Impulsive Fractional Differential Equations Research Articles

The existence of positive solutions for φ-Hilfer fractional differential equation with random impulses and boundary value conditions

In this paper, we consider a class of impulsive fractional differential equations involving φ-Hilfer fractional derivatives. Since the φ-Hilfer fractional derivative depends on the kernel φ, for a suitable choice of the kernel φ, our results in this paper can be applied to most of the fractional derivatives. First, we give a correspondence in this article between our problem and a Volterra-type integral equation. Then, sufficient conditions are given to ensure the existence of positive solutions and by applying the cone stretch and cone compression immobilization theorem and the Green function to study the existence of positive solutions for the problem. Finally, an example is given to illustrate the viability of the main theories.

α-Confluent-hyper-geometric stability of ξ-Hilfer impulsive nonlinear fractional Volterra integro-differential equation

The purpose of this work is to investigate the necessary conditions for the existence and uniqueness of solutions, and to introduce a new idea of α-confluent-hyper-geometric stability of an impulsive fractional differential equation with ξ-Hilfer fractional derivative. We use the Diaz–Margolis fixed point theorem to achieve this and illustrate the result with an example.

Impulsive fractional differential equations with state-dependent delay involving the Caputo-Hadamard derivative

In this paper, we investigate the existence of solutions for a class of initial value problems for impulsive Caputo-Hadamard fractional differential equations with state-dependent delay.

The explicit solution of a class of higher order impulsive fractional differential equation involving Hadamard fractional derivative

In this paper, we obtain the explicit solution of a class of singular impulsive fractional differential equation of order [Formula: see text] involving the Hadamard fractional derivatives. In particular, the result is given in the form of necessary and sufficient conditions about the explicit solution. The proof of the main theorem is obtained by converting boundary value problems into integral equations and applying analytical techniques. Finally, some examples explain the application of the main theorem in this paper.

The new general solution for a class of fractional-order impulsive differential equations involving the Riemann-Liouville type Hadamard fractional derivative

<abstract><p>In this paper, the new general solution for a class of higher-order impulsive fractional differential equations (IFDEs) involving the Riemann-Liouville (R-L) type Hadamard fractional derivative (FD) is presented. Specifically, the necessary and sufficient conditions of the solution are obtained by converting boundary value problems (BVPs) into integral equations and applying analytical techniques. The results in the paper provide a new method for converting BVPs or initial value problems (IVPs) for IFDEs to integral equations. Finally, some examples are devoted to explaining the application of the theorem.</p></abstract>

Existence and stability results for a coupled system of impulsive fractional differential equations with Hadamard fractional derivatives

<abstract><p>The purpose of this study is to give some findings on the existence, uniqueness, and Hyers-Ulam stability of the solution of an implicit coupled system of impulsive fractional differential equations possessing a fractional derivative of the Hadamard type. The existence and uniqueness findings are obtained using a fixed point theorem of the type of Kransnoselskii. In keeping with this, many forms of Hyers-Ulam stability are examined. Ultimately, to support main results, an example is provided.</p></abstract>

A study on controllability for Hilfer fractional differential equations with impulsive delay conditions

<abstract><p>This article focuses on the controllability of a Hilfer fractional impulsive differential equation with indefinite delay. We analyze our major outcomes using fractional calculus, the measure of non-compactness and a fixed-point approach. Finally, an example is provided to show the theory.</p></abstract>

Forced oscillation of impulsive fractional partial differential equations

In this paper, we investigate the forced oscillation of impulsive fractional partial differential equations. Some sufficient conditions are established which guarantee that the oscillation of the solutions by applying the integral transformation technique and differential inequality method. Also an example is shown to illustrate the effectiveness of our main results.

Existence and Ulam Type Stability for Impulsive Fractional Differential Systems with Pure Delay

Through literature retrieval and classification, it can be found that for the fractional delay impulse differential system, the existence and uniqueness of the solution and UHR stability of the fractional delay impulse differential system are rarely studied by using the polynomial function of the fractional delay impulse matrix. In this paper, we firstly introduce a new concept of impulsive delayed Mittag–Leffler type solution vector function, which helps us to construct a representation of an exact solution for the linear impulsive fractional differential delay equations (IFDDEs). Secondly, by using Banach’s and Schauder’s fixed point theorems, we derive some sufficient conditions to guarantee the existence and uniqueness of solutions of nonlinear IFDDEs. Finally, we obtain the Ulam–Hyers stability (UHs) and Ulam–Hyers–Rassias stability (UHRs) for a class of nonlinear IFDDEs.

A study on the mild solution of special random impulsive fractional differential equations

In this article, we deal with mild solution of special random impulsive fractional differential equations. Initially, we present the existence of the mild solution via Leray-Schauder fixed point method. After that, we establish the exponential stability of the system. Finally, we give examples to illustrate the effectiveness of the theoretical results.

CLASSICAL SOLUTIONS FOR A BVP FOR A CLASS IMPULSIVE FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS

In this paper, we investigate a BVP for a class impulsive fractional partial differential equations. We propose a new topological approach to prove the existence of at least one classical solution and at least two nonnegative classical solutions. The arguments are based upon recent theoretical results.

A Numerical Computation for an Impulsive Fractional Differential Equation with a Deviated Argument

Symmetry analysis is an effective tool for understanding differential equations, particularly when analyzing equations derived from mathematical concepts. This paper is concerned with an impulsive fractional differential equation (IFDE) with a deviated argument. We implement two techniques, the Adomian decomposition method (ADM) and the fractional differential transform method (FDTM), for solving IFDEs. In these schemes, we obtain the solutions in the form of a convergent power series with easily computed components. This paper also discusses the existence and uniqueness of solutions using the Banach contraction principle. This paper presents a numerical comparison between the two methods for solving IFDEs. We illustrate the proposed methods with a few examples and find their numerical solutions. Moreover, we show the graph of numerical solutions via MATLAB. The numerical results demonstrate that the ADM approach is quite accurate and readily implemented.

Sensitivity analysis for a fractional stochastic differential equation with $$S^{p}$$-weighted pseudo almost periodic coefficients and infinite delay

In this paper, we study the parameter sensitivity analysis a class of fractional impulsive stochastic differential equation with \(S^{p}\)-weighted pseudo almost periodic coefficients and infinite delay in Hilbert spaces. Firstly, a more appropriate concept of piecewise \(S^{p}\)-weighted pseudo almost periodic in distribution for stochastic processes of class \(\infty \) is introduced. The existence of piecewise weighted pseudo almost periodic mild solutions in distribution is presented by using \(\alpha \)-order fractional resolvent operator theory, the stochastic analysis techniques and a fixed-point theorem. Secondly, the sensitivity properties of these infinite delay systems under non-Lipschitz conditions is also obtained. Finally, two examples are given for demonstration.

Mathematical analysis of impulsive fractional differential inclusion of pantograph type

In this article, we formulate fractional differential inclusion of pantograph type (IFDIP), incorporating impulsive behavior of the solution. The boundary conditions taken into account are nonlocal in nature. We will consider the convex problem and prove the Filippov–Wazewski‐type theorem. Moreover, existence of solution, uniqueness of a solution, and the topological properties of the solution's set will be examined for the problem under consideration. In the second part, the study will be confined to the second‐order impulsive fractional differential equation of pantograph type. For certain geometric characteristics of the solution's set, Aronszajn–Browder–Gupta‐type results will be explored for the newly introduced differential equation. Also, it will prove the existence of solution for the first‐order fractional differential equation of pantograph type having impulsive behavior of the solution.

INVESTIGATION OF INTEGRAL BOUNDARY VALUE PROBLEM WITH IMPULSIVE BEHAVIOR INVOLVING NON-SINGULAR DERIVATIVE

This paper is devoted to investigating a class of impulsive fractional order differential equations (FODEs) with integral boundary condition. For the proposed paper, we use non-singular type derivative of fractional order which has been introduced by Atangana, Baleanu and Caputo (ABC). The aforesaid type problems have numerous applications in fluid mechanics and hydrodynamics to model various problems of flow phenomenons. We establish some sufficient conditions for the existence and uniqueness of solution to the proposed problem by using classical fixed point results due to Banach and Krasnoselskii. Further, on using tools of the nonlinear analysis, sufficient conditions are developed for Hyers–Ulam (HU) type stability results. A pertinent example is given to justify our results.

Existence Results of Fuzzy Delay Impulsive Fractional Differential Equation by Fixed Point Theory Approach

The main aim of this article is to study controllability and existence of solution of fuzzy delay impulsive fractional nonlocal integro-differential equation in the sense of Caputo operator. The existence and uniqueness of the solution have been carried out with the help of the Banach fixed point theorem. Moreover, for fuzzy fractional differential equations (FFDEs) driven by the Liu process, this present work introduced a concept of stability in credibility space. Finally, efficient examples are presented to demonstrate the main theoretical findings.

Numerical solution of fractional dynamical systems with impulsive effects

This paper proposes an effective numerical scheme for solving impulsive fractional differential equations. For this purpose, Hermite interpolation is used to approximate fractional-order integrals. The proposed methods convergence analysis is studied in detail by bounding the approximation error. Finally, the application and performance of the presented method are illustrated in two practical examples, including the impulsive control of the family of Lorenz systems, and the obtained results are compared with an existing method.

Convergence on iterative learning control of Hilfer fractional impulsive evolution equations

AbstractIn this paper, impulsive fractional differential equations with Hilfer fractional derivatives of order and type is considered. Convergence analysis of ‐type and ‐type open‐loop iterative learning scheme is studied in the sense of ‐norm. Examples are provided to explain the theory developed.

Impulsive Consensus of Fractional-Order Takagi–Sugeno Fuzzy Multiagent Systems With Average Dwell Time Approach and Its Applications: Achieving Finite-Time Consensus

The Takagi–Sugeno (T–S) fuzzy-based impulsive consensus problem of a fractional-order multiagent system (FOMAS) with an average dwell time (ADT) is investigated. FOMASs are nonlinear systems, and they are modeled as linear subsystems using a T–S fuzzy model. We study a T–S fuzzy FOMAS subject to a class of impulse time sequences with the ADT approach. In this article, an impulsive control scheme is proposed to make the tracking error converge in a finite-time consensus into a small neighborhood of origin. Based on impulsive fractional differential equations theory, the Lyapunov functional approach, and the ADT technique, an impulsive controller is designed to achieve finite-time consensus of a T–S fuzzy FOMAS. Finally, through numerical as well as practical examples, the effectiveness and superiority of the proposed approach are validated.

On Hilfer-Type Fractional Impulsive Differential Equations

Using the Schauder fixed point theorem, we prove the existence of impulsive fractional differential equations using Hilfer fractional derivative and nearly sectorial operators in this paper. We’ve gone over the two scenarios where the related semigroup is compact and noncompact for this purpose. We also go over an example to back up the main points.