Variable-order Fractional Derivative Research Articles

Review of approximation formulas for fractional derivatives of variable order

Fractional differential equations of variable order, depending on a time or space variable, are successfully used to study time and/or spatial dynamics. The purpose of this review is to consider the latest research and results related to the basic definitions and approximation formulas for fractional derivatives of variable order. The review begins with an examination of existing definitions based on various physical and practical knowledge. The following are formulas for discretizing fractional derivatives, since they play a key role in numerical modeling, providing an effective means of describing complex systems with long-term memory, heterogeneous parameters and non-local interactions. Their use not only improves the accuracy of models, but also facilitates the numerical solution of differential equations, which is essential for analyzing and predicting the behavior of real processes in various fields of science and engineering. This review is intended to help readers select appropriate definitions and discretization formulas to effectively solve specific physical and engineering problems.

Dynamic Effect of Green Building Concept for Sustainable Cities under Climate Change via Variable-Order Fractional Derivatives

Dynamic Effect of Green Building Concept for Sustainable Cities under Climate Change via Variable-Order Fractional Derivatives

The Ulam Stability of High-Order Variable-Order φ-Hilfer Fractional Implicit Integro-Differential Equations

This study investigates the initial value problem of high-order variable-order φ-Hilfer fractional implicit integro-differential equations. Due to the lack of the semigroup property in variable-order fractional integrals, solving these equations presents significant challenges. We introduce a novel approach that approximates variable-order fractional derivatives using a piecewise constant approximation method. This method facilitates an equivalent integral representation of the equations and establishes the Ulam stability criterion. In addition, we explore higher-order forms of fractional-order equations, thereby enriching the qualitative and stability results of their solutions.

A finite difference method for elliptic equations with the variable-order fractional derivative

A finite difference method for elliptic equations with the variable-order fractional derivative

Application of Newton’s polynomial interpolation scheme for variable order fractional derivative with power-law kernel

This paper, offers a new method for simulating variable-order fractional differential operators with numerous types of fractional derivatives, such as the Caputo derivative, the Caputo–Fabrizio derivative, the Atangana–Baleanu fractal and fractional derivative, and the Atangana–Baleanu Caputo derivative via power-law kernels. Modeling chaotical systems and nonlinear fractional differential equations can be accomplished with the utilization of variable-order differential operators. The computational structures are based on the fractional calculus and Newton’s polynomial interpolation. These methods are applied to different variable-order fractional derivatives for Wang–Sun, Rucklidge, and Rikitake systems. We illustrate this novel approach’s significance and effectiveness through numerical examples.

A study on variable-order delay fractional differential equations: existence, uniqueness, and numerical simulation via a predictor corrector algorithm

In this study, we adapted a predictor-corrector technique to simulate delay differential equations incorporating variable-order Caputo-type fractional derivatives. We addressed the existence and uniqueness of solutions for the studied models. Then, we presented numerical simulation of some delay differential equations with variable-order fractional derivatives to demonstrate the efficiency of the used technique. Various periodic and chaotic characteristics of the studied models are observed for some variable-orders from the performed graphical simulations. The used technique can be modified and extended to handle different classes of initial value problems which involve variable-order fractional derivatives.

On the variable-order fractional derivatives with respect to another function

AbstractIn this paper, we present various concepts concerning generalized fractional calculus, wherein the fractional order of operators is not constant, and the integral kernel depends on a function. We observe that in the case of variable order, the concepts are distinct, and we present relations between them. Formulas for approximating fractional derivatives are provided, involving only integer-order derivatives. Finally, we conclude the work with some simulations to exemplify the method.

Numerical study of variable order model arising in chemical processes using operational matrix and collocation method

This article introduces the fractional variable order (VO) Gray–Scott model using the notion of VO fractional derivative in the Caputo sense. An efficient numerical method has been designed based on the Vieta–Lucas polynomial and the spectral collocation method for solving this model. The designed technique converts the concerned model into a nonlinear algebraic system of equations, which can be solved by Newton’s iterative method. In this article, we have illustrated the convergence analysis of the approximation and shown that a high order of convergence can be achieved despite a smaller number of approximations. A few numerical results are presented in order to verify the reliability and accuracy of the demonstrated scheme. The results of absolute errors for the considered Gray–Scott model with its exact solution show that the technique is very suitable for finding the solutions to the said kind of complex physical problem.

On Numerical Simulations of Variable-Order Fractional Cable Equation Arising in Neuronal Dynamics

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 containing variable-order fractional operators have been extensively resorted for modeling such phenomena accurately. In this paper, we consider the two-dimensional fractional cable equation with the Caputo variable-order fractional derivative in the time direction, which is preferable for describing neuronal dynamics in biological systems. A point-wise scheme, namely, the Crank–Nicolson finite difference method, along with a group-wise scheme referred to as the explicit decoupled group method are proposed to solve the problem under consideration. The stability and convergence analyses of the numerical schemes are provided with complete details. To demonstrate the validity of the proposed methods, numerical simulations with results represented in tabular and graphical forms are given. A quantitative analysis based on the CPU timing, iteration counting, and maximum absolute error indicates that the explicit decoupled group method is more efficient than the Crank–Nicolson finite difference scheme for solving the variable-order fractional equation.

Impulsive Control of Variable Fractional-Order Multi-Agent Systems

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 order of the fractional derivative changes at update times. We study a case for which the exchanged information between agents occurs only at initially given update times. Two types of linear variable-order Caputo fractional models are studied. We consider both multi-agent systems without a leader and multi-agent systems with a leader. In the case of multi-agent systems without a leader, two types of models are studied. The main difference between the models is the fractional derivative describing the dynamics of agents. In the first one, a Caputo fractional derivative with respect to another function and with a continuous variable order is applied. In the second one, the applied fractional derivative changes its constant order at each update time. Mittag–Leffler stability via impulsive control is defined, and sufficient conditions are obtained. In the case of the presence of a leader in the multi-agent system, the dynamic of the agents is described by a Caputo fractional derivative with respect to an increasing function and with a constant order that changes at each update time. The leader-following consensus via impulsive control is defined, and sufficient conditions are derived. The theoretical results are illustrated with examples. We show with an example the leader’s influence on the consensus.

A Comprehensive Study of the Langevin Boundary Value Problems with Variable Order Fractional Derivatives

The authors investigate Langevin boundary value problems containing a variable order Caputo fractional derivative. After presenting the background for the study, the authors provide the definitions, theorems, and lemmas that are required for comprehending the manuscript. The existence of solutions is proved using Schauder’s fixed point theorem; the uniqueness of solutions is obtained by adding an additional hypothesis and applying Banach’s contraction principle. An example is provided to demonstrate the results.

The Stability of Solutions of the Variable-Order Fractional Optimal Control Model for the COVID-19 Epidemic in Discrete Time

This article introduces a discrete-time fractional variable order over a SEIQR model, incorporated for COVID-19. Initially, we establish the well-possedness of solution. Further, the disease-free and the endemic equilibrium points are determined. Moreover, the local asymptotic stability of the model is analyzed. We develop a novel discrete fractional optimal control problem tailored for COVID-19, utilizing a discrete mathematical model featuring a variable order fractional derivative. Finally, we validate the reliability of these analytical findings through numerical simulations and offer insights from a biological perspective.

Variable-Order Fractional Linear Systems with Distributed Delays—Existence, Uniqueness and Integral Representation of the Solutions

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 fractional systems, which is “almost” equivalent to the considered variable-order system in an appropriate sense. This approach allows us to replace the study of the initial problem (IP) for variable-order fractional systems with the study of an IP for these one-parameter families of constant-order fractional systems. We prove that the initial problem for the variable-order fractional system with a discontinuous initial function possesses a unique continuous solution on the half-axis when the function describing the variable order of differentiation is locally bounded, Lebesgue integrable and has an appropriate decomposition similar to the Lebesgue decomposition of functions with bounded variation. The obtained results lead to the existence and uniqueness of a fundamental matrix for the studied variable-order fractional homogeneous system. As an application of the obtained results, we establish an integral representation of the solutions of the studied IP.

A Convergent Legendre Spectral Collocation Method for the Variable-Order Fractional-Functional Optimal Control Problems

In this paper, a numerical method is applied to approximate the solution of variable-order fractional-functional optimal control problems. The variable-order fractional derivative is described in the type III Caputo sense. The technique of approximating the optimal solution of the problem using Lagrange interpolating polynomials is employed by utilizing the shifted Legendre–Gauss–Lobatto collocation points. To obtain the coefficients of these interpolating polynomials, the problem is transformed into a nonlinear programming problem. The proposed method offers a significant advantage in that it does not require the approximation of singular integral. In addition, the matrix differentiation is calculated accurately and efficiently, overcoming the difficulties posed by variable-order fractional derivatives. The convergence of the proposed method is investigated, and to validate the effectiveness of our proposed method, some examples are presented. We achieved an excellent agreement between numerical and exact solutions for different variable orders, indicating our method’s good performance.

Qualitative Analysis of Fractional Stochastic Differential Equations with Variable Order Fractional Derivative

Qualitative Analysis of Fractional Stochastic Differential Equations with Variable Order Fractional Derivative

STUDY ON THE INITIAL BOUNDARY VALUE PROBLEM FOR AFRACTIONAL DIFFERENTIAL EQUATION WITH A FRACTIONALDERIVATIVE OF VARIABLE ORDER

This article studies the convergence of a numerical method for solving an initial-boundary value problem of a fractional differential equation with a variable order of the fractional derivative. In the generalized fractional differential filtration equation with a transitional filtration law in heterogeneous porous media, it is assumed that the order of the fractional derivative depends on the spatial variable. The main attention is paid to the development and theoretical justification of a method that provides high accuracy and efficiency of calculations with a variable order of the fractional derivative. For the numerical solution, an approximation was developed that combines the finite difference method for the time derivative and the finite element method for the spatial variable. The fractional derivative of variable order in the sense of Caputo is approximated by a formula of second order in time. The convergence of the constructed method is proven with order O(τ2+hk+1) for the case α(x)ϵ(0,1). The results of computational experiments for various functions of the order of the fractional derivative are presented, confirming the reliability of the theoretical analysis. The conclusions drawn emphasize the importance and relevance of the further development of numerical methods for fractional differential equations of variable order in modern mathematics and applied sciences, including the modeling of complex processes.

A pseudo-spectral approach for optimal control problems of variable-order fractional integro-differential equations

<p>Nonlinear optimal control problems governed by variable-order fractional integro-differential equations constitute an important subgroup of optimal control problems. This group of problems is often difficult or impossible to solve analytically because of the variable-order fractional derivatives and fractional integrals. In this article, we utilized the expansion of Lagrange polynomials in terms of Chebyshev polynomials and the power series of Chebyshev polynomials to find an approximate solution with high accuracy. Subsequently, by employing collocation points, the problem was transformed into a nonlinear programming problem. In addition, variable-order fractional derivatives in the Caputo sense were represented by a new operational matrix, and an operational matrix represented fractional integrals. As a result, the mentioned integro-differential optimal control problem becomes a nonlinear programming problem that can be easily solved with the repetitive optimization method. In the end, the proposed method is illustrated by numerical examples that demonstrate its efficiency and accuracy.</p>

α-FRACTAL FUNCTION WITH VARIABLE PARAMETERS: AN EXPLICIT REPRESENTATION

In this paper, new results on the [Formula: see text]-fractal function with variable parameters are presented. The Weyl–Marchaud variable order fractional derivative of an [Formula: see text]-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 [Formula: see text]-fractal function with variable parameters is evaluated for various intervals in the prescribed domain. Finally, an explicit structure for the [Formula: see text]-fractal function is provided using the base [Formula: see text] representation of numbers.

Analytical solutions for a class of variable-order fractional Liu system under time-dependent variable coefficients

In this article, we present a nonlinear model of the Liu system that includes fractional derivatives of variable-order. Due to the nonlocality of the dynamical system, we introduce the fractional derivative with power laws, exponential decay laws, and generalized Mittag-Leffler functions as kernels. We provide a detailed analysis of the existence and uniqueness of the proposed model, as well as the stability of these equations. Due to the existence of time-varying fractional derivatives, the proposed variable-order fractional system exhibits more complex characteristics and more degrees of freedom than an integer or conventional constant fractional-order chaotic Liu oscillator. Different chaotic behaviors can be obtained by using different smooth functions defined within the interval (0,1] as variable orders for fractional derivatives in the simulations. Furthermore, simulations demonstrate that fractional chaotic systems with variable orders can be synchronized.

Numerical Treatments for Crossover Cancer Model of Hybrid Variable-Order Fractional Derivatives

Numerical Treatments for Crossover Cancer Model of Hybrid Variable-Order Fractional Derivatives