Variable-order Fractional Derivative Research Articles

Variable–Order Model of Cardiac Fibrillation

Atrial fibrillation is a cardiac disorder marked by rapid and disorganized electrical activity, leading to atrial mechanical dysfunction. The alterations in electrophysiological properties during this arrhythmia are not solely attributed to electrical remodeling, structural changes in atrial tissue are also involved. This work aims to formulate a mathematical model for fibrillatory electrical conduction through the implementation of variable-order fractional derivatives. The adoption of such an operator is intended to represent the process of structural remodeling, which has been related to the course of atrial fibrillation. Simulations are performed using a simplified model of the ionic kinetics of the cardiac cell membrane, which allows for distinct electrophysiological properties through its parameterization. For the variable order, a fluctuating function is adopted that can be interpreted as the progression of structural remodeling when the order decreases, and the reverse process when the order increases. Fibrillatory propagation is initiated by generating reentrant conduction, also known as a rotor. We observed that, on the one hand, electrical conduction becomes chaotic as the fractional order decreases, and persists under such dynamics while the order increases. On the other hand, rotational activity persists during the fluctuation of the fractional order. Such mesoscopic outcomes depend on the sensitivity of the microscopic electrophysiological properties to the variations of the fractional order. Moreover, both propagation patterns can be associated with the known course of clinical AF. These results suggest that the fractional order model of cardiac electrophysiology may provide insight into the AF underlying mechanisms.

Ulam-Type Stability Results for Variable Order Ψ-Tempered Caputo Fractional Differential Equations

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 fractional derivative leads to difficulties in the study of the existence of corresponding differential equations. In this paper, we introduce approximate piecewise constant approximation of the variable order of the considered fractional derivative and approximate solutions of the given initial value problem. Then, we investigate the existence and the Ulam-type stability of the approximate solution of the variable order Ψ-tempered Caputo fractional differential equation. As a partial case of our results, we obtain results for Ulam-type stability for differential equations with a piecewise constant order of the Ψ-tempered Caputo fractional derivative.

Enhanced shifted Jacobi operational matrices of derivatives: spectral algorithm for solving multiterm variable-order fractional differential equations

This paper presents a new way to solve numerically multiterm variable-order fractional differential equations (MTVOFDEs) with initial conditions by using a class of modified shifted Jacobi polynomials (MSJPs). As their defining feature, MSJPs satisfy the given initial conditions. A key aspect of our methodology involves the construction of operational matrices (OMs) for ordinary derivatives (ODs) and variable-order fractional derivatives (VOFDs) of MSJPs and the application of the spectral collocation method (SCM). These constructions enable efficient and accurate numerical computation. We establish the error analysis and the convergence of the proposed algorithm, providing theoretical guarantees for its effectiveness. To demonstrate the applicability and accuracy of our method, we present five numerical examples. Through these examples, we compare the results obtained with other published results, confirming the superiority of our method in terms of accuracy and efficiency. The suggested algorithm yields very accurate agreement between the approximate and exact solutions, which are shown in tables and graphs.

The application of a novel variable-order fractional calculus on rheological model for viscoelastic materials

Viscoelastic materials have become popular for novel multifunctional materials in emerging modern technologies, which have complex elastic-viscous-plastic behaviors. The variable-order fractional derivative has been gradually applied to characterize such variable memory effects of complex physical systems, while existing definitions are somewhat controversial. The new Scarpi’s variable-order fractional calculus has rigorous mathematical logic compared with the existed definitions. However, its practicality and feasibility in engineering are ambiguous due to the fewer applications up to now. In this paper, we aimed at introducing the Scarpi’s variable-order fractional derivative into the rheological applications of viscoelastic materials. Based on the Scarpi’s variable-order fractional derivative, the fractional Zener model and fractional viscoelastic-visoplastic models are established, and the parameters sensitivity on stress-strain relation is discussed. Then, the proposed models are then applied to depict different rheological behaviors of polymers at various temperatures and strain rates, and the results show that they have excellent agreement for experimental data compared to the reference model. It is observed that the order increases linearly with strain of the viscoelastic model and decreases exponentially with strain of the viscoplastic model. Furthermore, the physical significance of the corresponding fractional order parameters is investigated. It is found that they have the variation relation with temperatures and strain rates. It reveals that the variable fractional order can be regarded as an index to characterize the evolution of mechanical behaviors of polymers.

Study of fractional variable order COVID-19 environmental transformation model

Abstract In this study, we explore the epidemic spread of the coronavirus using the Caputo fractional variable order derivative as variable order derivative provides a natural extension to classical as well as fractional order derivatives. Using the variable order derivatives in investigation of biological models of infectious diseases is an important area of research in the current time. Using the fixed point technique, we discuss the existence and uniqueness of solution to the corona virus infectious disease 2019 environmental transformation model. In order to demonstrate the existence and novelty of our findings, we examine the results numerically and graphically with the help of Euler’s method. There are several graphs provided that are related to different variable orders.

Variable-Order Fractional Scale Calculus

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 discussed. Fractional variable-order systems of autoregressive–moving-average type are introduced and exemplified. The corresponding transfer functions are obtained and used to find the corresponding impulse responses.

Existence and Uniqueness Results for a Pantograph Boundary Value Problem Involving a Variable-Order Hadamard Fractional Derivative

This paper discusses the problem of the existence and uniqueness of solutions to the boundary value problem for the nonlinear fractional-order pantograph equation, using the fractional derivative of variable order of Hadamard type. The main results are proved through the application of fractional calculus and Krasnoselskii’s fixed-point theorem. Moreover, the Ulam–Hyers–Rassias stability of the nonlinear fractional pantograph equation is analyzed. To conclude this paper, we provide an example illustrating our findings and approach.

Existence and Uniqueness of Variable-Order φ-Caputo Fractional Two-Point Nonlinear Boundary Value Problem in Banach Algebra

Using variable-order fractional derivatives in differential equations is essential. It enables more precise modeling of complex phenomena with varying memory and long-range dependencies, improving our ability to describe real-world processes reliably. This study investigates the properties of solutions for a two-point boundary value problem associated with φ-Caputo fractional derivatives of variable order. The primary objectives are to establish the existence and uniqueness of solutions, as well as explore their stability through the Ulam-Hyers concept. To achieve these goals, Banach’s and Krasnoselskii’s fixed point theorems are employed as powerful mathematical tools. Additionally, we provide numerical examples to illustrate results and enhance comprehension of theoretical findings. This comprehensive analysis significantly advances our understanding of variable-order fractional differential equations, providing a strong foundation for future research. Future directions include exploring more complex boundary value problems, studying the effects of varying fractional differentiation orders, extending the analysis to systems of equations, and applying these findings to real-world scenarios, all of which promise to deepen our understanding of Caputo fractional differential equations with variable order, driving progress in both theoretical and applied mathematics.

The exact solutions of the variable-order fractional stochastic Ginzburg–Landau equation along with analysis of bifurcation and chaotic behaviors

This article explores the exact solutions of the variable order fractional derivative of the stochastic Ginzburg–Landau equation (GLE) using the G′G2-expansion method with the assistance of Matlab R2021a software. The paper presents three key aspects that contribute to its novelty: (1) Our study introduces and examines the variable order fractional derivative of the stochastic Ginzburg–Landau equation (VOFDSGLE) for the first time, demonstrating our best cognitive effort. We successfully obtain a significant number of exact solutions and provide illustrative examples and visual representations of the VOFDSGLE. These obtained solutions have the potential to offer enhanced availability and practicality in understanding the mechanisms behind the complex physical phenomena observed in various fields. (2) Additionally, we investigate the phase portraits of the variable-order fractional stochastic Ginzburg–Landau equation under a specific condition. We analyze the associated sensitivity and chaotic behaviors, which have not been examined in prior studies on stochastic Ginzburg–Landau equation (Mohammed et al., 2021; Alhojilan and Ahmed, 2023). This exploration adds a unique contribution to the existing literature and offers valuable insights into the dynamics of the system. (3) Our numerical observations underscore the significant impact of multiplicative noise on the solutions. The progressive degradation of patterns and the tendency towards planar surfaces indicate the disruption caused by increasing noise intensity. Moreover, the convergence of solution magnitudes to zero underlines the stabilizing influence of the multiplicative noise. Furthermore, we advance the existing knowledge regarding the response of the stochastic GLE model to small noise, offering valuable insights into the system’s transition behaviors in the energy landscape.

Genocchi polynomials for variable-order time fractional Fornberg–Whitham type equations

In this research, a kind of non-singular variable-order fractional derivative is utilized to define two types of variable-order fractional Fornberg–Whitham equations. The shifted Genocchi polynomials are employed to construct two operational matrix approaches for solving these equations. More precisely, a new variable-order fractional derivative matrix is obtained for these polynomials, and used with the collocation technique for solving these equations. In fact, using this approaches, solving such problems converts into solving simple systems of algebraic equations. Some examples are solved to illustrate the applicability and accuracy of the proposed algorithms.

A fully spectral tau method for a class of linear and nonlinear variable-order time-fractional partial differential equations in multi-dimensions

We present a linear and nonlinear comprehensive class of variable-order (VO) time-fractional partial differential equations in multi-dimensions with Caputo VO fractional operator. This class comprises, as special cases, several linear and nonlinear equations of important applications in diverse fields, such as, the VO time-fractional telegraph equation, the VO time-fractional diffusion-wave equation and the VO time-fractional Klein–Gordon equation. A novel and accurate spectral tau method based on the shifted Gegenbauer polynomials (SGPs) is presented for the numerical treatment of the aforementioned class of equations. Actually, applying the tau method for solving this class of equations is very difficult, especially in the presence of the nonlinear term of these equations and the nature of the VO derivatives. To overcome this difficulty, new operational matrices for the VO fractional derivative and the approximation of the multiplication of the space–time Kronecker product vectors in multi-dimension are derived. These new matrices have the ability to remove the aforementioned difficulties that facing for applying the tau method. Several linear and nonlinear numerical examples are examined and compared in multi-dimensions with other methods to verify the validity, applicability and accuracy of the proposed methodology.

Mixed Poisson-Gaussian noise reduction using a time-space fractional differential equations

Images are frequently corrupted by various sorts of mixed or unrecognized noise, including mixed Poisson-Gaussian noise, rather than just a single kind of noise. In this work, we propose a time-space fractional differential equation to remove mixed Poisson-Gaussian noise. Combining fixed- and variable-order fractional derivatives allows us to maintain an image's high- and low-frequency components while eliminating noise. The current model, although primarily intended for mixed noise reduction, can indeed be utilized with great efficacy on images that have been solely degraded by Gaussian noise. In addition to this, a stable discretization strategy is presented. The illustrative results demonstrate that our scheme performs better than earlier models, reduces the staircase effect, and is applicable to electron microscopy and CT images.

A mathematical model for precise predicting microbial propagation based on solving variable‐order fractional diffusion equation

Continued mortality from Covid‐19 disease and environmental considerations resulting from the cemetery, landfill, wastewater treatment plants, and mass personal protective equipment (PPE) disposal sites demonstrate the importance of carefully studying how microorganisms spread in and across the soil, surface water, and groundwater. In this research, fractional diffusion equation for different functions has been solved using fractional derivative form with variable‐order (VO) to predict the diffusion of the SARS‐CoV‐2 virus. Comparison of the obtained results with the available experimental data and mathematical calculations shows that the use of VO form can predict the viral behavior more accurately. The main advantage of considering a VO fractional derivatives instead of a constant‐order fractional derivative or classical derivative is accurately predicting the behavior of propagation of microorganisms over large time and dimensional scales. In addition, to the specified SARS‐CoV‐2 virus, the method presented in this study will be able to investigate the diffusion pattern of all environmental pollutants at the nanoscale/microscale, including carcinogenic compounds, such as biogenic amines in aqueous and porous media. The obtained results manifest a high accuracy of mathematical approach which provides helpful information to urban and health planners/managers worldwide to manage other possible outbreaks.

Fast numerical scheme for the time-fractional option pricing model with asset-price-dependent variable order

We provide a fast numerical technique for a time-fractional option pricing model with asset-price-dependent variable order. Due to the complicated variable-order fractional derivative and its related fast approximations, the temporal coefficients are coupled with the inner product of the finite element method and lose monotonicity, which introduces uncommon difficulties in numerical analysis. In addition, the Riemann-Liouville fractional operators are often used in option pricing models, but its variable-order case gets far less attention than the corresponding Caputo-type problems. We prove error estimates for the proposed fast method and show that the computational cost is almost linear with respect to the temporal steps, which is much faster than the quadratic growth of the time-stepping solver. Numerical experiments are performed to illustrate the theoretical findings.

Stable Computational Techniques for the Advection–Dispersion Variable Order Model

In this paper, a novel approximation to the Caputo time fractional derivative of variable order (VO) [Formula: see text] ([Formula: see text]) is established. Since time fractional derivatives are integral, with a weakly singular kernel the discretization on the uniform mesh may not lead to satisfactory accuracy. So, numerical approximation is constructed on nonuniform meshes based on Newton interpolation polynomial. As a result, the novel approximation can be viewed as the modification of the existing method [Shen, S., Liu, F., Chen, J., Turner, I. and Anh, V. [2012] “Numerical techniques for the variable order time fractional diffusion equation,” Appl. Math. Comput. 218(22), 10861–10870]. The truncation errors and some significant properties of coefficients are analyzed. Furthermore, we derived Scheme-I and Scheme-II with second-order accuracy in spatial direction to solve the mobile–immobile advection–dispersion model (MIMADM) based on the VO Caputo derivative. The designed scheme converts the considered problems into a linear system of equations. Moreover, the stability properties of Scheme-I are discussed in detail. Ultimately, the proposed scheme is examined on several text examples demonstrating the high precision and effectiveness of the scheme. Also, comparative study between the outcomes achieved by implementing the suggested scheme with other numerical scheme in the existing literature is provided and also the acquired numerical results of these applications indicate that the suggested technique performs extremely well in terms of reliability and capability.

Jacobian spectral collocation method for spatio-temporal coupled Fokker-Planck equation with variable-order fractional derivative

Fractional Fokker–Planck equation plays an important role in describing anomalous dynamics. In this paper, we consider a spectral method for the spatio-temporal coupled fractional Fokker-Planck equation with variable-order fractional derivative. After introducing a suitable intermediate variable, the original equation can be split into two parts and decoupled. Then a spectral collocation method is employed to the resulting equation. Specifically, the points collocated both in temporal and spatial domains are related to fractional Jacobi functions, and the so-called fractional interpolation functions are used as basis functions for temporal semi-discretization. A recurrence relation is derived for fast generation of the fractional differentiation matrix. Regularity analysis for the solution as well as the convergence estimates for the spectral approximation are established for the constant-order case. Several numerical tests demonstrate the efficiency of the method and support the theoretical results.

Modeling Approaches to Permeability of Coal Based on a Variable-Order Fractional Derivative

In the past decades, many permeability models have been proposed to characterize the coal permeability evolution in the elastic state. Considering that the coal near the mining working face is often in the plastic or post-peak failure state, it is significant to characterize the coal permeability evolution in the plastic and post-peak failure stages for safe mining. In this study, two permeability models are developed using a variable-order fractional derivative to characterize the coal permeability evolution during the whole process of elastic, plastic, and post-peak failure stages. The results indicate that both models have the ability to better describe the coal permeability evolution during the whole process in areas near the mining working face. In addition, the physical and mechanical interpretation of the variable-order function is given as an indicator of fracture development, indicating the sensitivity of stress variation to coal fracture development. Moreover, the relationships among the effective stress, damage variable, and post-peak permeability of coal are discussed.

Homotopy perturbation method for solving time-fractional nonlinear Variable-Order Delay Partial Differential Equations

The homotopy perturbation method is extend to derive the approximate solution of the variable order fractional partial differential equations with time delay. The variable order fractional derivative is taken in the Caputo sense. An approximation formula of the Caputo derivative of fractional variable order is presented in terms of standard (integer order) derivatives only. Then the original problem will be transformed into a systems of partial differential equations with delay. By employing the homotopy perturbation method the explicit approximate solutions are found. The error and convergence analysis of the homotopy perturbation method has been discussed for the applicability of the method. The absolute errors and the approximate solutions are presented graphically and by tables at the values of various variable fractional order. From the results of the illustrated examples, we can Judge that the homotopy perturbation method is very effective, and simple accelerates the rapid convergence of the solution.

On the variable order fractional calculus of fractal interpolation functions

In this paper, a new idea of exploring the variable order fractional calculus on the fractal interpolation functions is addressed. The studies of constant order fractional calculus on the fractal functions are generalized to the case of variable order. The Riemann–Liouville variable order fractional integral (/derivative) and the Weyl–Marchaud variable order fractional derivative are investigated over the different kinds of fractal interpolation functions (FIFs). Furthermore, the necessary conditions of variable fractional order p(z) defined on the interval $$[x_0,x_N]$$ are also derived. It is observed that under the derived conditions, both the Riemann–Liouville variable order fractional integral (/derivative) and the Weyl–Marchaud variable order fractional derivative of FIFs are again FIFs interpolating new data set.

Finite difference scheme for a non-linear subdiffusion problem with a fractional derivative along the trajectory of motion

Abstract The article is devoted to the construction and study of a finite-difference scheme for a one-dimensional diffusion–convection equation with a fractional derivative with respect to the characteristic of the convection operator. It develops the previous results of the authors from [5, 6] in the following ways: the differential equation contains a fractional derivative of variable order along the characteristics of the convection operator and a quasi-linear diffusion operator; a new accuracy estimate is proved, which singles out the dependence of the accuracy of mesh scheme on the curvature of the characteristics.