In this paper, we consider the asymptotic stability of linear constant coeecient delay diierential-algebraic equations and of-methods, Runge-Kutta methods and linear multistep methods applied to these systems.

In this paper an analysis is provided of nonlinear monotonicity and boundedness properties for linear multistep methods. Instead of strict monotonicity for arbitrary starting values we shall focus on generalized monotonicity or boundedness with Runge-Kutta starting procedures. This allows many multistep methods of practical interest to be included in the theory. In a related manner, we also con...

To obtain high order integration methods for ordinary differential equations which combine to some extent the advantages of RungeKutta methods on one hand and linear multistep methods on the other, the use of “modified multistep” or “hybrid” methods has been proPosedIll, PI, 131. In this paper formulae are derived for methods which use one extra intermediate point than in the previously pub lis...

In this paper, an interpolation method for solving linear diierential equations was developed using multiquadric scheme. Unlike most iterative formula , this method provides a global interpolation formulae for the solution. Numerical examples show that this method ooers a higher degree of accuracy than Runge-Kutta formula and the iterative multistep methods developed by Hyman (1978).

This paper deals with the convergence and stability of linear multistep methods for impulsive differential equations. Numerical experiments demonstrate that both the mid-point rule and twostep BDFmethod are of order p 0when applied to impulsive differential equations. An improved linear multistep method is proposed. Convergence and stability conditions of the improved methods are given in the p...

Recently an implicit method has been presented for solving first order singular initial value problem. The method is extended to solve second or higher order problems having a singular point. The method presents more correct result than those obtained by the implicit Euler and second order implicit Runge-Kutta (RK2) methods. The method is illustrated by suitable examples.

The shape of plasmonic nanostructures such as silver and gold is vital to their physical and chemical properties and potential applications. Recently, preparation of complex nanostructures with rich function by chemical multistep methods is the hotspot of research. In this review we introduce three typical multistep methods to prepare silver nanostructures with well-controlled shapes, including...

In this article, conditions for the preservation of quadratic and Hamiltonian invariants by numerical methods which can be written as B-series are derived in a purely algebraical way. The existence of a modified invariant is also investigated and turns out to be equivalent, up to a conjugation, to the preservation of the exact invariant. A striking corollary is that a symplectic method is forma...

Some new higher algebraic order symmetric various-step methods are introduced. For these methods a direct formula for the computation of the phase-lag is given. Basing on this formula, calculation of free parameters is performed to minimize the phase-lag. An explicit symmetric multistep method is presented. This method is of higher algebraic order and is fitted both exponentially and trigonomet...

Abstract During embryogenesis, optic vesicles, the eye primordium, develop from diencephalon via a controlled multistep process of organogenesis that has been difficult to reconstruct in humans. Using iPSC-derived human brain organoids, we demystify complexities and demonstrate organoids can innately forebrain-associated bilaterally symmetric vesicles. These next-generation with relevant cell d...