scholarly journals Linear Multistep Methods for Impulsive Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
X. Liu ◽  
M. H. Song ◽  
M. Z. Liu
Keyword(s):  
Differential Equations ◽  
Numerical Experiments ◽  
Multistep Methods ◽  
Impulsive Differential Equations ◽  
Multistep Method ◽  
Linear Multistep Methods ◽  
Linear Multistep Method ◽  
Convergence And Stability ◽  
Improved Methods ◽  
Bdf Method

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 two-step BDF method are of orderp=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 paper. Numerical experiments are given in the end to illustrate the conclusion.

scholarly journals Exponential mean-square stability properties of stochastic linear multistep methods

2021 ◽  
Vol 47 (4) ◽  
Author(s):  
Evelyn Buckwar ◽  
Raffaele D’Ambrosio
Keyword(s):  
Numerical Experiments ◽  
Multistep Methods ◽  
Multistep Method ◽  
Linear Multistep Methods ◽  
Linear Multistep Method ◽  
Mean Square ◽  
Stability Properties ◽  
Mean Square Stability ◽  
And Diffusion ◽  
Exponential Mean Square Stability

AbstractThe aim of this paper is the analysis of exponential mean-square stability properties of nonlinear stochastic linear multistep methods. In particular it is known that, under certain hypothesis on the drift and diffusion terms of the equation, exponential mean-square contractivity is visible: the qualitative feature of the exact problem is here analysed under the numerical perspective, to understand whether a stochastic linear multistep method can provide an analogous behaviour and which restrictions on the employed stepsize should be imposed in order to reproduce the contractive behaviour. Numerical experiments confirming the theoretical analysis are also given.

DYNAMICS OF LINEAR MULTISTEP METHODS FOR DELAY DIFFERENTIAL EQUATIONS

2004 ◽  
Vol 14 (01) ◽  
pp. 329-336
Author(s):  
HONGJIONG TIAN ◽  
QIAN GUO
Keyword(s):  
Multistep Methods ◽  
Multistep Method ◽  
Linear Multistep Methods ◽  
Linear Multistep Method ◽  
Step Size ◽  
Time Step ◽  
True Solution ◽  
Delay Differential ◽  
Stable Linear ◽  
Period Two Solutions

In this paper we study the relationship between the asymptotic behavior of a numerical simulation by linear multistep method and that of the true solution itself for fixed step sizes. The numerical method is viewed as a dynamical system in which the step size acts as a parameter. Numerical stability of linear multistep method for nonlinear delay differential equation is investigated and we prove that A-stable linear multistep methods are NP-stable. It is shown that a consistent zero-stable linear multistep method does not admit spurious fixed points. The existence of spurious period-two solutions in the time-step is also studied. Finally we give a simple example to illustrate instability of the spurious period-two solutions.

scholarly journals Classical Theory of Linear Multistep Methods for Volterra Functional Differential Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Yunfei Li ◽  
Shoufu Li
Keyword(s):  
Differential Equations ◽  
Numerical Experiments ◽  
Multistep Methods ◽  
Functional Differential Equations ◽  
Linear Multistep Methods ◽  
Interpolation Theory ◽  
Initial Value ◽  
Classical Stability ◽  
Functional Differential ◽  
Delay Differential

Based on the linear multistep methods for ordinary differential equations (ODEs) and the canonical interpolation theory that was presented by Shoufu Li who is exactly the second author of this paper, we propose the linear multistep methods for general Volterra functional differential equations (VFDEs) and build the classical stability, consistency, and convergence theories of the methods. The methods and theories presented in this paper are applicable to nonneutral, nonstiff, and nonlinear initial value problems in ODEs, Volterra delay differential equations (VDDEs), Volterra integro-differential equations (VIDEs), Volterra delay integro-differential equations (VDIDEs), etc. At last, some numerical experiments verify the correctness of our theories.

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