A class of boundary value methods for the complex delay differential equation

International Journal of Soft Computing, Mathematics and Control (IJSCMC) Vol.6, No.2/3, August 2017
DOI:10.5121/ijscmc.2017.6302 15
A CLASS OF BOUNDARY VALUE METHODS FOR THE
COMPLEX DELAY DIFFERENTIAL EQUATION
Shifeng Wu
Department of Computer Science, Guangdong Polytechnic Normal University,
Guangzhou, 510665, China
ABSTRACT
In this paper, a class of boundary value methods (BVMs) for delay differential equations (DDEs) is
considered. The delay dependent stable regions of the extended trapezoidal rules of second kind (ETR2s),
which are a class of BVMs, are displayed for the test equation of DDEs. Furthermore, it is showed ETR2s
cannot preserve the delay-dependent stability of the complex coefficient test equation considered. Some
numerical experiments are given to confirm the theoretical results.
AMS 2000 Mathematics Subject Classification: 65L20, 65M12
KEYWORDS
Delay differential equations; boundary value methods; delay-dependent stability; extended trapezoidal
rules of second kind
1. INTRODUCTION
The stability of numerical methods plays an important role in the numerical solution of initial
value problems (IVPs). During the past decade, most of the work on the asymptotic stability
for delay differential equations (DDEs) dealt with finding the stability region independently
of the delay term. Compared with the delay-independent analysis, the stability analysis for a
fixed value of the delay is much more difficult [1, 2, 3, 4]. In recent years, some studies have
been devoted to delay-dependent stability (see, for example, Zhao [5], J. Ma [6], Aceto[7, 8],
Abdelhameed[9] ).
It is less known that the control of the parasitic solutions is much easier if the problem is
transformed into an almost equivalent boundary value problem. Starting from such an idea, a new
class of multistep methods, called boundary value methods (BVMs), has been proposed and
analyzed in the last few years(see, for example, Amodio and Mazzia [10], Brugnano ,Iavernaro
and Trigiante [11], Brugnano and Trigiante[12,13], Brugnano [14]). BVMs no longer suffer the
limitations the Dahlquist barriers [15,16,17] and the impossibility to define stable, high-order
symplectic methods, but also permits us to avoid the difficulties when changing the stepsize.
Consequently, A-stable, essentially sympletic BVMs of any order are obtained. The extended
trapezoidal rules of second kind (ETR2s), which are a class of BVMs, are very important. In fact,
they result to be the methods to be chosen when approximating either continuous boundary value
problem [13] or Hamiltonian problems [10,13]. In particular, they are all "essentially" symplectic
methods [14].
∗Supported by the Foundation for Distinguished Young Talents in Higher Education of
Guangdong, China (Grant No. 2014KQNCX175).
†
Corresponding: fengtree@126.com

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In this paper, we investigate the delay-dependent stability of ETR2s, which are a class of BVMs,
for the test equation of complex DDEs. This paper is organized as follows. In section 2, the
stability concepts and definitions are introduction for the coefficient test equation. In section 3,
we deal with the ETR2s for the complex coefficient version of equation. Finally, in section 4,
several numerical experiments are given to confirm the theoretical results presented in previous
sections.
2. THE ETR2S FOR DDES
Consider the test equation for DDEs
where , the constants and is a continuous function. Since we are interested
in analyzing equation (2.1) for an arbitrary but fixed delay, we remark that, by using a scaling of
the time variable, we are able to bring equation (2.1) into the from
where and . Therefore, there is no loss of generality in performing the stability
analysis for equation (2.2).
We consider the following ETR2s [13,14], which are a class of boundary value methods, with
-boundary conditions. For , ETR2s have the following
general form:
and values
are needed. A set of additional equations is needed. Such equation can be derived by additional
methods having the same order as the main methods, and the stability properties of the global
method are inherited by the main formula (2.3). For more detail on the practical use of the
additional method, see [13].
The coefficients of (2.3) are determined so that the maximum possible order is
reached. In Table 1, the normalized coefficients of these methods, up to , are
displayed [13,14]. Also in this case, for , one obtains the based trapezoidal rule.
Table 3.1. Coefficients of ETR2s

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We apply the ETR2s (2.3) with constant stepsize to (2.2), with a positives integer.
This leads to
for and are needed. where for
and , can be be derived by additional methods
Next, the characteristic equation of (2.4) is given by
where is called the stability polynomial. In order to research the stability of the
numerical solution of (2.4), the following proposition is given [13].
Proposition 2.1 The following statements are equivalent:
(1) has zeros inside and zeros outside ,
(2) the discrete numerical solution of (2.4), with constant stepsize , satisfies
for all initial functions .
Lemma 2.2 For the ,where
, then
(i) the function is strictly monotonically increasing for when ;
(ii)the when .
Proof. Firstly, we prove (i).
For , changes to
The is negative and strictly monotonically decreasing function with respect to , so
is strictly monotonically increasing function.
For , changes to
It shows that the is negative and strictly monotonically decreasing function with respect
to , so is strictly monotonically increasing function.

18
For , changes to
A straightforward computation of (2.10) shows that the is negative and strictly
monotonically decreasing function with respect to , so is strictly monotonically
increasing function.
For , changes to
monotonically decreasing function with respect to , so is strictly monotonically
increasing function.
For , changes to
monotonically decreasing function, so is strictly monotonically increasing function.
Therefore, the are negative and strictly monotonically decreasing function.
Secondly, from (2.6)-(2.15), for

19
From (2.16)-(2.19) and , we get
(2.20)
So, we get the Lemma2.2.
3. THE COMPLEX COEFFICIENT CASE
In this section the stability definitions given in Section 2 are extended to the complex coefficient
version of the test equation (2.2). Hence, we assume in the continuation . We
denote by the stability region of the equation (2.2), that is the set of complex pairs such
that for all initial function . Now, for the numerical solution, the
following definitions are given by Guglielmi [3] and Bell [1].
Definition 3.1 The -stability region ( -stability region) of a numerical method for DDEs is the
set
where, for a given positive integer , is the set of the pair of complex numbers
such that the discrete numerical solution of (2.2), with constant stepsize
and a positive integer, satisfies for all initial functions
.
Definition 3.2 The numerical method for DDEs is -stable ( -stable ) if .
3.1. The true stability region
To analyze the stability region of (2.2), we consider the stability set (with and fixed)
and represent it in the -plane. The parametric equations of the boundary of the stability region
(cf. [1]) are
where
(3.1)
with
(3.2)
for the parameter varies in . Observe that the boundary is the ßum" of a circle
centered in the origin with the radius and of the transcendental curve ,
which does not depend on .
The form of the stability region in the complex -plane is reported in Fig 3.1 for a given complex
value of .

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3.2. Extended analysis of ETR2s for the complex DDEs
We focus attention on the complex coefficient version of the characteristic polynomial (2.5) and
study the corresponding boundary locus, which we denote here by , we denote by
the restriction of the boundary locus of the obtained by fixing and , we denote
by and the previously defined and for the special
case of the or fixed . The straightforward discussion involves the explicit
determination of . By standard algebraic manipulation we obtain
Figure 3.1: An example of the stability set (shaded region) for the complex test equation (2.2), with
a section of the boundary curve.
with and given by (3.2), given by
. We observe that some interesting
relationships persist between the two sets and . In fact, by comparing (3.3) and
(3.1), we see that the numerical stability boundary is still given by the ’sum’ of the circle (3.1)
and of a regular curve which does not depend on . However, the all of
are unfortunately not -stable. We have the following Theorem.
Theorem 3.3 The (2.4)( ) is not -stable .
In order to prove this Theorem 3.3, we need the following Lemma 3.4.
Lemma 3.4 For fixed and the , ( ), such that and
. Then

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Proof. Let us compute the minimum distance of the set from the origin of the -plane.
By (3.1) we get
Since diverges for and
we have:
The same computation for the boundary locus generated by the , by (3.3),
and, by some algebraic manipulations and Lemma 2.2(i),
On this basis we can prove that
urthermore we observe that
(3.8)

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from Lemma 2.2(ii). However, by (3.5) and (3.7),
(3.9)
Now, by well know results concerning A-stability, we have that the origin of the -plane belongs
to both sets and . As a consequence we get
(3.10)
Finally, by (3.9)-(3.10), we are in a position to state the following relations:
for any and any considered in the hypotheses (see Fig.3.2).
The central result is an immediate consequence of Lemma 3.2.
4. NUMERICAL EXPERIMENTS
In order to illustrate the Theorem 3.3. Consider the complex coefficient test equation
(4.1)
where and . The (2.4) with
is applied to equation (4.1) for studying the numerical stability ( -stable). It
shows that the exact solution of (4.1) is asymptotically stable.
For the , equation (2.4) requires the additional initial conditions which can be
gotten by for and by additional initial equations
obtained by additional methods, and additional final conditions, which can be gotten by
additional final equations obtained by additional methods [13]. It can be shown that if the
additional methods are appropriately chosen, the stability properties of the global methods are
inherited by the main formula (2.4) [13].
As an example, we mention the fourth-order ETR2 ( )
which can be completed with the following additional equations:
obtained by additional fourth-order methods.
For the six-order ETR2 ( )
which can be completed with the following additional equations, obtained from sixth order
methods,

23
For each ETR2 previously considered, there exists an appropriate choice of the additional
methods [13]. The numerical solution (2.4) for the complex equation (4.1) is given in Fig 4.1, Fig
4.2 and Fig 4.3 for the different .

24
Fig 4.1 and Fig4.2 give numerical solution of ETR2s (different stepsize ) for (4.1). However,
the numerical solutions of Fig 4.1 and Fig 4.2 are not stable for the different ETR2s . Fig
4.1 and Fig 4.3 give numerical solution of ETR2s (different order ) for (4.1). Fig 4.1-4.3
demonstrate that the ETR2s cannot preserve the asymptotic stability of its exact solution when
. So, the ETR2s are no -stable. These numerical examples confirm our theoretical findings.
5. CONCLUSIONS
The delay dependent stable regions of the extended trapezoidal rules of second kind
(ETR2 ), which are a class of BVMs, are displayed for the test equation of DDEs.
Furthermore, It is showed ETR2s for cannot preserve the delay-dependent stability of the
complex coefficient test equation considered. Before concluding the section 3 and section 4, we
point out that the ETR2 for has not yet been studied. However, ETR2 are a
class of BVMs and constructed using the same law. If the coefficients of the ETR2 for
are given, we conjecture that the ETR2s for has the same result as the
ETR2 for in this paper, which ETR2 for can not preserve the delay-
dependent stability of the complex coefficient test equation considered. This will be one of the
next steps we need to do.
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25
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