Fractional Neutral Differential Equations Research Articles

Ulam-Hyers-Stability for nonlinear fractional neutral differential equations

We discuss Ulam-Hyers stability, Ulam-Hyers-Rassias stability and Generalized Ulam-Hyers-Rassias stability for a class of nonlinear fractional functional differential equations with delay involving Caputo fractional derivative by using Picard operator. An example is also given to show the applicability of our results.

Oscillation properties of solutions of fractional neutral differential equations

In this study, we consider a class of fractional neutral differential equations. We are going to give some new theorems that they complete and improve a number of results in the literature. Then we give an example to illustrating the main results.

Ulam-Hyers-Stability for Nonlinear Fractional Neutral Differential Equations

Ulam-Hyers-Stability for Nonlinear Fractional Neutral Differential Equations

Ulam-Hyers-Mittag-Leffler stability for nonlinear fractional neutral differential equations

In this paper, first we discuss two existence and uniqueness results for a class of nonlinear fractional functional differential equations with delay involving Caputo fractional derivatives with respect to the Chebyshev and Bielecki norms. Second, we use the Picard operator to establish Ulam-Hyers-Mittag-Leffler stability results on a compact interval. Finally, two examples are provided to illustrate our results. Bibliography: 29 titles.

Faedo–Galerkin approximate solutions of a neutral stochastic fractional differential equation with finite delay

In this work, we study a class of neutral stochastic fractional differential equation in an arbitrary separable Hilbert space H. We obtain an associated integral equation and then consider a sequence of approximate integral equations obtained by projection of considered associated integral equation onto finite dimensional space. The existence and uniqueness of solutions to every approximate integral equation are obtained by using Banach fixed point theorems and analytic semigroup theory. We show the convergence of the solutions by using Faedo–Galerkin approximations and give an example to show the effectiveness of the main theory. Finally, we provide the conclusion at the end.

Neutral Fractional Stochastic Differential Equations Driven by Rosenblatt Process

Neutral Fractional Stochastic Differential Equations Driven by Rosenblatt Process

A new class of operational matrices method for solving fractional neutral pantograph differential equations

This paper uses new fractional integration operational matrices to solve a class of fractional neutral pantograph delay differential equations. A fractional-order function space is constructed where the exact solution lies in, and a set of orthogonal bases are given. Using them, we reduce the fractional delay differential equation to algebraic equations and get the approximate solution. Finally, we give the Legendre operational matrix of fractional integration to solve the equation as an example and show the efficiency of the method.

Transportation inequalities for fractional stochastic functional differential equations driven by fractional Brownian motion

Using the method of Girsanov transformation, we establish the transportation inequalities, with respect to the uniform distance, for the law of the mild solution of fractional stochastic functional differential equations driven by fractional Brownian motion. The case of neutral fractional stochastic functional differential equations are also investigated.

Controllability of a Stochastic Neutral Functional Differential Equation Driven by a fBm

In this paper, we consider a class of Sobolev-type fractional neutral stochastic differential equations driven by fractional Brownian motion with infinite delay in a Hilbert space. When α>1-H, by the technique of Sadovskii’s fixed point theorem, stochastic calculus and the methods adopted directly from deterministic control problems, we study the approximate controllability of the stochastic system.

Existence and exponential stability for neutral stochastic fractional differential equations with impulses driven by Poisson jumps

The paper is mainly concerned with a class of neutral stochastic fractional integro-differential equation with Poisson jumps. First, the existence and uniqueness for mild solution of an impulsive stochastic system driven by Poisson jumps is established by using the Banach fixed point theorem and resolvent operator. The exponential stability in the pth moment for mild solution to neutral stochastic fractional integro-differential equations with Poisson jump is obtained by establishing an integral inequality.

Existence results for hybrid fractional neutral differential equations

We discuss the existence of solutions of initial value problems for a class of hybrid fractional neutral differential equations. To prove the main results, we use a hybrid fixed point theorem for the sum of three operators. We also derive the dependence of a solution on the initial data and present an example to illustrate the results.

Non-Instantaneous Impulsive Fractional Neutral Differential Equations with State-Dependent Delay

Non-Instantaneous Impulsive Fractional Neutral Differential Equations with State-Dependent Delay

Periodic solutions of neutral fractional differential equations

AbstractWe characterize the existence of periodic solutions for some abstract neutral functional fractional differential equations with finite delay when the underlying space is a UMD space.

Existence of nonoscillatory solutions for fractional neutral differential equations

In this paper, we present sufficient conditions for the existence of nonoscillatory solutions of the following fractional neutral functional differential equation D t α [ x ( t ) − c x ( t − τ ) ] + ∑ i = 1 m P i ( t ) x ( t − σ i ) = 0 , t ≥ t 0 , where D t α is Liouville fractional derivatives of order α ∈ [ 1 , + ∞ ) on the half-axis, c , τ , σ i ∈ ( 0 , + ∞ ) , P i ∈ C ( [ t 0 , + ∞ ) , R ) , m ≥ 1 is an integer.

S-Asymptotically $$\varvec{\omega }$$ ω -Periodic Mild Solutions of Neutral Fractional Differential Equations with Finite Delay in Banach Space

In this paper, we present some results concerning the existence and uniqueness of S-asymptotically $$\omega $$ -periodic solutions for a class of nonlinear fractional differential equations with finite delay in Banach space X. These results are new even in the case of $$X=\mathbb {R}^n$$ . Moreover, we apply the above results to study the S-asymptotically $$\omega $$ -periodicity for the fractional-order diffusion equations and the fractional-order autonomous neural networks with delay.

A Compact Difference Scheme for Solving Fractional Neutral Parabolic Differential Equation with Proportional Delay

A linearized compact finite difference scheme is constructed for solving the fractional neutral parabolic differential equation with proportional delay. By the energy method, the unconditional stability of the scheme is proved, and the convergence order of the scheme is proved to be O(τ2-α+h4). A numerical test is also conducted to validate the accuracy and efficiency of the numerical algorithm.

Symmetric-periodic solutions for some types of generalized neutral equations

The existence of symmetric-periodic outcomes for a class of fractional differential equations has been increasingly studied. Such study has used various methods such as fixed point theory, critical point theory, and approximation theory. In this work, we study the m-pseudo almost automorphic (m-P $$\Lambda \Lambda$$ ) outcomes for a category of fractional neutral differential equations. To satisfy this aim, we introduce composition results under suitable conditions and employ them to establish some extant outcomes using interpolation theory mixed with fixed point technique. Examples are illustrated.

Dependence Analysis of the Solutions on the Parameters of Fractional Neutral Delay Differential Equations

AbstractIn this paper, we discuss the dependence of the solutions on the parameters (order, initial function, right-hand function) of fractional neutral delay differential equations (FNDDEs). The corresponding theoretical results are given respectively. Furthermore, we present some numerical results that support our theoretical analysis.

Monotone iterative technique for neutral fractional differential equation with infinite delay

In this paper, we employ the method of lower and upper solutions coupled with the monotone iterative technique to obtain existence and uniqueness of extremal mild solutions to neutral fractional differential equation with infinite delay in an ordered Banach space. The results are obtained by using the theory of fractional calculus, monotone iterative technique, measure of noncompactness, and semigroup theory. Finally, we discuss an example to illustrate our results. Copyright © 2016 John Wiley & Sons, Ltd.

Faedo–Galerkin Approximation of Solution for a Nonlocal Neutral Fractional Differential Equation with Deviating Argument

In this work, we study a class of nonlocal neutral fractional differential equations with deviated argument in the separable Hilbert space. We obtain an associated integral equation and then, consider a sequence of approximate integral equations. We investigate the existence and uniqueness of the mild solution for every approximate integral equation by virtue of the theory of analytic semigroup theory via the technique of Banach fixed point theorem. Next we demonstrate the convergence of the solutions of the approximate integral equations to the solution of the associated integral equation. The Faedo–Galerkin approximation of the solution is studied and demonstrated some convergence results. Finally, we give an example.