Properties Of Fractional Calculus Research Articles

Ground state solutions for the fractional impulsive differential system with ψ‐Caputo fractional derivative and ψ–Riemann–Liouville fractional integral

This article examines a new family of (p,q)‐Laplacian type nonlinear fractional impulsive differential coupled equations involving both the ‐Caputo fractional derivative and –Riemann–Liouville fractional integral. With the help of Nehari manifold in critical point theory and fractional calculus properties, we obtain the existence of at least one nontrivial ground state solution for the coupled system with some natural and easily verifiable superlinear conditions on the nonlinearity.

On the analysis of fractional calculus operators with bivariate Mittag Leffler function in the kernel

Bivariate Mittag-Leffler (ML) functions are a substantial generalization of the univariate ML functions, which are widely recognized for their significance in fractional calculus. In the present paper, our initial focus is to investigate the fractional calculus properties of the integral and derivative operators with kernels including the Bivariate ML functions. Further, certain fractional Cauchy-type problems including these operators are considered. Also the numerical approximations of the Caputo type derivative operator are investigated. The theoretical results are justified by applications on examples. Furthermore, the theory of applying the same operators with respect to arbitrary monotonic functions is analyzed in this research.

Research on the higher-order logic formalization of fractance element

Fractance element reflects the fractional order behavior of circuits, which can show the characteristics of the actual circuits. Higher-order logic theorem proving is based on the rigorous and correct mathematical theories. It becomes more and more important in the verifications of high-reliability systems. Fractance element is formalized using the proof of higher-order logic theorem in this paper. Firstly, the formalized model of fractional calculus which is based on Caputo definition is established in higher-order logic theorem proof tool. Then some properties of fractional calculus are proved, including the zero order property, the fractional differential of a constant and the consistency of fractional calculus and integer order calculus. Finally, fractance element and fractional differential circuit constituted by fractance element are formally analyzed. These formalizations demonstrate the effectiveness of the formal method in the analysis of fractance element.

Almost periodic quasi-projective synchronization of delayed fractional-order quaternion-valued neural networks

This paper examines the issue of almost periodic quasi-projective synchronization of delayed fractional-order quaternion-valued neural networks. First, using a direct method rather than decomposing the fractional quaternion-valued system into four equivalent fractional real-valued systems, using Banach’s fixed point theorem, according to the basic properties of fractional calculus and some inequality methods, we obtain that there is a unique almost periodic solution for this class of neural network with some sufficient conditions. Next, by constructing a suitable Lyapunov functional, using the characteristic of the Mittag-Leffler function and the scaling idea of the inequality, the adequate conditions for the quasi-projective synchronization of the established model are derived, and the upper bound of the systematic error is estimated. Finally, further use Matlab is used to carry out two numerical simulations to prove the results of theoretical analysis.

Comments for “Fractional Liu uncertain differential equation and its application to finance”

In this paper, some comments are proposed for the paper by Najafi and Taleghani (2022). Some properties for fractional Liu process and fractional calculus are used in the proof of existence and uniqueness results for fractional Liu uncertain differential equation (Theorem 7). We show that incorrect properties of fractional calculus is used, and lead to subsequent incorrect proof, furthermore, we make some corrections and present new results.

An Overview of Mathematical Modelling in Cancer Research: Fractional Calculus as Modelling Tool

Cancer is a complex disease, responsible for a significant portion of global deaths. The increasing prioritisation of know-why over know-how approaches in biological research has favoured the rising use of both white- and black-box mathematical techniques for cancer modelling, seeking to better grasp the multi-scale mechanistic workings of its complex phenomena (such as tumour-immune interactions, drug resistance, tumour growth and diffusion, etc.). In light of this wide-ranging use of mathematics in cancer modelling, the unique memory and non-local properties of Fractional Calculus (FC) have been sought after in the last decade to replace ordinary differentiation in the hypothesising of FC’s superior modelling of complex oncological phenomena, which has been shown to possess an accumulated knowledge of its past states. As such, this review aims to present a thorough and structured survey about the main guiding trends and modelling categories in cancer research, emphasising in the field of oncology FC’s increasing employment in mathematical modelling as a whole. The most pivotal research questions, challenges and future perspectives are also outlined.

Discrete Analogue of Fishburn’s Fractional-Order Stochastic Dominance

A stochastic dominance (SD) relation can be defined by two different perspectives: One from the view of distributions, and the other one from the view of expected utilities. In the early days, Fishburn investigated SD from the view of distributions, and we refer this perspective as Fishburn’s SD. One of his many results was the development of fractional-order SD for continuous distributions. However, discrete fractional-order SD cannot be directly generalized, because some properties of fractional calculus may not possess a discrete counterpart. In this paper, we develop a discrete analogue of fractional-order SD for discrete utilities from the view of distributions. We generalize the order of SD by Lizama’s fractional delta operator, show the preservation of SD hierarchy, and formulate the utility classes that are congruent with our SD relations. This work brings a message that some results of discrete SD cannot be directly generalized from continuous SD. We characterize the difference between discrete and continuous fractional-order SD, as well as the way to handle it for further applications in mathematics and computer science.

Generalized fractional calculus in Banach spaces and applications to existence results for boundary value problems

In this paper, we present the definitions of fractional integrals and fractional derivatives of a Pettis integrable function with respect to another function. This concept follows the idea of Stieltjes-type operators and should allow us to study fractional integrals using methods known from measure differential equations in abstract spaces. We will show that some of the well-known properties of fractional calculus for the space of Lebesgue integrable functions also hold true in abstract function spaces. In particular, we prove a general Goebel–Rzymowski lemma for the De Blasi measure of weak noncompactness and our fractional integrals.We suggest a new definition of the Caputo fractional derivative with respect to another function, which allows us to investigate the existence of solutions to some Caputo-type fractional boundary value problems. As we deal with some Pettis integrable functions, the main tool utilized in our considerations is based on the technique of measures of weak noncompactness and Mönch’s fixed-point theorem. Finally, to encompass the full scope of this research, some examples illustrating our main results are given.

Master–slave synchronization of fractional-order memristive MAM neural networks with parameter disturbances and mixed delays

In order to describe the system as realistically as possible, the fractional-order memristive multidirectional associative memory neural networks (FOMMAMNNs) with parameter disturbances, time-varying and distributed delays are established. Further, the master–slave synchronization is explored. Firstly, in the Filippov’s sense, some conditions are derived via the contraction mapping principle to guarantee the uniqueness of equilibrium point of the system. Secondly, from the perspective of reducing control costs, pinning controller and adaptive pinning controller are designed for the first time to induce synchronization of FOMMAMNNs. Moreover, some synchronization criteria are obtained by using the properties of fractional calculus, the appropriate Lyapunov candidate functions and fractional Barbalat’s lemma. Thirdly, a communication scheme based on chaotic synchronization is proposed to achieve the safe transmission of important information. Finally, two numerical simulations are given to illustrate the feasibility of the theoretical results and the security of the designed communication scheme.

Mittag-Leffler stabilization of anti-periodic solutions for fractional-order neural networks with time-varying delays

<abstract> <p>Mittag-Leffler stabilization of anti-periodic solutions for fractional-order neural networks with time-varying delays are investigated in the article. We derive the relationship between the fractional-order integrals of the state function with and without delays through the division of time interval, using the properties of fractional calculus, and initial conditions. Moreover, by constructing the sequence solution of the system function which converges to a continuous function uniformly with the Arzela-Asoli theorem, a sufficient condition is obtained to ensure the existence of an anti-periodic solution and Mittag-Leffler stabilization of the system. In the final, we verify the correctness of the conclusion by numerical simulation.</p> </abstract>

Analytical method for solving a time-conformable fractional telegraph equation

In this paper, we present an analytical method to solve a time-conformable fractional telegraph equation with three boundary conditions namely, Dirichlet, Neumann and Robin. This method based on Fourier method and conformable fractional calculus properties. We give three examples to validate this method.

On the matrix function $_pR_q(A, B; z)$ and its fractional calculus properties

The main objective of the present paper is to introduce and study the function $_pR_q(A, B; z)$ with matrix parameters and investigate the convergence of this matrix function. The contiguous matrix function relations, differential formulas and the integral representation for the matrix function $_pR_q(A, B; z)$ are derived. Certain properties of the matrix function $_pR_q(A, B; z)$ have also been studied from fractional calculus point of view. Finally, we emphasize on the special cases namely the generalized matrix $M$-series, the Mittag-Leffler matrix function and its generalizations and some matrix polynomials.

Estimate scheme for fractional order-dependent fixed-time synchronization on Caputo quaternion-valued BAM network systems with time-varying delays

In this paper, an estimate scheme for derivative order-dependent fixed-time synchronization (F-TS) on Caputo quaternion-valued BAM neural networks (QVBAMNNs) with time-varying delays is implemented. A novel fractional-order (FO) F-T stability lemma is established by the synthetic properties of fractional calculus. The appropriate feedback controllers are designed to realize the F-TS between the master-slave systems. Based on the Lyapunov functional method and multiple inequality techniques, the F-TS criterion of FOQVBAMNNs is obtained. Furthermore, the upper bound of settling time is predicted and controlled by the Caputo derivative order and system parameters, which is independent of the initial state. Finally, two simulation examples manifest the feasibility and validity of the results on F-TS and the control strategies.

Mixed ℋ ∞ and Passivity Analysis of Delayed Fractional-Order Complex Dynamical Networks with Hybrid Coupling

In this article, global asymptotic stability analysis, and mixed ℋ ∞ and passive control for a class of control fractional-order systems is investigated. Based on the fractional-order Lyapunov stability theorem and some properties of fractional calculus, we propose sufficient conditions to ensure the mixed ℋ ∞ and passivity performance. More relaxed conditions by employing the new type of augmented matrices by using Kronecker product terms can be handled, which can be introduced. The derived criteria are expressed in terms of linear matrix inequalities that which can be checked numerically using toolbox MATLAB. Finally, two numerical examples are provided to demonstrate the correctness of the proposed results.

Stability analysis of delayed fractional-order switched systems

This paper discusses stability of delayed fractional-order switched systems. First, two useful propositions are proposed based on Razumikhin approach together with properties of fractional calculus and analytical techniques, which can well solve the difficulties caused by distributed and discrete delays and fractional-order derivatives. Then, with the help of multiple Lyapunov function approach and dwell time technique, an effective way is developed to overcome the trouble caused by switching rules. A delay-dependent dwell time is derived to guarantee asymptotic stability under the hypothesis that each subsystem of delayed fractional-order switched systems is asymptotically stable. Finally, two illustrative examples further clarify the obtained conclusions.

Global Mittag-Leffler Stability and Global Asymptotic ω-Period for Fractional-Order Cohen–Grossberg Neural Networks with Time-Varying Delays

The dynamic behaviors for fractional-order Cohen–Grossberg neural networks with time-varying delays (FCGNND) are studied in this paper. By introducing the Mittag-Leffler (ML) function, based on properties of fractional calculus, the differential mean-value theorem and Arzela–Ascoli theorem, we give some sufficient theorems to determine the boundedness, global Mittag-Leffler stability (GMLS) and global asymptotical [Formula: see text]-periodicity (GAP) for FCGNND. Finally, a numerical example is given to verify the effectiveness of the theorems.

Properties of fractional calculus with respect to a function and Bernstein type polynomials

This paper is divided into two stages. In the first stage, we investigated a new approach for the ‐Riemann–Liouville fractional integral and the Faa di Bruno formula for the ‐Hilfer fractional derivative. In addition, we discussed other properties involving the ‐Hilfer fractional derivative and the ‐Riemann–Liouville fractional integral. In the second stage, Bernstein polynomials involving the function are investigated and the ‐Riemann–Liouville fractional integral and ‐Hilfer fractional derivative from the Bernstein polynomials are evaluated. We also discussed the relationship between the ‐Hilfer fractional derivative with Laguerre polynomials and hypergeometric functions, and a version of the fractional mean value theorem with respect to a function. Motivated by the Bernstein polynomials, the second stage uses the Bernstein polynomials to approximate the solution of a fractional integro‐differential equation with Hilfer fractional derivative and concluding with a numerical approach with its respective graph.

Fixed-time synchronization of fractional-order complex-valued neural networks with time-varying delay via sliding mode control

Taking into account fractional-order complex-valued neural networks with time-varying delay, the issue of achieving fixed-time synchronization is discussed in this paper. By utilizing the properties of fractional calculus and fractional-order comparison principle, an improved lemma is proposed to derive the fixed-time synchronization conditions. On the basis of sliding model control and Lyapunov stability theorem, an effective sliding mode surface is constructed, which only uses the synchronization error information of FOCVNNs and is composed of fractional and integer integral terms. Further, a suitable sliding model control is constructed, which makes synchronization error converges to zero in a fixed-time. Beyond that, several sufficient conditions are posed to guarantee fixed-time synchronization of the fractional-order complex-valued neural networks and the upper bound of synchronization settling time is estimated. Finally, two numerical simulations are given to demonstrate the effectiveness of the presented theoretical results.

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.