Properties Of Fractional Calculus Research Articles

OPERATIONAL MATRIX OF GENERALIZED BLOCK PULSE FUNCTIONS FOR SOLVING FRACTIONAL VOLTERRA–FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS

This article aims to introduce two effective numerical approaches for solving the mixed linear fractional Volterra-Fredholm integro-differential equation (FVIDE) and multi-higher order with initial conditions. Even though these approaches transfer the integro-differential equations into a system of algebraic equations through operational matrices of generalized block pulse functions, these problems can be transferred to a system of algebraic equations by expanding the solution's multi-highest order derivative through the block pulse functions (BPFs). The block pulse operational matrices for fractional-order integration are derived by expanding the Riemann-Liouville fractional integral for repeated fractional integration in block pulse functions (BPFs). Also, the fractional derivative is done by using the generalized operational matrices of BPFs with the fractional calculus properties in the first scheme. The methods are attractive in the calculation, and examples are supplied to explain their use. The estimated and exact outcomes are compared in a table. In addition, they decrease the error terms in the specified domain using the least squares-error technique. For this, most general programs are written in Python V.3.8.8 software (2021).

Asymptotic and Finite-Time Synchronization of Fractional-Order Memristor-Based Inertial Neural Networks with Time-Varying Delay

This paper emphasized on studying the asymptotic synchronization and finite synchronization of fractional-order memristor-based inertial neural networks with time-varying latency. The fractional-order memristor-based inertial neural network model is offered as a more general and flexible alternative to the integer-order inertial neural network. By utilizing the properties of fractional calculus, two lemmas on asymptotic stability and finite-time stability are provided. Based on the two lemmas and the constructed Lyapunov functionals, some updated and valid criteria have been developed to achieve asymptotic and finite-time synchronization of the addressed systems. Finally, the effectiveness of the proposed method is demonstrated by a number of examples and simulations.

MODELING THE TRANSMISSION PHENOMENA OF COVID-19 INFECTION WITH THE EFFECT OF VACCINATION VIA NONINTEGER DERIVATIVE UNDER REAL STATISTIC

The infection of coronavirus (COVID-19) is a dangerous and life-threatening disease which spread to almost all parts of the globe. We present a mathematical model for the transmission of COVID-19 with vaccination effects. The basic properties of fractional calculus are presented for the inspection of the model. We calculate the equilibria of the model and determined the reproduction number [Formula: see text]. Local asymptotic stability conditions for the disease-free are obtained which determines the conditions to stabilize the exponential spread of the disease. The nonlinear least-square procedure is utilized to parameterize the model from actual cases reported in Pakistan. By fixed point theory, we prove the existence of a unique solution. We also present numerical results to simulate virus transmission and compare the results with those of the Caputo derivative. We study the solution pathways of the COVID-19 system to provide effective control policies for the infection. Significant changes have been noticed by lowering the order of fractional derivative.

Analysis of fractional-order dynamics of dengue infection with non-linear incidence functions

The infection of dengue is a devastating mosquito-borne infection around the globe that affects human health, social and economic sectors in low-income areas. Therefore, policymakers and health experts are trying to point out better policies to reduce these losses and provide better information for the development of vaccination and medication. Here, we formulated a compartmental model for the transmission phenomena of dengue fever with nonlinear forces of infection through fractional derivative. We established several results related to the solution of our dengue model by using the basic properties of fractional calculus. We determined the basic reproduction number of our fractional-order system, symbolized by [Formula: see text]. We established the local asymptotic stability of the infection-free equilibrium of our dengue system for [Formula: see text], and proved that the infection-free equilibrium is globally asymptotically stable without vaccination. The threshold dynamics [Formula: see text] is tested through partial rank correlation coefficient method to notice the importance of parameters in the transmission of dengue infection. In addition, we have shown the impact of memory on the basic reproduction number numerically with the variation of different parameters. We conclude that the biting rate, recruitment rate of mosquitoes and index of memory are the most sensitive factors, which can effectively lower the level of dengue fever. The dynamical behavior of the proposed fractional system is presented through a numerical scheme to explore the overall transmission process. We predict that the fractional-order model can explore more accurately and preciously the intricate dengue disease transmission model rather than the integer-order derivative.

Controllability of higher-order fractional damped stochastic systems with distributed delay

In this paper, the controllability analysis is proposed for both linear and nonlinear higher-order fractional damped stochastic dynamical systems with distributed delay in Hilbert spaces which involve fractional Caputo derivative of different orders. Based on the properties of fractional calculus, the fixed point technique, and the construction of controllability Gramian matrix, we establish the controllability results for the considered systems. Finally, examples are constructed to illustrate the applicability of obtained results.

Katugampola Fractional Calculus With Generalized k−Wright Function

In this article, we present some properties of the Katugampola fractional integrals and derivatives. Also, we study the fractional calculus properties involving Katugampola Fractional integrals and derivatives of generalized k−Wright function nΦkm(z).

Output feedback finite‐time dissipative control for uncertain nonlinear fractional‐order systems

AbstractThis paper solves the robustly finite‐time dissipativity control problems for uncertain nonlinear fractional‐order systems (FOS). Firstly, by using some basic mathematical transformations associated with linear matrix inequality (LMI) techniques, a novel condition for the existence of output feedback controllers is established to ensure the closed‐loop FOS is finite‐time stabilizable. Then, based on the proposed stabilization criterion combined with some well‐known properties of fractional calculus, the finite‐time dissipativity control problem for nonlinear FOS subject to uncertainties is studied for the first time. Two numerical examples are provided to illustrate the effectiveness of the proposed results.

Finite-Time Projective Synchronization of Caputo Type Fractional Complex-Valued Delayed Neural Networks

This paper focuses on investigating the finite-time projective synchronization of Caputo type fractional-order complex-valued neural networks with time delay (FOCVNNTD). Based on the properties of fractional calculus and various inequality techniques, by constructing suitable the Lyapunov function and designing two new types controllers, i.e., feedback controller and adaptive controller, two sufficient criteria are derived to ensure the projective finite-time synchronization between drive and response systems, and the synchronization time can effectively be estimated. Finally, two numerical examples are presented to verify the effectiveness and feasibility of the proposed results.

Containment control of fractional discrete-time multi-agent systems with nonconvex constraints

This paper is devoted to the containment control problem of fractional discrete-time multi-agent systems (FDTMASs) with nonconvex velocity and control input constraints. First, distributed containment controllers with nonlinear projection algorithms are proposed for all followers. Then, some sufficient conditions are deduced during the convergence analysis of the closed-loop systems. By using Lyapunov stability theory and properties of fractional calculus, it proved that the containment control of the systems could be achieved with the distributed containment controller. Finally, some numerical examples are given to show the correctness of the proposed theoretical results.

Non-separation method-based robust finite-time synchronization of uncertain fractional-order quaternion-valued neural networks

In this paper, robust finite-time synchronization (F-TS) issue is addressed for a class of uncertain fractional-order quaternion-valued neural networks by employing non-separation method instead of separation method. First, a general fractional differential inequality is developed to provide new insight into the research about finite-time stability and synchronization of fractional-order systems. Next, quaternion-valued feedback controller and quaternion-valued adaptive controller are designed. On the basis of the newly developed inequality, quaternion inequality techniques, together with the properties of fractional calculus and reduction to absurdity, some easily-verified algebraic criteria for robust F-TS are established, and the settling time for robust F-TS is explicitly reckoned, which depends on not only the controller parameters but also the initial values and order of the considered systems. Eventually, numerical results are provided to substantiate our robust F-TS criteria.

Some Existence and Stability Results of Hilfer-Hadmard Fractional Implicit Differential Equation in a Weighted Space

This paper studies a nonlinear fractional implicit differential equation (FIDE) with boundary conditions involving a Hilfer-Hadamard type fractional derivative. We establish the equivalence between the Cauchy-type problem (FIDE) and its mixed type integral equation through a variety of tools of some properties of fractional calculus and weighted spaces of continuous functions. The existence and uniqueness of solutions are obtained. Further, the Ulam-Hyers and Ulam-Hyers-Rassias stability are discussed. The arguments in the analysis rely on Schaefer fixed point theorem, Banach contraction principle and generalized Gronwall inequality. At the end, an illustrative example will be introduced to justify our results.

An Approximation to Zeros of the Riemann Zeta Function Using Fractional Calculus

In this document, as far as the authors know, an approximation to the zeros of the Riemann zeta function has been obtained for the first time using only derivatives of constant functions, which was possible only because a fractional iterative method was used. This iterative method, valid for one and several variables, uses the properties of fractional calculus, in particular the fact that the fractional derivatives of constants are not always zero, to find multiple zeros of a function using a single initial condition. This partly solves the intrinsic problem of iterative methods that if we want to find N zeros it is necessary to give N initial conditions. Consequently, the method is suitable for approximating nontrivial zeros of the Riemann zeta function when the absolute value of its imaginary part tends to infinity. The deduction of the iterative method is presented, some examples of its implementation, and finally 53 different values near to the zeros of the Riemann zeta function are shown.

An investigation on existence and uniqueness of solution for Integro differential equation with fractional order

Present article deals with some theoretical results regarding the solution of integro differential equation with integral boundary condition. Main idea is to covert the integro differential equation into corresponding integral equation. Then we adopt a suitable transformation to implement Banach contraction and Schaefer-Krasnoselskii fixed point Theorem. Finally by properties of fractional calculus and fixed point theory we derive the main results. An example is considered for the verification of theoretical work.

The Fractional View Analysis of Polytropic Gas, Unsteady Flow System

Generally, the differential equations of integer order do not properly model various phenomena in different areas of science and engineering as compared to differential equations of fractional order. The fractional-order differential equations provide the useful dynamics of the physical system and thus provide the innovative and effective information about the given physical system. Keeping in view the above properties of fractional calculus, the present article is related to the analytical solution of the time-fractional system of equations which describe the unsteady flow of polytropic gas dynamics. The present method provides the series form solution with easily computable components and a higher rate of convergence towards the targeted problem’s exact solution. The present techniques are straightforward and effective for dealing with the solutions of fractional-order problems. The fractional derivatives are expressed in terms of the Caputo operator. The targeted problems’ solutions are calculated using the Adomian decomposition method and variational iteration methods along with Shehu transformation. In the current procedures, we first applied the Shehu transform to reduce the problems into a more straightforward form and then implemented the decomposition and variational iteration methods to achieve the problems’ comprehensive results. The solution of the nonlinear equations of unsteady flow of a polytropic gas at various fractional orders of the derivative is the core point of the present study. The solution of the proposed fractional model is plotted via two- and three-dimensional graphs. It is investigated that each problem’s solution-graphs are best fitted with each other and with the exact solution. The convergence of fractional-order problems can be observed towards the solution of integer-order problems. Less computational time is the major attraction of the suggested methods. The present work will be considered a useful tool to handle the solution of fractional partial differential equations.

Containment control for delayed fractional multiple agent systems in Riemann–Liouville sense

This paper analyses containment control (CC) of Riemann–Liouville fractional multiple agent systems with time delays. A reliable and simple method is addressed to deal with CC, some simple sufficient conditions are proposed by utilising traditional Lyapunov direct method, properties of fractional calculus and analytical technique. The main advantage of our method is that the first-order derivative of Lyapunov function can be taken to avoid complicated calculation, which may overcome well the difficulties arising from fractional derivatives and time delays. In the end, numerical examples are offered to further demonstrate the reliability of our conclusions.

Cluster Synchronization of Fractional-Order Nonlinearly-Coupling Community Networks With Time-Varying Disturbances and Multiple Delays

This paper investigates the adaptive cluster synchronization of fractional-order complex networks with internal and coupling delays as well as time-varying disturbances via the fractional-order hybrid controllers. To be more practical, the unknown disturbances and dynamical behaviors of nodes are assumed to be nonidentical. Based on the properties of fractional calculus and the fractional-order comparison principle, sufficient conditions are derived to guarantee the cluster synchronization of two kinds of fractional-order nonlinear dynamical systems, which extends the results in existing literatures. Numerical simulations are presented to show the effectiveness of our theoretical results.

Global attractiveness and consensus for Riemann–Liouville’s nonlinear fractional systems with mixed time-delays

• Riemann–Liouville nonlinear fractional systems with mixed time-delays are studied. • A reliable and simple method is developed to handle attractiveness and consensus. • The method overcomes the difficulty arising from fractional derivatives and delays. • Several simple algebraic criteria based on linear matrix inequalities are presented. • Illustrative examples further show the effectiveness and reliability of main results. This article deals with attractiveness and consensus for Riemann–Liouville’s nonlinear fractional systems with mixed time-delays (RL-NFSMDs). A reliable and simple method is adopted to achieve global attractiveness in terms of traditional Lyapunov direct approach, properties of fractional calculus and analytical technique. As a straightforward application of our proposed method, global consensus analysis for RL fractional multiple agent systems is considered and several algebraic criteria are presented by means of graph theory. The method permits one to calculate first-order derivative for the corresponding Lyapunov function and may deal with well the trouble brought from fractional derivatives and time-delays. Finally, illustrative examples are given to further clarify the reliability and validity of our results.

ESTIMATION OF FRACTAL DIMENSION OF FRACTIONAL CALCULUS OF THE HÖLDER CONTINUOUS FUNCTIONS

In the present paper, fractal dimension and properties of fractional calculus of certain continuous functions have been investigated. Upper Box dimension of the Riemann–Liouville fractional integral of continuous functions satisfying the Hölder condition of certain positive orders has been proved to be decreasing linearly. If sum of order of the Riemann–Liouville fractional integral and the Hölder condition equals to one, the Riemann–Liouville fractional integral of the function will be Lipschitz continuous. If the corresponding sum is strictly larger than one, the Riemann–Liouville fractional integral of the function is differentiable. Estimation of fractal dimension of the derivative function has also been discussed. Finally, the Riemann–Liouville fractional derivative of continuous functions satisfying the Hölder condition exists when order of the Riemann–Liouville fractional derivative is smaller than order of the Hölder condition. Upper Box dimension of the function has been proved to be increasing at most linearly.

English

A nonlinear algebraic equation system of two variables is numerically solved, which is derived from a nonlinear algebraic equation system of four variables, that corresponds to a mathematical model related to investment under conditions of uncertainty. The theory of investment under uncertainty scenarios proposes a model to determine when a producer must expand or close, depending on his income. The system mentioned above is solved using a fractional iterative method, valid for one and several variables, that uses the properties of fractional calculus, in particular the fact that the fractional derivatives of constants are not always zero, to find solutions of nonlinear systems.

Passivity Analysis of Fractional-Order Neural Networks with Time-Varying Delay Based on LMI Approach

In this paper, we study the problem of passivity analysis of fractional-order neural networks (FONNs) with a time-varying delay. By using the Razumikhin fractional-order theorem, we first derive an improved sufficient criterion for asymptotic stability of FONNs with a bounded time-varying delay. Then, based on the proposed stability criterion and some auxiliary properties of fractional calculus, a delay-dependent condition is established to ensure the passivity of the considered system. These conditions are order-dependent and in the form of linear matrix inequalities, which therefore can be efficiently solved in polynomial time by using the existing convex algorithms. Some numerical examples are provided to show the effectiveness of the obtained results.