Variable Fractional Order Research Articles

On the Variable Order Fractional Calculus Characterization for the Hidden Variable Fractal Interpolation Function

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–Marchaud variable order fractional calculus are investigated for hidden variable fractal interpolation function. Moreover, the conditions for the variable fractional order μ on a specified range are also derived. It is observed that, under certain conditions, the Riemann–Liouville and the Weyl–Marchaud variable order fractional calculus of the hidden variable fractal interpolation function are again the hidden variable fractal interpolation functions interpolating the new data set.

Time-Delay Fractional Variable Order Adaptive Synchronization and Anti-Synchronization between Chen and Lorenz Chaotic Systems Using Fractional Order PID Control

In this research work, time-delay adaptive synchronization and adaptive anti-synchronization of chaotic fractional order systems are analyzed via the Caputo fractional derivative, and the prob-lem of synchronization and anti-synchronization of chaotic systems of variable fractional order is solved by using the fractional order PID control law, the adaptive laws of variable-order frac-tional calculus, and a control law deduced from Lyapunov’s theory extended to systems of time-delay variable-order fractional calculus. In this research work, two important problems are solved in the control area: The first problem is described in which deals with syn-chro-nization of chaotic systems of adaptive fractional order with time delay, this problem is solved by using the fractional order PID control law and adaptative laws. The second problem is de-scribed in which deals with anti-synchronization of chaotic systems of adaptive frac-tional order with time delay, and this problem is solved by using the fractional order PID con-trol law and adaptative laws.

Performance evaluation of grid connected photovoltaic system using novel HTLB-WOA regulated VPFOTADF controller

The aim of this manuscript is the design of Hybrid Teaching Learning Based-Whale Optimization Algorithm -Variable Parameter Fractional Order Tilt Acceleration Derivative with Filter (HTLB-WOA regulated VPFOTADF) controller to reduce the harmonic distortion and improving performance of grid connected solar photovoltaic system. The solar photovoltaic system consists of components like photovoltaic array, interleaved fractional order boost converter, Reduced Switch Multilevel Inverter (RSMI) and HTLB-WOA regulated VPFOTADF controller. Interleaved fractional order boost converter is preferred over boost converter because of its low ripple voltage, faster transient response, high efficiency, low duty cycle, reduced EMC and improved reliability and stability. In this control strategy the control logic is formulated by using a Tilt Integral Derivative Controller (TIDC), whose control parameters are considered as a function of error to improve the robustness. The validation, better performance and superiority of HTLB-WOA regulated VPFOTADF is established by comparative result analysis with other state of the art controllers. HTLB-WOA regulated VPFOTADF along with Direct and Quadrature Control based Sinusoidal Pulse Width Modulation (DQC-SPWM) technique is implemented in this manuscript. This proposed model is analyzed for reduction in harmonic, improved system performance, better accuracy, improved stability, improved robustness and better capabilities to handle system uncertainties.

A variety of Gronwall Inequalities of Fractional Variable Order

A variety of Gronwall Inequalities of Fractional Variable Order

ANALYSIS AND IMPLEMENTATION OF NUMERICAL SCHEME FOR THE VARIABLE-ORDER FRACTIONAL MODIFIED SUB-DIFFUSION EQUATION

This paper addresses the numerical study of variable-order fractional differential equation based on finite-difference method. We utilize the implicit numerical scheme to find out the solution of two-dimensional variable-order fractional modified sub-diffusion equation. The discretized form of the variable-order Riemann–Liouville differential operator is used for the fractional variable-order differential operator. The theoretical analysis including for stability and convergence is made by the von Neumann method. The analysis confirmed that the proposed scheme is unconditionally stable and convergent. Numerical simulation results are given to validate the theoretical analysis as well as demonstrate the accuracy and efficiency of the implicit scheme.

Dynamic analysis of variable fractional order cantilever beam based on shifted Legendre polynomials algorithm

The shifted Legendre algorithm is used to solve the governing differential equation of variable order cantilever beam in time domain. According to the variable fractional model, the governing equations of viscoelastic polymer cantilever beam are built. The integer order and variable fractional order differential operator matrices are obtained based on the properties of shifted Legendre polynomials. The variable fractional differential equations are converted into algebraic equations and solved in time domain by operator matrix. The convergence analysis confirmed the high efficiency and accuracy of the proposed algorithm for solving differential equations. The deflection and stress of the polymer cantilever beams (PET and HDPE) under various external load (uniformly distributed and harmonic) are studied.

Multiterm Impulsive Caputo–Hadamard Type Differential Equations of Fractional Variable Order

In this study, we deal with an impulsive boundary value problem (BVP) for differential equations of variable fractional order involving the Caputo–Hadamard fractional derivative. The fundamental problems of existence and uniqueness of solutions are studied, and new existence and uniqueness results are established in the form of two fixed point theorems. In addition, Ulam–Hyers stability sufficient conditions are proved illustrating the suitability of the derived fundamental results. The obtained results are supported also by an example. Finally, the conclusion notes are highlighted.

Numerical Investigation of Ordinary and Partial Differential Equations with Variable Fractional Order by Bernstein Operational Matrix

Numerical Investigation of Ordinary and Partial Differential Equations with Variable Fractional Order by Bernstein Operational Matrix

Online public opinion prediction based on rolling fractional grey model with new information priority

The data of online public opinion updates quickly and new information plays a greater role than old information, this paper proposes a novel new information variable weight fractional rolling grey model for predicting online public opinion trends. Firstly, the new information variable weight buffer operator is proposed, and it follows the principle of “New information priority”. The new information is assigned larger weight under the action of the buffer operator. On the basis of the buffer operator, the new information variable weight fractional order accumulating generated operator is proposed, which has the dual functions of accumulation and weighting. In addition, the idea of metabolism is introduced to the model, which adds the latest online public opinion time point data and eliminates the earliest data. Finally, this model is applied to forecast the online public opinion trends with monotonically increasing sequence, monotonically decreasing sequence, and oscillation sequence. Compared with other models, the new information variable-weight fractional rolling grey model has better prediction performance in the online public opinion.

Numerical simulation of time variable fractional order mobile–immobile advection–dispersion model based on an efficient hybrid numerical method with stability and convergence analysis

In this paper, we propose a hybrid numerical method based on the weighted finite difference and the quintic Hermite collocation methods. The proposed method is used for solving the variable-order time fractional mobile–immobile advection–dispersion(VOMIM-AD) model, such that the discretization is done by applying collocation method with Hermite splines in the spatial direction and weighted finite difference method in the temporal direction. Provided examples confirm the studied stability and convergence properties of the proposed method. The obtained results from the graphical illustration and numerical simulations, in comparison with other methods in the literature, demonstrate that the reported method is very robust and accurate.

The Effect of Caputo Fractional Variable Difference Operator on a Discrete-Time Hopfield Neural Network with Non-Commensurate Order

In this work, we recall some definitions on fractional calculus with discrete-time. Then, we introduce a discrete-time Hopfield neural network (D.T.H.N.N) with non-commensurate fractional variable-order (V.O) for three neurons. After that, phase-plot portraits, bifurcation and Lyapunov exponents diagrams are employed to verify that the proposed discrete time Hopfield neural network with non-commensurate fractional variable order has chaotic behavior. Furthermore, we use the 0-1 test and C0 complexity algorithm to confirm and prove the results obtained about the presence of chaos. Finally, simulations are carried out in Matlab to illustrate the results.

A Fractional Variable Order Model of COVID-19 Pandemic

A Fractional Variable Order Model of COVID-19 Pandemic

The Discrete Fractional Variable-Order Tinkerbell Map: Chaos, 0–1 Test, and Entropy

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 fractional variable-order and establish the nature of the dynamics. In addition, the 0–1 test tool was used to detect chaos. Finally, the numerical results were confirmed using the approximate entropy.

Delay-dependent and order-dependent LMI-based sliding mode [formula omitted] control for variable fractional order uncertain differential systems with time-varying delay and external disturbance

This paper concentrates on the sliding mode H∞ control for variable fractional order uncertain differential systems with time-varying delay and external disturbance by using the delay-dependent and order-dependent LMI approach. Firstly, by designing a suitable variable fractional order sliding mode surface, an equivalent control law has been given and the new state-space model has been introduced. Secondly, by using a Lyapunov–Krasovskii function and constructing an appropriate variable fractional order inequality, a novel delay-dependent and order-dependent stability theorem of the nominal systems without external disturbance is proposed. Then, based on the above stability conditions and the H∞ performance, the delay-dependent and order-dependent stability conditions for the systems with external disturbance are discussed. Moreover, the reachability of the sliding mode surface has been presented. In order to stabilize the systems with uncertainties, the controller gain matrices are also derived with the help of the proposed stability criteria. All the results are in the form of linear matrix inequalities. Finally, two numerical examples are provided to verify the effectiveness of the introduced theoretical formulation.

A Mathematical Investigation of Sex Differences in Alzheimer’s Disease

Alzheimer’s disease (AD) is an age-related degenerative disorder of the cerebral neuro-glial-vascular units. Not only are post-menopausal females, especially those who carry the APOE4 gene, at a higher risk of AD than males, but also AD in females appears to progress faster than in aged-matched male patients. Currently, there is no cure for AD. Mathematical models can help us to understand mechanisms of AD onset, progression, and therapies. However, existing models of AD do not account for sex differences. In this paper a mathematical model of AD is proposed that uses variable-order fractional temporal derivatives to describe the temporal evolutions of some relevant cells’ populations and aggregation-prone amyloid-β fibrils. The approach generalizes the model of Puri and Li. The variable fractional order describes variable fading memory due to neuroprotection loss caused by AD progression with age which, in the case of post-menopausal females, is more aggressive because of fast estrogen decrease. Different expressions of the variable fractional order are used for the two sexes and a sharper decreasing function corresponds to the female’s neuroprotection decay. Numerical simulations show that the population of surviving neurons has decreased more in post-menopausal female patients than in males at the same stage of the disease. The results suggest that if a treatment that may include estrogen replacement therapy is applied to female patients, then the loss of neurons slows down at later times. Additionally, the sooner a treatment starts, the better the outcome is.

A New Numerical Approach for the Analysis of Variable Fractal and Fractional Order Differential Equations.

The variable fractional dimensions differential and integral operator overrides the phenomenon of the constant fractional order. This leads to exploring some new ideas in the proposed direction due to its varied applications in the recent era of science and engineering. The present papers deal with the replacement of the constant fractional order by variable fractional order in various fractal-fractional differential equations. An advanced numerical scheme is developed with the help of Lagrange three-point interpolation and further, it is employed for the solution of the proposed differential equations. However, the properties of these new operators are presented in detail. Finally, the error analysis is also conducted for the numerical scheme deployed. The results are validated by the suitable choice of applications to real-life problems. The well- known multi-step-Adams–Bashforth numerical scheme for classical differential equations is recovered when the non-integer order is one.

Seismic control of the structures with active tuned mass damper and variable fractional order fuzzy proportional–integral–derivative controller

The structural control of the building is a critical part of mitigating the catastrophic consequences of the earthquakes. The active control of structures has gained popularity in the construction industry because of its ability to adapt to different ground motion characteristics. The active tuned mass damper (ATMD) is one of the most versatile devices used for active structural control. This paper uses the ATMD and variable fractional order fuzzy PID (VFOFPID) controller for improving the performance of building structures subjected to horizontal ground motions. Having obtained the response of an 11-story shear building with this controller, the optimized control forces are determined through the pattern search algorithm and the performance indices of the system subjected to 100 different ground motions are evaluated. The performance indices are compared with those of uncontrolled as well as genetic fuzzy and fuzzy PID controlled structure. The results indicate that the VFOFPID controller proposed here can effectively reduce the drift and the absolute acceleration of the structure.

On the Solutions of a Quadratic Integral Equation of the Urysohn Type of Fractional Variable Order.

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 fractional constant order. The obtained new results are based on the Schauder’s fixed-point theorem and the Banach contraction principle with the help of piece-wise constant functions. Although the used methods are very powerful, they are not applied to the quadratic integral equation of the Urysohn type of fractional variable order. With this research we extend the applicability of these techniques to the introduced the Urysohn type model of fractional variable order. The applicability of the new results are demonstrated by providing Ulam–Hyers stability criteria and an example. Moreover, the presented results lead to future progress and expansion of the theory of fractional-order models, as well as of the concept of entropy in the framework of fractional calculus. Further, an example is constructed to demonstrate the reasonableness and effectiveness of the observed results.

Numerical simulation using the non-standard weighted average FDM for 2Dim variable-order Cable equation

The fractional variable-order (VO) two-dimensional (2Dim) Cable equation is one of the most significant types of anomalous subdiffusion equations that emerge strongly in spiny neural dendrites and is solved by using an accurate numerical technique in this study. The non-standard weighted average finite difference approach is a simple proposed technique (NSWAFDM). The proposed method’s stability is investigated in depth by using the John von Neumann stability analysis approach. The arbitrary weight factor and multiple discretization schemes of the VO derivative are used to develop a specific stability criterion. To demonstrate the usefulness and accuracy of the suggested technique, numerical examples are offered.

Delay-dependent and order-dependent stability and stabilization analysis of variable fractional order uncertain differential systems with time-varying delay via linear matrix inequality approach

This paper concentrates on the robust stability and stabilization analysis of variable fractional order uncertain differential systems with time-varying delay. Firstly, by using a suitable Lyapunov–Krasovskii function and constructing an appropriate variable fractional order inequality, a novel delay-dependent and order-dependent stability theorem of the nominal systems is proposed. Then, based on the above stability conditions, the robust delay-dependent and order-dependent stability conditions for the uncertain systems are discussed. Moreover, in order to stabilize the nominal and uncertain systems, state feedback controllers are also derived with the help of the presented stability criteria. All the results are in the form of linear matrix inequalities. Finally, two numerical examples are provided to verify the effectiveness of the introduced theoretical formulation.