Variable-order Fractional Operators Research Articles

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.

Examining avascular tumour growth dynamics: A variable-order non-local modelling perspective

This study investigates the growth of an avascular tumour described through the interchange of mass among its constituents and the production of inelastic distortions induced by growth itself. A key contribution of this research examines the role of non-local diffusion arising from the complex and heterogeneous tumour micro-environment. In our context, the non-local diffusion is enhanced by a variable-order fractional operator that incorporates crucial information about regions of limited nutrient availability within the tissue. Our research also focuses on the identification of an evolution law for the growth-induced inelastic distortions recast through the identification of non-conventional forces dual to suitable kinematic descriptors associated with the growth tensor. The establishment of such evolution law stems from examining the dissipation inequality and subsequently determining a posteriori connections between the inelastic distortions and the source/sink terms in the mass balance laws. To gain insights into the dynamics of tumour growth and its response to the proposed modelling framework, we first study how the variables governing the tissue evolution are affected by the introduction of the new growth law. Second, we investigate how regions of limited diffusion within the tumour, encoded into a fractional operator of variable-order, influence its growth.

Pore-pressure and stress-coupled creep behavior in deep coal: Insights from real-time NMR analysis

Understanding the variations in microscopic pore-fracture structures (MPFS) during coal creep under pore pressure and stress coupling is crucial for coal mining and effective gas treatment. In this manuscript, a triaxial creep test on deep coal at various pore pressures using a test system that combines in-situ mechanical loading with real-time nuclear magnetic resonance (NMR) detection was conducted. Full-scale quantitative characterization, online real-time detection, and visualization of MPFS during coal creep influenced by pore pressure and stress coupling were performed using NMR and NMR imaging (NMRI) techniques. The results revealed that seepage pores and microfractures (SPM) undergo the most significant changes during coal creep, with creep failure gradually expanding from dense primary pore fractures. Pore pressure presence promotes MPFS development primarily by inhibiting SPM compression and encouraging adsorption pores (AP) to evolve into SPM. Coal enters the accelerated creep stage earlier at lower stress levels, resulting in more pronounced creep deformation. The connection between the micro and macro values was established, demonstrating that increased porosity at different pore pressures leads to a negative exponential decay of the viscosity coefficient. The Newton dashpot in the ideal viscoplastic body and the Burgers model was improved using NMR experimental results, and a creep model that considers pore pressure and stress coupling using variable-order fractional operators was developed. The model’s reasonableness was confirmed using creep experimental data. The damage-state adjustment factors ω and β were identified through a parameter sensitivity analysis to characterize the effect of pore pressure and stress coupling on the creep damage characteristics (size and degree of difficulty) of coal.

Numerical Investigation of the Fractional Oscillation Equations under the Context of Variable Order Caputo Fractional Derivative via Fractional Order Bernstein Wavelets

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 fractional Bernstein polynomials. The oscillation equation describes electrical circuits and exhibits a wide range of nonlinear dynamical behaviors. The proposed variable order model is of current interest in a lot of application areas in engineering and applied sciences. The purpose of this study is to analyze the behavior of the fractional force-free and forced oscillation equations under the variable-order fractional operator. The basic idea behind using the approximation technique is that it converts the proposed model into non-linear algebraic equations with the help of collocation nodes for easy computation. Different cases of the proposed model are examined under the selected variable order parameters for the first time in order to show the precision and performance of the mentioned scheme. The dynamic behavior and results are presented via tables and graphs to ensure the validity of the mentioned scheme. Further, the behavior of the obtained solutions for the variable order is also depicted. From the calculated results, it is observed that the mentioned scheme is extremely simple and efficient for examining the behavior of nonlinear random (constant or variable) order fractional models occurring in engineering and science.

A Variable-Order Fractional Constitutive Model to Characterize the Rate-Dependent Mechanical Behavior of Soft Materials

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 parameters and clear physical meaning, a variable-order fractional model is proposed to accurately describe and predict the rate-dependent mechanical behavior of soft materials. In this work, the discrete variable-order fractional operator enables the predicted stress response to be entirely consistent with the whole stress history and the fractional order’s path-dependent values. The proposed model is further implemented in a numerical form and applied to predict several typical soft materials’ tensile and compressive deformation behavior. Our research indicates that the proposed variable-order fractional constitutive model is capable of predicting the nonlinear rate-dependent mechanical behavior of soft materials with high accuracy and more convinced reliability in comparison with the existing fractional models, where the fractional order contains a constant initial order to depict the initial elastic response and a linear variable-order function to account for the strain hardening behavior after acrossing the yield point.

Predicting the creep behavior of concrete at high temperature using the variable-order fractional model

PurposeCreep behavior of concrete at high temperature has become a major concern in building structures, such as factories, bridges, tunnels, airports and nuclear buildings. Therefore, a simple and accurate prediction model for the high-temperature creep behavior of concrete is crucial in engineering applications.Design/methodology/approachIn this paper, the variable-order fractional operator is introduced to capture the high-temperature creep behavior of concrete. By assuming that the variable-order function is a linear function with time, the proposed model benefits from the advantages of both formal simplicity and the physical significance for macroscopic intermediate materials. The effectiveness of the model is demonstrated by data fitting with existing experimental results of high-temperature creep of two representative concretes.FindingsThe results show that the proposed model fits well with the experimental data, and the value of order is increasing with the increase of the applied stress levels, which meets the fact that higher stress can accelerate the rate of creep. Furthermore, the relationship between the model parameters and loading conditions is deeply analyzed. It is found that the material coefficients are constant at a constant temperature, while the order function parameters are determined by the applied stress levels. Finally, the variable-order fractional model can be further written into a general equation of time and applied stress.Originality/valueThis paper provides a simple and practical variable-order fractional model for predicting the creep behavior of concrete at high temperature.

Variable-order fractional dynamic behavior of viscoelastic damping material

Abstract Viscoelastic damping material has been widely used in engineering machinery to absorb vibration and noise. Dynamic behavior of the viscoelastic damping material is mainly affected by temperature and frequency. Classical dynamic behavior equations of the viscoelastic damping material have complex structures with multiple and ambiguous parameters. So a novel variable-order fractional constitutive model (VOFC) is established based on the variable-order fractional operator. Then the viscoelastic dynamic equations are derived by Laplace transform of the VOFC model. The dynamic thermomechanical analysis tests by the three-point bending mode are carried out at variable temperatures and frequencies, and the frequency spectrum of the dynamic behavior, i.e. the loss modulus, storage modulus and loss factor are obtained. Against the test data, the VOFC model is compared with classical models such as the integer-order Maxwell model, constant fractional-order Kelvin-Voigt model, constant fractional-order Maxwell model and constant fractional-order standard linear solid model. The results show that the VOFC model can describe dynamic behavior of the viscoelastic damping material at different temperatures and frequencies more accurately. And the VOFC model has simpler structure and only two parameters with clearly physical meaning. Finally, a calculation software is designed and developed, which simplifies the process of predicting viscoelastic dynamic behavior in a wide frequency range.

Optimal control of hybrid variable-order fractional coronavirus (2019-nCov) mathematical model; numerical treatments

A novel coronavirus is a serious global issue and has a negative impact on the economy of Egypt. According to the publicly reported data, the first case of the novel corona virus in Egypt was reported on 14 February 2020. Total of 96753 cases were recorded in Egypt from the beginning of the pandemic until the eighteenth of August, where 96, 581 individuals were Egyptians and 172 were foreigners. Recently, many mathematical models have been considered to better understand coronavirus infection. Most of these models are based on classical integer-order derivatives which can not capture the fading memory and crossover behavior found in many biological phenomena. Therefore, we study the coronavirus disease in this paper by exploring the dynamics of COVID-19 infection using new variable-order fractional derivatives. This paper presents an optimal control problem of the hybrid variable-order fractional model of Coronavirus. The variable-order fractional operator is modified by an auxiliary parameter in order to satisfy the dimensional matching between the both sides of the resultant variable-order fractional equations. Existence, uniqueness, boundedness, positivity, local and global stability of the solutions are proved. Two control variables are considered to reduce the transmission of infection into healthy people. To approximate the new hybrid variable-order operator, Grünwald-Letnikov approximation is used. Finite difference method with a hybrid variable-order operator and generalized fourth order Runge-Kutta method are used to solve the optimality system. Numerical examples and comparative studies for testing the applicability of the utilized methods and to show the simplicity of these approximation approaches are presented. Moreover, by using the proposed methods we can concluded that, the model given in this paper describes well the confirmed real data given by WHO about Egypt.

Neural Approximators for Variable-Order Fractional Calculus Operators (VO-FC)

The paper presents research on the approximation of variable-order fractional operators by recurrent neural networks. The research focuses on two basic variable-order fractional operators, i.e., integrator and differentiator. The study includes variations of the order of each fractional operator. The recurrent neural network architecture based on GRU (Gated Recurrent Unit) cells functioned as a neural approximation for selected fractional operators. The paper investigates the impact of the number of neurons in the hidden layer, treated as a hyperparameter, on the quality of modeling error. Training of the established recurrent neural network was performed on synthetic data sets. Data for training was prepared based on the modified Grünwald-Letnikov definition of variable-order fractional operators suitable for convenient numerical computing without memory effects. The research presented in this paper showed that recurrent network architecture based on GRU-type cells can satisfactorily approximate targeted simple yet functional variable-order fractional operators with minor modeling errors. In addition, the research also compares the presented solution with basic and recurrent neural networks that utilize Tapped Delay Lines (TDL) in their structure. The presented solution is a novel approach to the approximation of VO-FC operators. It has the advantage of automatic selection of neural approximator parameters by optimization based on data customized for specific requirements.

A full-stage creep model for rocks based on the variable-order fractional calculus

An accurate characterization of rocks for the modelling of creep is an essential step toward ensuring the safety with respect to reliability of underground rock engineering. In this work, a novel variable fractional order rheological model is established to describe the full-stage creep behavior of rocks. With simpler form and fewer parameters, the proposed model is verified to have the ability for the description of trimodal creep behavior of rocks. The evolution of mechanical properties during creep is quantitatively described by the variable fractional order including the creep hardening, softening and the transition part between them. Moreover, the physical significance of fractional order is further explored by the commonly accepted competition mechanism of creep hardening and softening for brittle creep, which indicates that the mechanical property of rocks during creep is harder with lower water contents but softens faster with larger applied stresses. Furthermore, the linear form of the variable order function is determined by applying the new variable order fractional operator. It is shown that the slope of order function confirms the creep strain rate and the intercept of order function primarily affects the critical time for entering the nonlinear creep phase. Finally, the rising tendency of fractional order reveals that the accelerating creep is a continuous softening process of mechanical properties since the larger values of fractional order exhibits the property of rocks is more viscous.

An original perspective on variable-order fractional operators for viscoelastic materials

An original perspective on variable-order fractional operators for viscoelastic materials

Applications of Distributed-Order Fractional Operators: A Review.

Distributed-order fractional calculus (DOFC) is a rapidly emerging branch of the broader area of fractional calculus that has important and far-reaching applications for the modeling of complex systems. DOFC generalizes the intrinsic multiscale nature of constant and variable-order fractional operators opening significant opportunities to model systems whose behavior stems from the complex interplay and superposition of nonlocal and memory effects occurring over a multitude of scales. In recent years, a significant amount of studies focusing on mathematical aspects and real-world applications of DOFC have been produced. However, a systematic review of the available literature and of the state-of-the-art of DOFC as it pertains, specifically, to real-world applications is still lacking. This review article is intended to provide the reader a road map to understand the early development of DOFC and the progressive evolution and application to the modeling of complex real-world problems. The review starts by offering a brief introduction to the mathematics of DOFC, including analytical and numerical methods, and it continues providing an extensive overview of the applications of DOFC to fields like viscoelasticity, transport processes, and control theory that have seen most of the research activity to date.

Computation of traveling wave solution for nonlinear variable-order fractional model of modified equal width equation

<abstract> The Variable-order fractional operators (VO-FO) have considered mathematically formalized recently. The opportunity of verbalizing evolutionary leading equations has led to the effective application to the modeling of composite physical problems ranging from mechanics to transport processes, to control theory, to biology. In this paper, find the closed form traveling wave solutions for nonlinear variable-order fractional evolution equations reveal in all fields of sciences and engineering. The variable-order evolution equation is an impressive mathematical model describes the complex dynamical problems. Here, we discuss space-time variable-order fractional modified equal width equation (MEWE) and used exp $ (-\phi(\xi)) $ method in the sense of Caputo fractional-order derivative. Based on variable order derivative and traveling wave transformation convert equation into nonlinear ordinary differential equation (ODE). As a result, constructed new exact solutions for nonlinear space-time variable-order fractional MEWE. It clearly shows that the nonlinear variable-order evolution equations are somewhat natural and efficient in mathematical physics. </abstract>

Modeling Contacts and Hysteretic Behavior in Discrete Systems Via Variable-Order Fractional Operators

Abstract The modeling of nonlinear dynamical systems subject to strong and evolving nonsmooth nonlinearities is typically approached via integer-order differential equations. In this study, we present the possible application of variable-order (VO) fractional operators to a class of nonlinear lumped parameter models that have great practical relevance in mechanics and dynamics. Fractional operators are intrinsically multiscale operators that can act on both space- and time-dependent variables. Contrarily to their integer-order counterpart, fractional operators can have either fixed or VO. In the latter case, the order can be function of either independent or state variables. We show that when using VO equations to describe the response of dynamical systems, the order can evolve as a function of the response itself; therefore, allowing a natural and seamless transition between widely dissimilar dynamics. Such an intriguing characteristic allows defining governing equations for dynamical systems that are evolutionary in nature. Within this context, we present a physics-driven strategy to define VO operators capable of capturing complex and evolutionary phenomena. Specific examples include hysteresis in discrete oscillators and contact problems. Despite using simplified models to illustrate the applications of VO operators, we show numerical evidence of their unique modeling capabilities as well as their connection to more complex dynamical systems.

Applications of variable-order fractional operators: a review.

Variable-order fractional operators were conceived and mathematically formalized only in recent years. The possibility of formulating evolutionary governing equations has led to the successful application of these operators to the modelling of complex real-world problems ranging from mechanics, to transport processes, to control theory, to biology. Variable-order fractional calculus (VO-FC) is a relatively less known branch of calculus that offers remarkable opportunities to simulate interdisciplinary processes. Recognizing this untapped potential, the scientific community has been intensively exploring applications of VO-FC to the modelling of engineering and physical systems. This review is intended to serve as a starting point for the reader interested in approaching this fascinating field. We provide a concise and comprehensive summary of the progress made in the development of VO-FC analytical and computational methods with application to the simulation of complex physical systems. More specifically, following a short introduction of the fundamental mathematical concepts, we present the topic of VO-FC from the point of view of practical applications in the context of scientific modelling.

A cardinal method to solve coupled nonlinear variable-order time fractional sine-Gordon equations

In this study, a computational approach based on the shifted second-kind Chebyshev cardinal functions (CCFs) is proposed for obtaining an approximate solution of coupled variable-order time-fractional sine-Gordon equations where the variable-order fractional operators are defined in the Caputo sense. The main ideas of this approach are to expand the unknown functions in tems of the shifted second-kind CCFs and apply the collocation method such that it reduces the problem into a system of algebraic equations. To algorithmize the method, the operational matrix of variable-order fractional derivative for the shifted second-kind CCFs is derived. Meanwhile, an effective technique for simplification of nonlinear terms is offered which exploits the cardinal property of the shifted second-kind CCFs. Several numerical examples are examined to verify the practical efficiency of the proposed method. The method is privileged with the exponential rate of convergence and provides continuous solutions with respect to time and space. Moreover, it can be adapted for other types of variable-order fractional problems straightforwardly.

Approximation methods for solving fractional equations

Abstract In this review paper, we are mainly concerned with the numerical methods for solving fractional equations, which are divided into the fractional differential equations (FDEs), time-fractional, space-fractional and space-time-fractional partial differential equations (FPDEs), fractional integro-differential equations (FIDEs) and delay fractional differential and/or fractional partial differential equations (DFDE/DFPDEs). The concept of the variable-order fractional operators will also be reviewed. At the same time, the techniques for improving the accuracy and computation and storage are also introduced.

Modeling the dynamics of nutrient–phytoplankton–zooplankton system with variable-order fractional derivatives

We extended the nutrient–phytoplankton–zooplankton model involving variable-order fractional differential operators of Liouville–Caputo, Caputo–Fabrizio and Atangana–Baleanu. Variable-order fractional operators permits model and describe accurately real world problems, for example, diffusion or spread of nutrients or species in different states. Particularly, we model the interaction of nutrient phytoplankton and its predator zooplankton. The variable-order fractional numerical scheme based on the fundamental theorem of fractional calculus and the Lagrange polynomial interpolation was consider. Numerical simulation results are provided for illustrating the effectiveness and applicability of the algorithm to solve variable-order fractional differential equations.

FPGA implementation and control of chaotic systems involving the variable-order fractional operator with Mittag–Leffler law

This paper presents the simulation and control implementation on a Field Programmable Gate Array (FPGA) for a class of variable-order fractional chaotic systems by using sliding mode control strategy. Four different fractional variable-order chaotic systems via Atangana–Baleanu–Caputo fractional-order derivative were considered; Dadras, Aizawa, Thomas and 4 Wings attractors. A methodology has been developed to construct variable-order fractional chaotic systems using LabVIEW® software for its implementation in the National Instruments myRio-1900 (Xilinx FPGA Z-7010)® device. The variable-order fractional differential equations and the control law were solved using the variable-order Adams algorithm. Finally, simulation results show that FPGA provides high-speed realizations with the desired accuracy and demonstrate the effectiveness of the proposed sliding mode control.

Fractional description of mechanical property evolution of soft soils during creep

The motion of pore water directly influences mechanical properties of soils, which are variable during creep. Accurate description of the evolution of mechanical properties of soils can help to reveal the internal behavior of pore water. Based on the idea of using the fractional order to reflect mechanical properties of soils, a fractional creep model is proposed by introducing a variable-order fractional operator, and realized on a series of creep responses in soft soils. A comparative analysis illustrates that the evolution of mechanical properties, shown through the simulated results, exactly corresponds to the motion of pore water and the solid skeleton. This demonstrates that the proposed variable-order fractional model can be employed to characterize the evolution of mechanical properties of and the pore water motion in soft soils during creep. It is observed that the fractional order from the proposed model is related to the dissipation rate of pore water pressure.