Caputo Fractional Differential Research Articles

A New Nonlinear Integral Inequality with a Tempered Ψ–Hilfer Fractional Integral and Its Application to a Class of Tempered Ψ–Caputo Fractional Differential Equations

In this paper, the tempered Ψ–Riemann–Liouville fractional derivative and the tempered Ψ–Caputo fractional derivative of order n−1<α<n∈N are introduced for Cn−1–functions. A nonlinear version of the second Henry–Gronwall inequality for integral inequalities with the tempered Ψ–Hilfer fractional integral is derived. By using this inequality, an existence and uniqueness result and a sufficient condition for the non-existence of blow-up solutions of nonlinear tempered Ψ–Caputo fractional differential equations are proved. Illustrative examples are given.

New Results on the Solvability of Abstract Sequential Caputo Fractional Differential Equations with a Resolvent-Operator Approach and Applications

This paper aims to establish the existence and uniqueness of mild solutions to abstract sequential fractional differential equations. The approach employed involves the utilization of resolvent operators and the fixed-point theorem. Additionally, we investigate a specific example concerning a partial differential equation incorporating the Caputo fractional derivative.

Controllability Results for $$\psi $$-Caputo Fractional Differential Systems with Impulsive Effects

Controllability Results for $$\psi $$-Caputo Fractional Differential Systems with Impulsive Effects

Contributions to the Numerical Solutions of a Caputo Fractional Differential and Integro-Differential System

The primary goal of this research is to offer an efficient approach to solve a certain type of fractional integro-differential and differential systems. In the Caputo meaning, the fractional derivative is examined. This system is essential for many scientific disciplines, including physics, astrophysics, electrostatics, control theories, and the natural sciences. An effective approach solves the problem by reducing it to a pair of algebraically separated equations via a successful transformation. The proposed strategy uses first-order shifted Chebyshev polynomials and a projection method. Using the provided technique, the primary system is converted into a set of algebraic equations that can be solved effectively. Some theorems are proved and used to obtain the upper error bound for this method. Furthermore, various examples are provided to demonstrate the efficiency of the proposed algorithm when compared to existing approaches in the literature. Finally, the key conclusions are given.

Bang-Bang Property and Time-Optimal Control for Caputo Fractional Differential Systems

In this paper, by using the controllability method, a bang-bang property and a time optimal control problem for time fractional differential systems (FDS) are considered. First, we formulate our problem and prove the existence theorem. We then state and prove the bang-bang theorem. Finally, we state the optimality conditions that characterize the optimal control. Some application examples are given to illustrate our results.

Medical image segmentation model based on caputo fractional differential

Medical image segmentation model based on caputo fractional differential

Ulam-Type Stability Results for Variable Order Ψ-Tempered Caputo Fractional Differential Equations

An initial value problem for nonlinear fractional differential equations with a tempered Caputo fractional derivative of variable order with respect to another function is studied. The absence of semigroup properties of the considered variable order fractional derivative leads to difficulties in the study of the existence of corresponding differential equations. In this paper, we introduce approximate piecewise constant approximation of the variable order of the considered fractional derivative and approximate solutions of the given initial value problem. Then, we investigate the existence and the Ulam-type stability of the approximate solution of the variable order Ψ-tempered Caputo fractional differential equation. As a partial case of our results, we obtain results for Ulam-type stability for differential equations with a piecewise constant order of the Ψ-tempered Caputo fractional derivative.

Cyber secure consensus of fractional order multi-agent systems with distributed delays: Defense strategy against denial-of-service attacks

Leader-follower cyber secure consensus problem in cybersecurity is crucial as it directly addresses the need for secure and stable communication in a networked system, which is paramount in safeguarding against cyber threats like denial-of-service attacks. In this article, by utilizing control strategy with distributed delays in denial-of-service attack's presence for discrete-time multi-agent system of fractional-order, the leader-following cyber secure consensus problem is examined. Nonlinear functions have been assumed in the leader and follower equation of the system to study variations due to this function in the system. As multi-agent system is networked environment, in response to numerous threats, their security control becomes critically desirable, for example, denial of service. The consensus performance may be destabilized by resulting topologies which are caused by Denial-of-Service attacks. The connectivity between agents is destroyed, especially under connectivity-broken attacks. In order to overcome these problems, the strategy of novel defense which consists of consensus (with distributed delays) control is introduced. By using Caputo fractional difference operator sufficient criteria which includes the condition in terms LMI is derived and by using Lyapunov function approach, average dwell time and algebraic graph theory for security of addressed system's cyber secure consensus this is done for determining the obtained system of error's stability. Finally, by showing some numerical examples on the introduced systems, the effectiveness of the obtained results is determined.

Global Mittag-Leffler boundedness of nabla discrete-time fractional-order fuzzy complex-valued molecular models of mRNA and protein in regulatory mechanisms

The multi-stable dynamics behavior for molecular models of mRNA and proteins in regulatory mechanisms has received increasing attention based on these biological findings. In this work, we focus on the global Mittag–Leffler boundedness (GMLB) for discrete-time fractional-order fuzzy delayed complex-valued genetic regulatory networks (DTFFDCVGRNs). To guarantee GMLB of the addressed molecular models, various sufficient criteria, including algebraic conditions and the condition in terms of LMI, are derived based on Caputo fractional difference operator using the Lyapunov function approach and the discrete Laplace transform. In addition, estimates of various global Mittag–Leffler attraction sets (GMLASs) known as convergence balls are provided. Finally, a numerical example demonstrates the proposed theoretical results.

State estimation for discrete-time fractional-order neural networks with time-varying delays and uncertainties

In this paper, the state estimation issue for a class of discrete-time fractional-order neural networks (DFNNs) with time-varying delays and uncertainties is investigated. Based on the theoretical results of Mittag-Leffler function and Caputo fractional difference, a novel discrete-time inequality is established, which has generality compared with the previous results. By designing suitable estimator, exploiting Lyapunov method and combining inequality we establish, sufficient conditions ensuring the global asymptotical stability of estimation error system are obtained through linear matrix inequalities (LMIs). Moreover, we further discuss the case of DFNNs without uncertainties. In the end, numerical examples are utilized to validate availability of our theoretical results.

Series Solution Method for Solving Sequential Caputo Fractional Differential Equations

Computing the solution of the Caputo fractional differential equation plays an important role in using the order of the fractional derivative as a parameter to enhance the model. In this work, we developed a power series solution method to solve a linear Caputo fractional differential equation of the order q,0<q<1, and this solution matches with the integer solution for q=1. In addition, we also developed a series solution method for a linear sequential Caputo fractional differential equation with constant coefficients of order 2q, which is sequential for order q with Caputo fractional initial conditions. The advantage of our method is that the fractional order q can be used as a parameter to enhance the mathematical model, compared with the integer model. The methods developed here, namely, the series solution method for solving Caputo fractional differential equations of constant coefficients, can be extended to Caputo sequential differential equation with variable coefficients, such as fractional Bessel’s equation with fractional initial conditions.

Estimating Approximate Solutions of Non-Linear Caputo Fractional Differential Equations with Forcing Functions Using Picard's Iteration Method

Estimating Approximate Solutions of Non-Linear Caputo Fractional Differential Equations with Forcing Functions Using Picard's Iteration Method

Approximation of sequential fractional systems of Liouville–Caputo type by discrete delta difference operators

We present the Liouville–Caputo fractional difference method for the numerical evolution of the sequential differential equation of fractional order. Meanwhile, some binomial coefficients are considered in discrete fractional calculus to find and determine the corresponding sequence of continuous fractional order equations. Finally, a standard numerical test is offered in detail to demonstrate the validity of the main theorem.

System of Caputo Fractional Differential Equations with Applications to Predator and Prey Model

In this research article, we provide a methodology to solve the three systems of qth order linear Caputo fractional differential equations, where 0 < q < 1. Since the Caputo derivative is in the convolution form, we can apply the Laplace transform technique. The solution of the two linear system can be used as a tool to study the stability of the equilibrium solution of the Lotka-Volterra predator-prey model. We have referenced three system SIR model in this work. Due to the global nature of the Caputo derivative, the solution obtained is closer to the real data than the integer derivative.

Qualitative Behaviour of a Caputo Fractional Differential System

Qualitative Behaviour of a Caputo Fractional Differential System

A computational approach for a system of coupled distributed-order fractional Klein–Gordon–Schrödinger equations

In this study, a system of coupled distributed-order fractional Klein–Gordon–Schrödinger equations is introduced. The distributed-order fractional derivative is generated based on the Caputo fractional differentiation. The discrete Chebyshev polynomials are handled to construct computational approach for this system. To do this, fractional derivatives matrices (distribute-order and classical) of the expressed discrete polynomials are obtained. The intended approach is based on approximating the imaginary and real parts of the system solution by the mentioned polynomials and applying the extracted operational matrices, along with employing the collocation technique. The constructed scheme transforms the solution of the original system into solving a related algebraic system. Three test problems are examined to confirm the adequacy of the expressed collocation procedure.

Existence of Extremal Solutions for Caputo Fractional Differential Equations with Bounded Delay

Existence of Extremal Solutions for Caputo Fractional Differential Equations with Bounded Delay

Complexity analysis and discrete fractional difference implementation of the Hindmarsh–Rose neuron system

Nowadays, multistability assessment in discrete nonlinear fractional difference systems is an emotive topic. This paper introduces a novel discrete, nonequilibrium, memristor-based Hindmarsh–Rose neuron (HRN) with the Caputo fractional difference scheme. Furthermore, in a setting involving commensurate and incommensurate scenarios, the complex structure of the proposed discrete fractional model, which includes its multistability, concealed chaos and hyperchaotic attractor, is examined using a wide range of computational approaches, such as Lyapunov exponents, phase portraits, bifurcation illustrations, and the 0–1 evaluation emergence of synchronization. These evolving properties imply that the fractional discrete HRN has an undetected multistability. Ultimately, an elaborate investigation is performed to confirm the existence of unpredictability employing approximation entropy (ApEn) and the ℂ0 measurements. It is demonstrated that whenever the immediate synchronization indicates that it happens, the neurons’ interactions modify, recurring in decreasing fractional orders. Furthermore, minimizing the derivative order increases the incidence of explosions in the synchronization manifold, which is opposite to the behaviour of one nerve cell.

Algorithm for Approximate Solving of a Nonlinear Boundary Value Problem for Generalized Proportional Caputo Fractional Differential Equations

In this paper an algorithm for approximate solving of a boundary value problem for a nonlinear differential equation with a special type of fractional derivative is suggested. This derivative is called a generalized proportional Caputo fractional derivative. The new algorithm is based on the application of the monotone-iterative technique combined with the method of lower and upper solutions. In connection with this, initially, the linear fractional differential equation with a boundary condition is studied, and its explicit solution is obtained. An appropriate integral fractional operator for the nonlinear problem is constructed and it is used to define the mild solutions, upper mild solutions and lower mild solutions of the given problem. Based on this integral operator we suggest a scheme for obtaining two monotone sequences of successive approximations. Both sequences consist of lower mild solutions and lower upper solutions of the studied problem, respectively. The monotonic uniform convergence of both sequences to mild solutions is proved. The algorithm is computerized and applied to a particular example to illustrate the theoretical investigations.

Discrete fuzzy complex-valued function and complex fuzzy Caputo fractional difference equations

In this paper, we introduce a notion of complex compact intervals on which Minkowski addition and scalar multiplication operators are well defined to form the space of complex fuzzy numbers whose α-cuts are complex intervals. Some basic fuzzy partial ordering relations among the complex numbers with “≤C”, the complex intervals with “≪”, the complex fuzzy numbers with “⪯C˜” and the discrete fuzzy complex-valued functions with ≲ are studied. Based on these fundamental properties, we introduce a concept of Caputo fractional sum and difference for the discrete fuzzy complex-valued functions and establish some of their basic results. Moreover, by introducing the complex monotone iterative technique combined with the method of upper and lower solutions, the existence of extremal solutions for discrete complex fuzzy Caputo fractional difference equations is investigated. In each section, several examples are presented to show the feasibility of our obtained results.