associated with variable-order derivatives. We critically assess existing definitions, identifying those that are consistent w...

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...

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...

This study investigates the initial value problem of high-order variable-order $\phi$-Hilfer fractional implicit integro-differential equations. Due to the lack of the semigroup property in variable-order fractional integrals, solving these equations...

In this paper, a hybrid variable-order mathematical model for multi-vaccination COVID-19 is analyzed. The hybrid variable-order derivative is defined as a linear combination of the variable-order integral of Riemann–Liouville and the variable-order C...

We study the variable-order fractional abstract Cauchy problem ${}^S{D}_0^{\alpha \left(t\right)}u\left(t\right)=Au\left(t\right)+f\left(t\right)$, $u\left(0\right)=u_0$, in a Banach space X, where A is sectorial and $\alpha \left(t\right):[0,\infty )\to (0,1)$ is a Scarpi transition function. Using the Sonine-pair structure of...

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...

This study presents an algorithmically efficient approach to address the complexities associated with nonlocal variable-order operators characterized by diverse definitions. The proposed method employs integro spline quasi interpolation to approximat...

The analysis of the complexity of dynamical systems generally presents a major dilemma between including the complexity of memory effects and being computationally efficient. For this purpose, this study proposes an unusual three-dimensional variable...

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...

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&...

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 investigate chaos control and hybrid synchronization in a variable-order fractional Arneodo system by treating the differentiation order $\alpha \left(t\right)$ as a closed-loop control variable. A hybrid chaos indicator, combining a tracking error with a wind...

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...

In this paper, a novel four-dimensional variable-order fractional chaotic system is presented through the use of the Liouville–Caputo variable-order fractional derivative to represent the time-varying memory properties of nonlinear dynamical be...

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...

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...

In this study, we develop a novel variable-order fractional extension of an eight-dimensional (8D) hyperchaotic differential equation system modeled via the Liouville–Caputo operator. Moving beyond constant fractional-order models, our system 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...

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...

This paper is concerned with the problem of tracking control for a class of variable-order fractional uncertain system. In order to realize the global robustness of systems, two types of controllers are designed by the global sliding-mode control met...

To accurately describe the deformation characteristics of clay under long-term cyclic load, based on fractional calculus theory, elastoplastic theory and the basic element model, a variable-order fractional dynamic model designed to predict accumulat...

In this paper, we investigate the stability analysis for a class of novel variable-order Caputo fractional-order dual switching systems. First, the short memory principle is adopted to construct the studied system model, where the Caputo fractional o...

Although the literature presents promising techniques for the control of integer-order systems, control and synchronizing fractional systems still need further improvement to ensure their robustness and convergence time. This study aims to address th...

In this paper, a finite difference scheme is proposed for the variable-order time-fractional sub-diffusion equation, achieving second-order accuracy in time and sixth-order accuracy in space. For spatial discretization, a newly constructed operator $\mathcal{A}$...

In this paper, we present a new highly efficient numerical algorithm for nonlinear variable-order space fractional reaction–diffusion equations. The algorithm is based on a new method developed by using the Gaussian quadrature pole rational app...

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...

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...

In this paper, the cluster synchronization of a variable-order fractional community network with nonidentical dynamics is investigated. For achieving the cluster synchronization, intermittent controllers are designed, and the sufficient conditions wi...

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 this work, a new 4D chaotic system is presented with Liouville–Caputo fractional derivatives, including constant-order (C) and variable-order (V). In this regard, the paper examines the equilibrium of the system, local stability, and dissipa...

In this study, we analyze the solution characteristics and dynamics of a variable-order fractional (V-OF) Arneodo system using the Liouville–Caputo fractional operator with variable order. The V-OF operator is used to describe the time-dependen...

In this paper, a class of variable-order fractional interval systems (VO-FIS) in which the system matrices are affected by the fractional order is investigated. Firstly, the sufficient conditions for robust stability of a VO-FIS with a unified order...

This paper studies a nonlinear Schrödinger equation with a variable-order time-fractional derivative. Because classical L1 and L1-2 schemes are not directly applicable to variable-order fractional operators, an improved L1-2 discretization with...

Fractional-order chaotic systems have received increasing attention over the past few years due to their ability to effectively model memory and complexity in nonlinear dynamics. Nonetheless, most of the research conducted so far has been on constant...

This paper develops and rigorously analyzes a fully discrete upwind interior penalty discontinuous Galerkin (IPDG) scheme for simulating solute transport in two-dimensional variable-order fractional mobile–immobile media. The temporal variable-...

Sintered nano-silver paste is widely used in electronic packaging due to its excellent thermal and electrical conductivity. A phenomenological variable-order fractional constitutive model has been developed to characterize the evolution of its mechan...

This article introduces a new numerical algorithm dedicated to solving the most general form of variable-order fractional partial differential models. Both the time and spatial order of derivatives are considered as non-constant values. A combination...

This paper introduces a novel memristor-based hyperchaotic system in which the integer-order derivatives are replaced by a variable-order fractal-fractional operator. The dynamical properties of the system, including equilibrium points, Lyapunov expo...

This study explores the variable-order fractional Nosé–Hoover system, investigating the evolution of its chaotic and stable states under variable-order derivatives. Variable-order derivatives introduce greater complexity and adaptability...

Human motivation is governed by a long-memory cognitive process in which the depth of temporal integration—how far into the past the system draws upon accumulated experience—is not fixed, but dynamically compressed under cognitive stress....

A numerical technique was developed for solving nonlocal nonlinear stochastic delayed differential equations driven by fractional variable-order Brownian noise. Error analysis of the proposed technique was performed and discussed. The method was appl...

The quadratic Lyapunov inequality of Aguila-Camacho, Duarte-Mermoud and Gallegos has been extended operator by operator—most recently to the generalised Hattaf mixed operator, which subsumes the Atangana–Baleanu, Hadamard, generalised pro...

In this article, we first obtain an embedding result for the Sobolev spaces with variable-order, and then we consider the following Schrödinger–Kirchhoff type equations ${\left(a+b{\int }_{\mathsf{\Omega }\times \mathsf{\Omega }}\frac{{|\xi \left(x\right)-\xi \left(y\right)|}^{{}^p}}{{|x-y|}^{N+ps(x,y)}}dxdy\right)}^{p-1}{(-\Delta )}_p^{s(·)}\xi +\lambda V\left(x\right){\left|\xi \right|}^{p-2}\xi =f(x,\xi ),x\in \mathsf{\Omega },\xi =0,x\in$...

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,...

This paper is concerned with the numerical solution of the variable-order time fractional advection–diffusion–reaction equation (VO-TFADRE) in two space dimensions. We first propose a Crank–Nicolson (C-N) discretization scheme based...

This manuscript is devoted to using Bernoulli polynomials to establish a new spectral method for computing the approximate solutions of initial and boundary value problems of variable-order fractional differential equations. With the help of the afor...