Fractional-order Delay Differential Equations Research Articles

Comment on “On the oscillation of fractional-order delay differential equations with constant coefficients” [Commun Nonlinear Sci 19(11) (2014) 3988–3993

In this paper, some necessary and sufficient conditions are established for the oscillation of fractional-order delay differential equations with constant coefficients in the form:x′(t)+pD-αx(t)+qx(t-τ)=0,0<α<1,where D-αx(t) denotes the Liouville fractional derivatives on half-axis R+, p,q and τ are real constants. And also some oscillation results for the fractional-order neutral delay differential equations with constant coefficients in the form:-D-α(x(t)+rx(t-δ))+qx(t-τ)=0,0<α<1,where D-αx(t) denotes the Liouville fractional derivatives on half-axis R+,q, r,τ and δ are real constants. The results obtained here correct and improve some known results in Bolat (2014).

Modified Laguerre Wavelets Method for delay differential equations of fractional-order

In this article, Laguerre Wavelets Method (LWM) is proposed and combined with steps Method to solve linear and nonlinear delay differential equations of fractional-order. Computational work is fully supportive of compatibility of proposed algorithm and hence the same may be extended to other physical problems also. A very high level of accuracy explicitly reflects the reliability of this scheme for such problems.

Ulam–Hyers–Mittag-Leffler stability of fractional-order delay differential equations

In this paper, we first prove two existence and uniqueness results for fractional-order delay differential equation with respect to Chebyshev and Bielecki norms. Secondly, we prove the above equation is Ulam–Hyers–Mittag-Leffler stable on a compact interval. Finally, two examples are also provided to illustrate our results.

On the oscillation of fractional-order delay differential equations with constant coefficients

In this manuscript, some oscillation results are given including sufficient conditions or necessary and sufficient conditions for the oscillation of fractional-order delay differential equations with constant coefficients. For this, α-exponential function which is a kind of functions that play the same role of the classical exponential functions and Laplace transformation formulations of fractional-order derivatives are used.

The Oscillation of a Class of the Fractional-Order Delay Differential Equations

Several oscillation results are proposed including necessary and sufficient conditions for the oscillation of fractional-order delay differential equations with constant coefficients, the sufficient or necessary and sufficient conditions for the oscillation of fractional-order delay differential equations by analysis method, and the sufficient or necessary and sufficient conditions for the oscillation of delay partial differential equation with three different boundary conditions. For this,α-exponential function which is a kind of functions that play the same role of the classical exponential functions of fractional-order derivatives is used.

Stability analysis of a class of fractional delay differential equations

In this paper we analyse stability of nonlinear fractional order delay differential equations of the form $D^{\alpha} y(t) = a f\left(y(t-\tau)\right) - b y(t)$ , where D α is a Caputo fractional derivative of order 0 < α ≤ 1. We describe stability regions using critical curves. To explain the proposed theory, we discuss fractional order logistic equation with delay.

Analytical and numerical methods for the stability analysis of linear fractional delay differential equations

In this paper, several analytical and numerical approaches are presented for the stability analysis of linear fractional-order delay differential equations. The main focus of interest is asymptotic stability, but bounded-input bounded-output (BIBO) stability is also discussed. The applicability of the Laplace transform method for stability analysis is first investigated, jointly with the corresponding characteristic equation, which is broadly used in BIBO stability analysis. Moreover, it is shown that a different characteristic equation, involving the one-parameter Mittag-Leffler function, may be obtained using the well-known method of steps, which provides a necessary condition for asymptotic stability. Stability criteria based on the Argument Principle are also obtained. The stability regions obtained using the two methods are evaluated numerically and comparison results are presented. Several key problems are highlighted.

A numerical algorithm for stability testing of fractional delay systems

This paper presents an effective numerical algorithm for testing the BIBO stability of fractional delay systems described by fractional-order delay-differential equations. It is based on using Cauchy's integral theorem and solving an initial-value problem. The algorithm has a reliable result which is illustrated by several examples, and hence is practically useful in the analysis and design of feedback control for both integer- and fractional-order systems having time delays.