Preservation of stability and oscillation of Euler-Maclaurin method for differential equation with piecewise constant arguments of alternately advanced and retarded type
Wang et al. Journal of Inequalities and Applications (2015) 2015:165
DOI 10.1186/s13660-015-0685-5
RESEARCH
Open Access
Preservation of stability and oscillation of
Euler-Maclaurin method for differential
equation with piecewise constant arguments
of alternately advanced and retarded type
Qi Wang* , Xiaoming Wang, Yucheng Xie and Ling Chen
*
Correspondence:
School of Applied Mathematics,
Guangdong University of
Technology, Guangzhou, 510006,
P.R. China
Abstract
This paper is devoted to stability and oscillation analysis of Euler-Maclaurin method
for differential equation with piecewise constant arguments
u (t) = au(t) + bu(2[(t + 1)/2]). The necessary and sufficient conditions under which the
numerical stability regions contain the analytical stability regions are given. Moreover,
the conditions of oscillation for the Euler-Maclaurin method are obtained. We show
that the Euler-Maclaurin method preserves the oscillation of the exact solution. In
addition, the connection between stability and oscillation are discussed theoretically
and numerically. Finally, some numerical examples are also provided.
MSC: Primary 65L07; secondary 65L20
Keywords: Euler-Maclaurin method; stability; oscillation; piecewise constant
arguments
1 Introduction
The theory of differential equation with piecewise constant arguments (EPCA) was initiated in [, ], which provided a mathematical instrument to applied science [, ]. These
systems have been under intensive investigation for the last twenty years. They describe
hybrid dynamical systems and combine properties of both differential and difference equations. For example, applying the explicit linear multistep method to differential equation
u (t) = f (u(t)), we have
un+ = un + h c f (un ) + c f (un– ) ,
where h is stepsize and un is approximation to u(t) at tn . By integration, we can see that
the above difference equation is equivalent to the following EPCA:
t
t
h
+ c f u
– h ,
u (t) = c f u
h
h
© 2015 Wang et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License
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indicate if changes were made.
Wang et al. Journal of Inequalities and Applications (2015) 2015:165
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so EPCA has a similar structure to the difference equation. In the present paper we shall
consider the following EPCA:
t+
,
u (t) = au(t) + bu
u() = u ,
()
where a, b and u are real constants, b = for a = and [·] denotes the greatest integer
function. Differential equation of this form has stimulated considerable interest and has
been studied by Cooke and Wiener [], Jayasree and Deo [], Wiener and Aftabizadeh [].
In this type of equation the argument deviation η(t) = t – [(t + )/] is a piecewise linear
period function with period . Also, η(t) is negative for t ∈ [n – , n) and positive for
t ∈ [n, n + ). Thus () is advanced type on [n – , n) and retarded type on [n, n + ).
Therefore () is EPCA of alternately advanced and retarded type.
There exists an extensive literature dealing with EPCA, for instance, the existence and
uniqueness of the solution of a class of first order nonhomogeneous advanced impulsive
EPCA were considered in [], the stability property of first order EPCA of generalized
type (EPCAG) was addressed in [], oscillation of exact solution of EPCA with retarded
and advanced arguments was discussed in []. In [], the authors constructed Green’s
function to the linear operator of boundary value EPCA and obtained some comparison
results for the same differential equation. The general theory and basic results for EPCA
have been thoroughly developed in the book of Wiener [].
In contrast to the study on the qualitative behavior of EPCA, the research on the numerical solution of EPCA has become a hot issue recently. Numerical stability of many kinds
of EPCA was addressed in [–]. Numerical oscillation of θ -methods and Runge-Kutta
methods for equation x (t) + ax(t) + a x([t – ]) = was investigated in [, ], respectively. Moreover, stability and oscillation of numerical solution for EPCA with [t + /]
and [(t + )/] were considered in [, ], respectively. Numerical methods in the above
mentioned papers involve θ -methods, Runge-Kutta methods, linear multistep method
and Galerkin methods. However, to the best of our knowledge, very few results concerning Euler-Maclaurin method were obtained (see []). The authors of [] investigated
the stability of Euler-Maclaurin method for a linear neutral EPCA. Different from [], in
the present paper, we study the stability and oscillation of the numerical solution in the
Euler-Maclaurin method for (). Whether the numerical method preserves stability and
oscillation and the connection between stability and oscillation are also investigated.
The rest of this paper is arranged as follows. In Section , we propose some useful concepts and results for stability and oscillation of the exact solution. In Section , we obtain
a discrete equation by applying the Euler-Maclaurin method to (), then the asymptotic
stability, oscillation and non-oscillation of numerical method for () are considered. In
Section , we discuss the preservation properties of Euler-Maclaurin method. The conditions under which the analytical stability regions are contained in the numerical stability
regions are obtained, and it is proved that the Euler-Maclaurin method can preserve oscillation of the exact solution. In Section , we obtain a lot of connections between stability
and oscillation. Finally, some numerical examples are reported in Section .
2 Stability and oscillation of exact solution
Definition [] A solution of () on [, ∞) is a function u(t) which satisfies the following
conditions:
Wang et al. Journal of Inequalities and Applications (2015) 2015:165
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(i) u(t) is continuous on [, ∞),
(ii) the derivative u (t) exists at each point t in [, ∞), with the possible exception of
the points t = n – for n ∈ N, where one-sided derivatives exist,
(iii) () is satisfied on each interval [n – , n + ) for n ∈ N.
Theorem [] Assume that a, b and u ∈ R, then () has on [, ∞) a unique solution u(t)
given by
t+
() [ ]
u
u(t) = η(t)
(–)
for a = and
t+
() [ ]
u(t) = η(t)
u
(–)
for a = , where
(t) = eat + eat – a– b,
(t) = + bt,
η(t) = t –
t+
.
Theorem [] The solution u(t) = of () is asymptotically stable (u(t) → as t → ∞)
if and only if any one of the following conditions is satisfied:
–
a(ea + )
< b < –a
(ea – )
b>–
a(ea + )
(ea – )
b<
for a > ,
b < –a
or
for a < ,
for a = .
Definition A nontrivial solution of () (...truncated)
