A Legendre-based computational method for solving a class of Itô stochastic delay differential equations

Numerical Algorithms - This paper provides a numerical method for solving a class of Itô stochastic delay differential equations (SDDEs). The method’s novelty is its use of the spectral...     

On the asymptotic stability ofθ-methods for delay differential equations

Stability regions of -methods for the linear delay differential test equations

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.     

Stability of linear impulsive Itô differential equations with bounded delays

We use the method of “model” equations to study the exponential p-stability (2 ≤ p < ∞) of the trivial solution with respect to the initial function for a linear impulsive system of Itô differential equations with bounded delays. The specific form of the equation and the method used permit one to analyze the stability of solutions starting from an arbitrary point of the half-line [0,∞) and obtain constructive sufficient conditions in terms of the parameters of the equations to be studied.     

Exponential stability of equidistant Euler–Maruyama approximations of stochastic differential delay equations

Our aim is to study under what conditions the exact and numerical solution (based on equidistant nonrandom partitions of integration time-intervals) to a stochastic differential delay equation (SDDE) share the property of mean-square exponential stability. Our approach is trying to avoid the use of Lyapunov functions or functionals. We show that under a global Lipschitz assumption an SDDE is exponentially stable in mean square if and only if for some sufficiently small stepsize Δ$\Delta$ the Euler–Maruyama (EM) method is exponentially stable in mean square. We then replace the global Lipschitz condition with a finite-time convergence condition and establish the same “if and only if” result. The important feature of this result is that it transfers the asymptotic problem into a finite-time convergence problem. Replacing the exact and EM numerical solution with stochastic processes, we also discuss whether a family of stochastic processes share the stability property. This new approach allows us to discuss (i) whether a family of SDDEs share the stability property, and (ii) whether an SDDE with variable time lag shares stability property with the modified EM method. As another application of this general approach we consider a linear SDDE with variable time lag and establish an “if and only if” result. It should also be mentioned that the questions addressed, results proved, as well as style of analysis borrow heavily from [14] but the computations involved to cope with time delay are nontrivial.     

The asymptotic stability of multistep multiderivative methods for systems of delay differential equations

IntroductionFor many years, many papers investigated the linear stabilit}' of delay differential equation(DDE) solvers and a significant number of important results have already been found for bothRunge-Kutta methods and linear multistep methods (see, for example, [l--8]). In this paper,we firstly consider stability of numerical methods with derivative for DDEs. It is shown thatA-stability of multistep multiderivative methods for ordinary differential equations (ODEs) isequit,alent to p-s…     

ℋ p stability of solutions of stochastic differential equations

Summary Solutions of systems of stochastic differential equations are shown to be stable in ^p under ^p perturbations of semimartingale differentials. Analogous results are obtained inp ^p when the solutions are not semimartingales but are only cadlag, adapted processes. Also, the solutions are shown to be stable under almost sure perturbations. These results are contrasted with the lack of stability under non- ^p perturbations, a result originally obtained by Wong and Zakai.This research was supported in part by NSF Grant No. MCS77-00095     

Hyers-Ulam-Rassias stability of κ-Caputo fractional differential equations

The paper is connected with the existence of solutions and Hyers-Ulam stability for a class of nonlinear fractional differential equations with κ-Caputo fractional derivative in boundary value problems. The existence and uniqueness results are obtained by utilizing the Banach fixed point theorem and Leray-Schauder nonlinear alternative theorem. In addition, two sufficient conditions to guarantee the Hyers-Ulam stability and the Hyers-Ulam-Rassias stability of boundary value problems of fractional differential equations are also presented. Finally, theoretical results are illustrated by two numerical examples.     

Hyers–Ulam stability of hypergeometric differential equations

Aequationes mathematicae - In the present paper by applying the series method we prove the Hyers–Ulam stability of the homogeneous hypergeometric differential equation in a subclass of...     

Families of Liapunov-Krasovskiį functionals and stability for functional differential equations

This paper discusses the stability of solutions of nonautonomous functional differential equations with infinite delay with respect to a parr of admissible phase spaces of Hale and Kato. A one-parameter family of Liapunov-Krasovskiį functional, together with some additional analysis, is used to prove new sufficient conditions of asymptotic and uniform asymptotic stability for such equations. It is also shown that the so-called Razumikhin condition is unessential when families of Liapunov-Krasovskiį functionals are used. Entrata in Redazione il 25 settembre 1997. Invited address at the Second Marrakesh International Conference on Differential Equations, Marrakesh, Morocco, June 1995.     

Numerical stability of Grünwald–Letnikov method for time fractional delay differential equations

BIT Numerical Mathematics - This paper is concerned with the numerical stability of time fractional delay differential equations (F-DDEs) based on Grünwald–Letnikov (GL) approximation...     

A class of new Magnus-type methods for semi-linear non-commutative Itô stochastic differential equations

Numerical Algorithms - In this paper, a class of new Magnus-type methods is proposed for non-commutative Itô stochastic differential equations (SDEs) with semi-linear drift term and...     

Multi-dimensional backward stochastic differential equations with one reflecting lower barrier of Itô diffusion type

This paper investigates a class of multi-dimensional stochastic differential equations with one reflecting lower barrier (RBSDEs in short), where the random obstacle is described as an Itô diffusion type of stochastic differential equation. The existence and uniqueness results for adapted solutions to such RBSDEs are established based on a penalization scheme and some higher moment estimates for solutions to penalized BSDEs under the Lipschitz condition and a higher moment condition on the coefficients. Finally, two examples are given to illustrate our theory and their applications.     

Oscillation of second-order Emden–Fowler neutral delay differential equations

We establish some new criteria for the oscillation of second-order Emden–Fowler neutral delay differential equations. We study the case of superlinear and the case of sublinear equations subject to various conditions. The results obtained show that the presence of a neutral term in a differential equation can cause or destroy oscillatory properties. Several examples are provided to illustrate the relevance of new theorems.     

Mean-square A-stable diagonally drift-implicit integrators of weak second order for stiff Itô stochastic differential equations

We introduce two drift-diagonally-implicit and derivative-free integrators for stiff systems of Itô stochastic differential equations with general non-commutative noise which have weak order 2 and deterministic order 2, 3, respectively. The methods are shown to be mean-square A-stable for the usual complex scalar linear test problem with multiplicative noise and improve significantly the stability properties of the drift-diagonally-implicit methods previously introduced (Debrabant and Rößler, Appl. Numer. Math. 59(3–4):595–607, 2009).