A variable fractional-order (VFO) fuzzy sliding mode controller is designed to control the speed of a permanent magnet synchronous motor (PMSM). First, a VFO sliding mode surface is established. Then, a VFO fuzzy sliding mode controller is designed,...
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...
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 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...
This work extends the analytical operation of the Riemann–RHPHilbert approach (RHA) for fractional-order nonlinear integrable systems under the solvable meaning of inverse scattering transform (IST) to variable-coefficient fractional-order nonl...
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...
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...
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...
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...
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
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...
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...
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...
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...
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...
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...
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...
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...
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
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 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, 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...
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...
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...
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 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 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...
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...
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 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...
Commutators of fractional integral operators play an important role in harmonic analysis due to their close connections with function regularity and partial differential equations. In this paper, we study higher-order commutators of fractional integr...
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 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...
This paper investigates a class of Hadamard variable-order fractional differential equations in which the fractional order is determined by the cumulative distribution function (CDF) of a continuous random variable. The proposed framework establishes...
The manuscript contains the new scientific results of theoretical modeling of longitudinal flow dynamics (creeping) of a viscoelastic fluid governed by a fractional-type rheological Maxwell model material within a pipe of variable cross-section. Stru...
In this article, we develop a compact finite difference scheme for a variable-order-time fractional-sub-diffusion equation of a fourth-order derivative term via order reduction. The proposed scheme exhibits fourth-order convergence in space and secon...
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 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...
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, 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...
In order to quantitatively describe the time-varying mechanical properties of asphalt mixture during creep process, a nonlinear viscoelastoplastic creep model was proposed, by using variable-order fractional calculus. The differential order of the va...
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...
We investigate chaos control and hybrid synchronization in a variable-order fractional Arneodo system by treating the differentiation order
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...
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...
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...
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...
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...
This paper introduces and explores the dynamics of a novel three-dimensional (3D) fractional map with hidden dynamics. The map is constructed through the integration of a discrete sinusoidal memristive into a discrete Duffing map. Moreover, a mathema...
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