Values Of Fractional Derivative Research Articles

Time-fractional lorenz type chaotic systems with asymmetric gaussian uncertainty: series solutions via extended He-Mohand algorithm in fuzzy-caputo sense

In this paper, modeling and analysis of 3D fuzzy-fractional Lorenz type systems is presented. System under-consideration includes classical Lorenz, Chen and Burke-Shaw chaotic systems. Asymmetrical Gaussian fuzzy logic with fractional calculus is applied to model complex systems with intricate patterns. The focus of this study is fuzzy-fractional modeling and simulations. For solution purpose, a hybrid perturbation method is introduced where standard homotopy perturbation method (HPM) is enhanced by incorporating Mohand transform in fuzzy-Caputo sense. This hybrid mechanism provides an efficient way to find solutions in fuzzy-fractional environment. Validity of obtained solutions is checked by computing residual errors, which ultimately confirms the convergence of applied methodology. The dynamical behavior of fuzzy-fractional chaotic models is analyzed through various 2-3D plots to represent the chaotic regions as well unpredictable trajectories at both upper and lower bounds. Fuzzy membership functions of 3D models at different values of fractional derivative are also demonstrated through 2D plots. Analysis reveals that extended hybrid methodology proves to be a valuable tool for researchers dealing with nonlinear chaotic fractional systems with fuzzy characteristics.

Application of the Optimal Homotopy Asymptotic Approach for Solving Two-Point Fuzzy Ordinary Differential Equations of Fractional Order Arising in Physics

This work focuses on solving and analyzing two-point fuzzy boundary value problems in the form of fractional ordinary differential equations (FFOBVPs) using a new version of the approximation analytical approach. FFOBVPs are useful in describing complex scientific phenomena that include heritable characteristics and uncertainty, and obtaining exact or close analytical solutions for these equations can be challenging, especially in the case of nonlinear problems. To address these difficulties, the optimal homotopy asymptotic method (OHAM) was studied and extended in a new form to solve FFOBVPs. The OHAM is known for its ability to solve both linear and nonlinear fractional models and provides a straightforward methodology that uses multiple convergence control parameters to optimally manage the convergence of approximate series solutions. The new form of the OHAM presented in this work incorporates the concepts of fuzzy sets theory and some fractional calculus principles to include fuzzy analysis in the method. The steps of fuzzification and defuzzification are used to transform the fuzzy problem into a crisp problem that can be solved using the OHAM. The method is demonstrated by solving and analyzing linear and nonlinear FFOBVPs at different values of fractional derivatives. The results obtained using the new form of the fuzzy OHAM are analyzed and compared to those found in the literature to demonstrate the method’s efficiency and high accuracy in the fuzzy domain. Overall, this work presents a feasible and efficient approach for solving FFOBVPs using a new form of the OHAM with fuzzy analysis.

PREDICTIVE CONTROL OF THE VARIABLE-ORDER FRACTIONAL CHAOTIC ECOLOGICAL SYSTEM

Since ecological systems are history-dependent, incorporating fractional calculus and especially variable order ones could significantly improve the emulation of these systems. Nonetheless, in the literature, no study considers ecological processes by variable-order fractional (VOF) model. This study is motivated by this issue. At first, we propose to extend a predator–prey mathematical model with VOF derivatives. The underlying assumption in the proposed model lies in considering values of fractional derivatives as time-varying functions instead of constant parameters. Some system’s dynamic features are investigated, and then the control of the proposed system is studied. To this end, a nonlinear model predictive control is offered for the VOF system. The necessary optimality and sufficient conditions for solving the nonlinear optimal control problem in the form of fractional calculus with variable-order derivative are formulated, and the controller’s design procedure is delineated. Finally, numerical simulations are performed to demonstrate the developed control technique’s effectiveness and performance for the VOF predator–prey model.

Exact Solutions of the 3D Fractional Helmholtz Equation by Fractional Differential Transform Method

In this work, we applied the fractional reduced differential transform method (FRDTM) to find the exact solutions of the three-dimensional fractional Helmholtz equation (FHE) and compared our outcomes with the tenth-order approximate solutions for diverse fractional orders. Different values of fractional derivatives are signified explicitly in three dimensions. Outcomes are denoted in the tables for nonfractional exact and approximate solutions and fractional approximate solutions for different fractional orders. Two examples are given to prove the proficiency of the suggested method, to resolve diverse categories of fractional partial differential equations.

On the development of variable-order fractional hyperchaotic economic system with a nonlinear model predictive controller

Abstract Mathematical modelling plays an indispensable role in our understanding of systems and phenomena. However, most mathematical models formulated for systems either have an integer order derivate or posses constant fractional-order derivative. Hence, their performance significantly deteriorates in some conditions. For the first time in the current paper, we develop a model of an economic system with variable-order fractional derivatives. Our underlying assumption is that the values of fractional derivatives are time-varying functions instead of constant parameters. The effects of variable-order time derivative into the economic system is studied. The dependency of the system's behaviour on the value of the fractional-order derivative is investigated. Afterwards, a nonlinear model predictive controller (NMPC) for hyperchaotic control of the system is suggested. The necessary optimality and sufficient conditions for solving the nonlinear optimal control problem (NOCP) of the NMPC in the form of fractional calculus with variable-order which is formulated as a two-point boundary value problem (TPBVP) are derived. Since the proposed methodology is a robust controller, the efficiency of the proposed controller in the presence of external bounded disturbances is examined. Simulation results show that not only does the presented control approach suppresses the related chaotic behaviour and stabilizes the close-loop system, but it also rejects the external bounded disturbances.

Application of non‐local and non‐singular kernel to an epidemiological model with fractional order

In this century when computers and other electronic devices have taken over the world, the rate at which malwares including viruses are infecting these devices has also started to increase rapidly. The developments of computer systems and its ever increasing uses require preventing strategies of malwares. This study aims to analyze an epidemiological model on the spread of computer viruses. We investigated the existence and uniqueness of solutions, examined the stability of the model, and performed numerical simulations. First, a model has been extended with the help of Atangana–Baleanu fractional derivative in Caputo sense. Then, the fixed point approach has been adopted to investigate the existence and uniqueness of solution of the model. Furthermore, the stability of the model has been examined with the help of the Hyers–Ulam stability method, and the model's numerical solutions have been obtained with the help of the Adam–Bashforth method. Lastly, the model's simulations have been performed considering various fractional derivative values.

Application of frequency stability criterion for analysis of dynamic systems with characteristic polynomials formed in j1/3 basis

This paper considers the stability of dynamical systems described by differential equations with fractional derivatives. In contrast to a number of works, where the differential equation describing the system may have a set of different values ​​of fractional derivatives, and the characteristic polynomial is formed on the basis of the least common multiple for the denominators of these indicators, this article proposes forming such a polynomial in a specific j¹/³ basis and studying the stability of systems with such fractional description based on the resulting rotation angles of Hn(jl/mω) vector at a frequency change from zero to infinity. This technique is similar to the investigation of system stability by frequency criteria used for a similar problem in describing the system by differential equations in integer derivatives. The application of characteristic polynomials formed in the j¹/³ basis for the description of the processes in dynamic systems and the analysis of the stability of such systems on the basis of the frequency criterion are the essence of the scientific novelty of this paper. The article contains the following sections: problem statement, work purpose, presentation of the research material, conclusions, list of references.

Effective Method for Solving Different Types of Nonlinear Fractional Burgers’ Equations

In this study, a relatively new method to solve partial differential equations (PDEs) called the fractional reduced differential transform method (FRDTM) is used. The implementation of the method is based on an iterative scheme in series form. We test the proposed method to solve nonlinear fractional Burgers equations in one, two coupled, and three dimensions. To show the efficiency and accuracy of this method, we compare the results with the exact solutions, as well as some established methods. Approximate solutions for different values of fractional derivatives together with exact solutions and absolute errors are represented graphically in two and three dimensions. From all numerical results, we can conclude the efficiency of the proposed method for solving different types of nonlinear fractional partial differential equations over existing methods.

A new design method for FIR notch filter using Fractional Derivative and swarm intelligence

In this paper, a new design method for the finite impulse response (FIR) notch filters using fractional derivative (FD) and swarm intelligence technique is presented. The design problem is constructed as a minimization of the magnitude response error w.r.t. filter coefficients. To acquire high accuracy of notch filter, fractional derivative (FD) is evaluated, and the required solution is computed using the Lagrange multiplier method. The fidelity parameters like passband error, notch bandwidth, and maximum passband ripple vary non-linearly with respect to FD values. Moreover, the tuning of appropriate FD value is computationally expensive. Therefore, modern heuristic methods, known as the constraint factor particle swarm optimization (CFI-PSO), which is inspired by swarm intelligence, is exploited to search the best values of FDs and number of FD required for the optimal solution. After an exhaustive analysis, it is affirmed that the use of two FDs results in 21% reduction in passband error, while notch bandwidth is slightly increased by 2% only. Also, it has been observed that, in the proposed methodology, at the most 66 iterations are required for convergence to optimum solution. To examine the performance of designed notch filter using the proposed method, it has been applied for removal of power line interference from an electrocardiography (ECG) signal, and the improvement in performance is affirmed.

Design of Bandpass and Bandstop Infinite Impulse Response Filters Using Fractional Derivative

In this paper, a new design method for digital bandpass and bandstop infinite impulse response filters with nearly linear-phase response is proposed. In this method, the phase response of an all-pass filter is optimized in the frequency domain to yield less passband error ${\text{Er}}_{p}$ and stopband error ${\text{Er}}_{s}$ with optimal stopband attenuation $A_{s}$ . To achieve high accuracy in passband and stopband regions, fractional derivative (FD) constraints are evaluated in the respective regions, and the filter coefficients are computed using the Lagrange multiplier method. The behavior of fidelity parameters measured in terms of ${\text{Er}}_{p}{,\text{Er}}_{s}$ , and phase error ${\text{Er}}_{{\text{ph}}}$ is multimodel w.r.t. FD values. Therefore, modern heuristic technique, known as cuckoo search optimization is used for determining the optimal value of FDs and reference frequency simultaneously to minimize the fitness function, which is constructed as a sum of the squared error in passband and stopband. The designed filter yields up to 60% reduction in ${\text{Er}}_{p}$ and ${\text{Er}}_{{\text{ph}}}$ in the case of bandpass filter. Meanwhile, the response of filter is not degraded due to the finite word length effect.

Invariant analysis of nonlinear time fractional Qiao equation

In this study, Lie’s invariant analysis method is applied to nonlinear time fractional Qiao equation with the Riemann–Liouville derivative. Fractional derivative value \(\alpha \) is considered for three cases and each cases symmetry generators are given. Using these generators group-invariant solutions in explicit form are obtained.

Homotopy analysis transform algorithm to solve time-fractional foam drainage equation

AbstractThis paper emphasizes on finding the solution for a foam drainageequation using the technique of modified homotopy analysis transform method (MHATM). MHATM is a new amalgamation of the homotopy analysis method and Laplace transform method with homotopy polynomial. Comparisons are made between the results of the proposed method for different values of fractional derivative α and exact solutions. Then, we analyze the results by numerical simulations, which demonstrate the simplicity and effectiveness of the present method.

Effect of power-law ionic conductances in the Hodgkin and Huxley model

An increasing number of results show that the voltage and spiking activity of a neuron follows scale free adaptation. Under such conditions, the voltage, or firing rate, of a neuron cannot be characterized by a unique time constant, instead these processes are characterized by power-laws. Power-law behaviors suggest that the components responsible for the voltage are strongly interacting across temporal scales, slow processes affect the rates of fast processes and vice versa. We recently introduced the fractional leaky integrate-and-fire model, which we have used to replicate the firing rate activity of adapting cortical neurons [1]. Our results have shown that spike rate adaptation can be modeled by a fractional derivative of low order. Thus suggesting, a strong interaction across conductances in cortical neurons. this work we decided to study the effects of power-law behavior in a biophysical model of spiking activity, the Hodgkin-Huxley model. The classical Hodgkin-Huxley model is described by CdVdt=−(gm(V−Erest)+∑i=1Ngi(V−Ei))+I. where C is the capacitance; V, voltage; gm, the membrane conductance; Erest, the resting potential; gi={gNa, gK } the sodium or potassium conductances with corresponding reversal potentials (Ei={ ENa, EK }); and I, the input curret. The conductances are gK=gK¯n4 and gNa=gNa¯m3h, with gK¯ and gNa¯ the maximum conductances. The gating variables n, m, and h are defined by the general equation dxdt=αxV,t1-x-βx(V,t)x where x={n, m, h}. To implement power-law behavior in either gating variable we substitute the classical derivative by the fractional derivative of order η using the Caputo definition dηfdtη=1Γn-η∫0tf(n)tt-uη+1-ndu where is the Gamma function, and n is the nearest integer larger than η. In our case, n = 1 because this models how the interaction of previous activity slows down the activation of the gating variables. The fractional derivative value is the result of integrating the activity of the function over all past activities weighted by a function that follows a power-law. The weighted values are called the memory trace. Thus, the fractional derivative provides information over all past activity, in contrast with the classical derivative that only takes into account the value of the function in the immediate previous time point. We have recently developed efficient ways to computationally solve these equations [2]. Our results show the emergence of a wide range of spiking behaviors (bursting, spiking and sub-threshold oscillations) in response to constant stimulation as a function of the fractional order in the different activation/inactivation variables. In the case of n, the neuron shows reduction if spiking response and emergence of sub-threshold oscillations. While fractional h results in bursting activity. This emergent richness in spiking activity, while only modeling two conductances, allows study of the overall effects of power-law behavior in neuronal activity. At the biophysical level, our results suggest that voltage dependent ion channels that deviate from a classical Markov process could increase the amount of spiking patterns in single cells, thus increasing their information content capacity.

Analytical solution for the time-fractional heat conduction equation in spherical coordinate system by the method of variable separation

In this paper, using the fractional Fourier law, we obtain the fractional heat conduction equation with a time-fractional derivative in the spherical coordinate system. The method of variable separation is used to solve the timefractional heat conduction equation. The Caputo fractional derivative of the order 0 < α ≤ 1 is used. The solution is presented in terms of the Mittag-Leffler functions. Numerical results are illustrated graphically for various values of fractional derivative.