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.     

New exact solutions of differential equations derived by fractional calculus

Fractional calculus generalizes the derivative and antiderivative operations dⁿ/dzⁿ of differential and integral calculus from integer orders n to the entire complex plane. Methods are presented for using this generalized calculus with Laplace transforms of complex-order derivatives to solve analytically many differential equations in physics, facilitate numerical computations, and generate new infinite-series representations of functions. As examples, new exact analytic solutions of differential equations, including new generalized Bessel equations with complex-power-law variable coefficients, are derived.     

Asymptotic stability of linear multistep methods for nonlinear neutral delay differential equations

This paper is concerned with the numerical solution to initial value problems of nonlinear delay differential equations of neutral type. We use A-stable linear multistep methods to compute the numerical solution. The asymptotic stability of the A-stable linear multistep methods when applied to the nonlinear delay differential equations of neutral type is investigated, and it is shown that the A-stable linear multistep methods with linear interpolation are GAS-stable. We validate our conclusions by numerical experiments.     

Asymptotic behavior of delay differential equations with instantaneously terms

This paper deals with scalar delay differential equation with instantaneously term

     

Asymptotic behavior of certain delay differential equations with forcing term

We study the asymptotic behavior of solutions of the following forced delay differential equation:

Some sufficient conditions that guarantee every solution of the equation to converge to zero are obtained. The results obtained are applied to some well-known delay differential ecological equations with forcing term.     

Simplified reproducing kernel method for fractional differential equations with delay

This paper is devoted to the numerical scheme for the delay initial value problems of a fractional order. The main idea of this method is to establish a novel reproducing kernel space that satisfies the initial conditions. Based on the properties of the new reproducing kernel space, the simplified reproducing kernel method (SRKM for short) is applied to obtain accurate approximation. The Schmidt orthogoralization process which requires a large number of calculation is less likely to be employed. Numerical experiments are provided to illustrate the performance of the method.     

About stability of delay differential equations with square integrable level of stochastic perturbations

The long term behavior of solutions of stochastic delay differential equations with a fading stochastic perturbations is investigated. It is shown that if the level of stochastic perturbations fades on the infinity, for instance, if it is given by square integrable function, then an asymptotically stable deterministic system remains to be an asymptotically stable (in mean square).     

Existence of fractional differential equations

Consider the fractional differential equation

Dαx=f(t,x),     

Asymptotic behavior of linear fractional stochastic differential equations with time-varying delays

In this paper we give a sufficient condition for the exponential asymptotic behavior of solutions of a general class of linear fractional stochastic differential equations with time-varying delays. Our obtained results allow us to employ the theories developed for the deterministic systems and to illustrate this, some examples are provided.