Stability Of Fractional Differential Equations Research Articles

New Results on the Ulam–Hyers–Mittag–Leffler Stability of Caputo Fractional-Order Delay Differential Equations

The author considers a nonlinear Caputo fractional-order delay differential equation (CFrDDE) with multiple variable delays. First, we study the existence and uniqueness of the solutions of the CFrDDE with multiple variable delays. Second, we obtain two new results on the Ulam–Hyers–Mittag–Leffler (UHML) stability of the same equation in a closed interval using the Picard operator, Chebyshev norm, Bielecki norm and the Banach contraction principle. Finally, we present three examples to show the applications of our results. Although there is an extensive literature on the Lyapunov, Ulam and Mittag–Leffler stability of fractional differential equations (FrDEs) with and without delays, to the best of our knowledge, there are very few works on the UHML stability of FrDEs containing a delay. Thereby, considering a CFrDDE containing multiple variable delays and obtaining new results on the existence and uniqueness of the solutions and UHML stability of this kind of CFrDDE are the important aims of this work.

Fractional Fourier Transform and Ulam Stability of Fractional Differential Equation with Fractional Caputo-Type Derivative

In this paper, we study the Ulam-Hyers-Mittag-Leffler stability for a linear fractional order differential equation with a fractional Caputo-type derivative using the fractional Fourier transform. Finally, we provide an enumeration of the chemical reactions of the differential equation.

Existence and Hyers–Ulam Stability for a Multi-Term Fractional Differential Equation with Infinite Delay

This paper is devoted to investigating one type of nonlinear two-term fractional order delayed differential equations involving Caputo fractional derivatives. The Leray–Schauder alternative fixed-point theorem and Banach contraction principle are applied to analyze the existence and uniqueness of solutions to the problem with infinite delay. Additionally, the Hyers–Ulam stability of fractional differential equations is considered for the delay conditions.

The Ulam Stability of Fractional Differential Equation with the Caputo-Fabrizio Derivative

The aim of this paper is to establish the Ulam stability of the Caputo-Fabrizio fractional differential equation with integral boundary condition. We also present the existence and uniqueness results of the solution for the Caputo-Fabrizio fractional differential equation by Krasnoselskii’s fixed point theorem and Banach fixed point theorem. Some examples are provided to illustrate our theorems.

Hyers-Ulam-Mittag-Leffler stability of fractional differential equations with two caputo derivative using fractional fourier transform

<abstract><p>In this paper, we discuss standard approaches to the Hyers-Ulam Mittag Leffler problem of fractional derivatives and nonlinear fractional integrals (simply called nonlinear fractional differential equation), namely two Caputo fractional derivatives using a fractional Fourier transform. We prove the basic properties of derivatives including the rules for their properties and the conditions for the equivalence of various definitions. Further, we give a brief basic Hyers-Ulam Mittag Leffler problem method for the solving of linear fractional differential equations using fractional Fourier transform and mention the limits of their usability. In particular, we formulate the theorem describing the structure of the Hyers-Ulam Mittag Leffler problem for linear two-term equations. In particular, we derive the two Caputo fractional derivative step response functions of those generalized systems. Finally, we consider some physical examples, in the particular fractional differential equation and the fractional Fourier transform.</p></abstract>

Existence and Ulam stability for fractional differential equations of mixed Caputo-Riemann derivatives

<abstract><p>In this paper, we study the existence, uniqueness, and stability theorems of solutions for a differential equation of mixed Caputo-Riemann fractional derivatives with integral initial conditions in a Banach space. Our analysis is based on an application of the Shauder fixed point theorem with Ulam-Hyers and Ulam-Hyers-Rassias theorems. A couple of examples are presented to illustrate the obtained results.</p></abstract>

Stability of Fractional Differential Equations with New Generalized Hattaf Fractional Derivative

This paper aims to study the stability of fractional differential equations involving the new generalized Hattaf fractional derivative which includes the most types of fractional derivatives with nonsingular kernels. The stability analysis is obtained by means of the Lyapunov direct method. First, some fundamental results and lemmas are established in order to achieve the goal of this study. Furthermore, the results related to exponential and Mittag–Leffler stability existing in recent studies are extended and generalized. Finally, illustrative examples are presented to show the applicability of our main results in some areas of science and engineering.

A fractional differential equation with multi-point strip boundary condition involving the Caputo fractional derivative and its Hyers\u2013Ulam stability

In this work, we investigate the existence, uniqueness, and stability of fractional differential equation with multi-point integral boundary conditions involving the Caputo fractional derivative. By utilizing the Laplace transform technique, the existence of solution is accomplished. By applying the Bielecki-norm and the classical fixed point theorem, the Ulam stability results of the studied system are presented. An illustrative example is provided at the last part to validate all our obtained theoretical results.

Fractional Fourier transform and stability of fractional differential equation on Lizorkin space

In the current study, we conduct an investigation into the Hyers–Ulam stability of linear fractional differential equation using the Riemann–Liouville derivatives based on fractional Fourier transform. In addition, some new results on stability conditions with respect to delay differential equation of fractional order are obtained. We establish the Hyers–Ulam–Rassias stability results as well as examine their existence and uniqueness of solutions pertaining to nonlinear problems. We provide examples that indicate the usefulness of the results presented.

On Hyers-Ulam-Rassias stability of fractional differential equations with Caputo derivative

On Hyers-Ulam-Rassias stability of fractional differential equations with Caputo derivative

Discrete almost maximal regularity and stability for fractional differential equations in Lp([0, 1], Ω)

The present paper is devoted to the study of discrete almost maximal regularity and stability of the difference schemes of nonhomogeneous fractional evolution equations. Using the discretization method of the fractional derivative proposed by Ashyralyev, which actually is the same as the Grünwald-Letnikov approximation for the fractional derivative, the discrete almost maximal regularity and stability of the implicit difference scheme in Lτnp([0,1],Ωn) spaces are established. For the explicit difference scheme, the expression of the solution is obtained. Then the discrete almost maximal regularity and stability of the explicit difference scheme in Lτnp([0,1],Ωn) spaces are achieved as well.

Existence and stability of fractional differential equations involving generalized Katugampola derivative

Present article deals with existence and stability results for a class of fractional differential equations involving generalized Katugampola derivative. Some fixed point theorems are used to obtain the results and enlightening examples of obtained result are also given.

On the Stability of Fractional Differential Equations Involving Generalized Caputo Fractional Derivative

This work presents the results of the global existence for fractional differential equations involving generalized Caputo derivative with the case of the fractional order derivative α∈1,2. In addition, the Ulam–Hyers–Mittag-Leffler stability of the given problems is also established.

Existence results and the Ulam Stability for fractional differential equations with hybrid proportional-Caputo derivatives

Existence results and the Ulam Stability for fractional differential equations with hybrid proportional-Caputo derivatives

On Hyers–Ulam Stability for Fractional Differential Equations Including the New Caputo–Fabrizio Fractional Derivative

In this paper, we study the stability in the sense of Hyers–Ulam for the following fractional differential equations including the new Caputo–Fabrizio fractional derivative: $$\begin{aligned} \left( ^{CF}D^{\alpha }y\right) \left( x\right) =f\left( x\right) \quad \qquad \quad \end{aligned}$$ and $$\begin{aligned} \left( ^{CF}D^{\alpha } y \right) \left( x\right) -\lambda y\left( x\right) =f\left( x\right) . \end{aligned}$$ Finally, two examples are given to illustrate our results.

Stability region of fractional differential systems with Prabhakar derivative

This paper analyzes the stability of fractional differential equations with Prabhakar derivative, which is a generalization of the fractional differential equation with Caputo and Riemann-Liouville derivative. As a result, the sufficient condition for asymptotic stability has been obtained by studying the eigenvalues of system related matrix and the position of these eigenvalues in the complex plane. To show the application of the results a two-dimensional predator–prey model has been studied with Prabhakar derivative and the dynamic behavior has been extracted. Then, a numerical method along with its error has been presented for solving differential equations with Prabhakar derivative. The predator–prey model simulation has been conducted by using this numerical method.

On Stability of Fractional Differential Equations with Lyapunov Functions

We discuss the asymptotic stability of autonomous nonlinear fractional order systems, in which the state equations contain integer derivative and fractional order. We use the Lyapunov's second method to derive some sufficient conditions to ensure asymptotic stability of nonlinear fractional order differential equations. We also give two examples in order to consolidate the obtained results.

Mittag-Leffler input stability of fractional differential equations and its applications

This paper addresses the Mittag-Leffler input stability of the fractional differential equations with exogenous inputs. We continuous the first note. We discuss three properties of the Mittag-Leffler input stability: converging-input converging-state, bounded-input bounded-state, and Mittag-Leffler stability of the unforced fractional differential equation. We present the Lyapunov characterization of the Mittag-Leffler input stability, and conclude by introducing the fractional input stability for delay fractional differential equations, and we provide its Lyapunov-Krasovskii characterization. Several examples are treated to highlight the Mittag-Leffler input stability.

Stability analysis of fractional differential equations with unknown parameters

In this paper, the stability of fractional differential equations (FDEs) with unknown parameters is studied. Using the graphical based D-decomposition method, the parametric stability analysis of FDEs is investigated without complicated mathematical analysis. To achieve this, stability boundaries are obtained firstly by a conformal mapping from s-plane to parameter space composed by unknown parameters of FDEs, and then the stability region set depending on the unknown parameters is found. The applicability of the presented method is shown considering some benchmark equations, which are often used to verify the results of a new method. Simulation examples show that the method is simple and give reliable stability results.

Stability of Fractional Differential Equations without Singular Kernal

In this paper, we establish four types of Ulam stability, namely Ulam-Hyers stability, generalized Ulam-Hyers stability, Ulam-Hyers-Rassias and generalized Ulam-Hyers-Rassias stability for differential equation of fractional order.