Taylor wavelet method for fractional delay differential equations

We present a new numerical method for solving fractional delay differential equations. The method is based on Taylor wavelets. We establish an exact formula to determine the Riemann–Liouville fractional integral of the Taylor wavelets. The exact formula is then applied to reduce the problem of solving a fractional delay differential equation to the problem of solving a system of algebraic equations. Several numerical examples are presented to show the applicability and the effectiveness of this method.

     

Robust stability of uncertain stochastic differential delay equations

In this paper we first discuss the robust stability of uncertain linear stochastic differential delay equations. We then extend the theory to cope with the robust stability of uncertain semi-linear stochastic differential delay equations. We shall also give several examples to illustrate our theory.     

On some fractional evolution equations

In this paper the solutions of some evolution equations with fractional orders in a Banach space are considered. Conditions are given which ensure the existence of a resolvent operator for an evolution equation in a Banach space.     

On stability in distribution of stochastic differential delay equations with Markovian switching

This paper provides a new sufficient condition for stability in distribution of stochastic differential delay equations with Markovian switching (SDDEs). It can be considered as an improvement to the result given by Yuan C. et al. in [6].     

On fractional integro-differential equations with state-dependent delay

In this paper we provide sufficient conditions for the existence of mild solutions for a class of fractional integro-differential equations with state-dependent delay. A concrete application in the theory of heat conduction in materials with memory is also given.     

Stabilization of hybrid neutral stochastic differential delay equations by delay feedback control

This paper is concerned with the problem of exponential mean-square stabilization of hybrid neutral stochastic differential delay equations with Markovian switching by delay feedback control. A delay feedback controller is designed in the drift part so that the controlled system is mean-square exponentially stable. We discussed two types of structure controls; that is, state feedback and output injection. The stabilization criteria are derived in terms of linear matrix inequalities.     

Solving fractional pantograph delay differential equations via fractional-order Boubaker polynomials

In the current study, we introduce fractional-order Boubaker polynomials related to the Boubaker polynomials to achieve the numerical result for pantograph differential equations of fractional order in any arbitrary interval. The features of these polynomials are exploited to construct the new fractional integration and pantograph operational matrices. Then these matrices and least square approximation method are used to reorganize the problem to a nonlinear equations system which can be resolved by means of the Newton’s iterative method. The brief discussion about errors of the used estimations is deliberated and, finally, some examples are included to demonstrate the validity and applicability of our method.

     

Exponential stability of a class of impulsive stochastic delay partial differential equations driven by a fractional Brownian motion

Several new criteria on exponential stability of a class of impulsive stochastic delay partial differential equations with a fractional Brownian motion are presented by employing the variation of parameters formula and using an inequality technique. The criteria we provided in this paper do not assume the condition that the corresponding continuous stochastic system is stable, so this criteria can be applied to stabilize unstable systems. Moreover, the exponential convergence rate is estimated. Some examples are given to illustrate our main results.     

On solving elliptic stochastic partial differential equations

A model elliptic boundary value problem of second order, with stochastic coefficients described by the Karhunen–Loève expansion is addressed. This problem is transformed into an equivalent deterministic one. The perturbation method and the method of successive approximations is analyzed. Rigorous error estimates in the framework of Sobolev spaces are given.     

Stability of linear stochastic delay differential equations with infinite Markovian switchings

This paper investigates the stability of linear stochastic delay differential equations with infinite Markovian switchings. Some novel exponential stability criteria are first established based on the generalized It formula and linear matrix inequalities. Then, a new sufficient condition is proposed for the equivalence of 4 stability definitions, namely, asymptotic mean square stability, stochastic stability, exponential mean square stability with conditioning, and exponential mean square stability. In particular, our results generalize and improve some of the previous results. Finally, two examples are given to illustrate the effectiveness of the proposed results.     

New criteria on exponential stability of neutral stochastic differential delay equations

Neutral stochastic differential delay equations (NSDDEs) have recently been studied intensively (see e.g. [V.B. Kolmanovskii, V.R. Nosov, Stability and Periodic Modes of Control Systems with Aftereffect, Nauka, Moscow, 1981; X. Mao, Exponential stability in mean square of neutral stochastic differential functional equations, Systems Control Lett. 26 (1995) 245–251; X. Mao, Razumikhin type theorems on exponential stability of neutral stochastic functional differential equations, SIAM J. Math. Anal. 28(2) (1997) 389–401; X. Mao, Stochastic Differential Equations and Their Applications, Horwood Publishing, Chichester, 1997]). More recently, Mao [Asymptotic properties of neutral stochastic differential delay equations, Stochastics and Stochastics Rep. 68 (2000) 273–295] provided with some useful criteria on the exponential stability for NSDDEs. However, the criteria there require not only the coefficients of the NSDDEs to obey the linear growth condition but also the time delay to be a constant. One of our aims in this paper is to remove these two restrictive conditions. Moreover, the key condition on the diffusion operator associated with the underlying NSDDE will take a much more general form. Our new stability criteria not only cover many highly non-linear NSDDEs with variable time delays but they can also be verified much more easily than the known criteria.     

Stability in distribution of stochastic differential delay equations with Markovian switching

In this paper we discuss stochastic differential delay equations with Markovian switching. Such an equation can be regarded as the result of several stochastic differential delay equations switching from one to another according to the movement of a Markov chain. The aim of this paper is to investigate the stability in distribution of the equations.     

The improved split-step backward Euler method for stochastic differential delay equations

A new, improved split-step backward Euler method is introduced and analysed for stochastic differential delay equations (SDDEs) with generic variable delay. The method is proved to be convergent in the mean-square sense under conditions (Assumption 3.1) that the diffusion coefficient g(x, y) is globally Lipschitz in both x and y, but the drift coefficient f(x, y) satisfies the one-sided Lipschitz condition in x and globally Lipschitz in y. Further, the exponential mean-square stability of the proposed method is investigated for SDDEs that have a negative one-sided Lipschitz constant. Our results show that the method has the unconditional stability property, in the sense, that it can well reproduce stability of the underlying system, without any restrictions on stepsize h. Numerical experiments and comparisons with existing methods for SDDEs illustrate the computational efficiency of our method.     

On global solutions to fractional functional differential equations with infinite delay in Fréchet spaces

In this paper, we investigate global uniqueness results for fractional functional differential equations with infinite delay in Fréchet spaces. We shall rely on a nonlinear alternative of Leray-Schauder type in Fréchet spaces due to Frigon and Granas. The results are obtained by using the α-resolvent family (S_α(t))_t≥0 on a complex Banach space X combined with the above-mentioned fixed point theorem. As an application, a controllability result with one parameter is also provided to illustrate the theory.     

Existence and uniqueness of mild solutions for abstract delay fractional differential equations

In this paper, by means of solution operator approach and contraction mapping theorem, the existence and uniqueness of mild solutions for a class of abstract delay fractional differential equations are obtained.     

Robust stability and controllability of stochastic differential delay equations with Markovian switching

In this paper, we investigate the almost surely asymptotic stability for the nonlinear stochastic differential delay equations with Markovian switching. Some sufficient criteria on the controllability and robust stability are also established for linear stochastic differential delay equations with Markovian switching.