R E S E A R C H
Open Access
On stability regions of the modified midpoint
method for a linear delay differential
equation
Jana Hrabalová
*and Petr Tomášek
*Correspondence: [email protected] Institute of Mathematics, Brno University of Technology, Technická 2, Brno, 616 69, Czech Republic
Abstract
The paper deals with stability regions of a certain discretization of a linear differential equation with constant delay. The main aim of the paper is to analyze the regions of asymptotic stability of the modified midpoint method applied to a linear differential equation with constant delay. Obtained results are compared with other known results, particularly for Euler discretization. The relation between asymptotic stability conditions in the discrete case and continuous case is discussed, too.
1 Introduction
Qualitative investigation of difference equations plays a key role in the numerical anal-ysis of differential equations. Particularly, the study of asymptotic stability of numerical schemes (including construction of stability regions) is based on the results for asymp-totic stability of difference equations. In this paper we deal with the necessary and suffi-cient asymptotic stability conditions for a certain discretization applied to the linear delay differential equation
y(t) =ay(t) +by(t–τ), t> , ()
wherea,b,τ∈R,τ > . In particular, we compare the obtained conditions with results known for another numerical discretization applied to () and also with asymptotic stabil-ity conditions for delay differential equation () itself. For this purpose, we mention several works, which deal with the asymptotic properties of () and some of its discrete counter-parts. First, we recall the papers of Andronov and Mayer [], and of Hayes [], where the necessary and sufficient conditions for asymptotic stability of () were derived. These con-ditions can be captured as follows:
a≤b< –a for allτ> ()
and, in addition to the previous,
|a|+b< forτ<arccos(–a/b) (b–a)/ .
Particularly, consideringa= , equation () turns to
y(t) =by(t–τ), t> ()
and it is asymptotically stable if and only if –π/ <bτ< .
The first result for a discrete case (related to ()) that we mention is the paper of Levin and May [], where the difference equation
y(n+ ) –y(n) +βy(n–) = , n= , , , . . . , ()
≥,∈Zwas investigated. The necessary and sufficient condition for asymptotic sta-bility of () is <β< cos(π/(+ )). A more general case of () in the form
y(n+ ) +αy(n) +βy(n–) = , n= , , , . . . , ()
≥,∈Zwas discussed by Kuruklis [].
Theorem Letα= ,β be arbitrary reals.Equation()is asymptotically stable if and only if|α|< (+ )/,and
|α|– <β<α+ – |α|cosφ/ forodd,
|α+β|< and |β|<α+ – |α|cosφ/ foreven,
whereφ∈(,π/(+ ))is a solution of the auxiliary equationsin(x)/sin((+ )x) = /|α|.
We note that forα= the necessary and sufficient condition for asymptotic stability of () becomes|β|< . We remark that the conditions in this famous result have an im-plicit form with respect to. Another equivalent set of conditions in an explicit form with respect tois introduced in [].
The analysis of this paper is based on the assertion by Čermák and Tomášek [], which formulates the necessary and sufficient asymptotic stability conditions for the difference equation
y(n+ ) +αy(n) +βy(n–) = , n= , , , . . . , ()
whereα,β∈Rand≥,∈Z.
Theorem Letα,βbe arbitrary reals such thatαβ= .
(i) Letbe even andβ(–α)/+< .Then()is asymptotically stable if and only if
|α|+|β|< . ()
(ii) Letbe even andβ(–α)/+> .Then()is asymptotically stable if and only if
either
or
|α|–|β|< <|α|+|β|, < arccosα
+β–
|αβ|
arccosα
–β+
|α| ()
holds.
(iii) Letbe odd andα< .Then()is asymptotically stable if and only if()holds. (iv) Letbe odd andα> .Then()is asymptotically stable if and only if either(),or
β< –α<|β|, < arcsin –α
–β
|αβ|
arccosα
–β+
|α| ()
holds.
Recently, Ren [] also gave an equivalent system of necessary and sufficient conditions for asymptotic stability of (), but his formulation needs to solve a nonlinear auxiliary equation, similarly to the result of Kuruklis mentioned above. A description of the stability boundary for () in terms of some straight lines and certain parametric curves can be found in Kipnis and Nigmatullin [].
We close the survey of the results for various linear difference equations with Dannan [], where a general form of a three-term difference equation
y(n+m) +αy(n) +βy(n–) = , n= , , , . . . ,
with positive integersm,was investigated.
The above mentioned results can be utilized to describe stability regions (i.e., sets of pairs (a,b)∈R×R, for which the given discretization is asymptotically stable considering given stepsize) for various numerical schemes, which solve an initial value problem for (). For more details about numerical background, methods and their stability theory, see,e.g., Bellen and Zennaro [] and in’ t Hout [].
The paper is focused on the asymptotic properties of a numerical scheme, which arises from () by use of the modified midpoint method. The numerical formula is derived by integration over two steps, where the integrals of terms on the right-hand side of () are approximated via the trapezoidal rule and the midpoint rule, respectively. The analysis is realized on the equidistant meshtn=nh,n= , , . . . with stepsizeh=τ/k, wherek≥ is
a positive integer. Such an efficient choice of stepsize makes the discretization formulae free of extra interpolation terms, which can arise from an appropriate approximation of the delayed term (see []). First, we apply the modified midpoint method to equation () to obtain a linear difference equation
Y(n+ ) – +ah –ahY(n) –
bh
–ahY(n–k+ ) = , n= , , . . . , ()
where the stepsizehsatisfiesah= . The valueY(n) then represents a numerical approxi-mation of solutionyof delay differential equation () at the nodal pointtn.
2 Main result
In this section we state the necessary and sufficient conditions for asymptotic stability of (). The analysis of () falls naturally into two parts according to the parity ofk. For an effective and clear formulation of the main result, we introduce the symbols
τ*(h) =h+ harcsin a+b
h
( +ah)|b|
arccos +a
h– bh
ah– ,
τ*(h) =h+ harccos a+b
h |( +ah)b|
arccos +a
h– bh |ah– | ,
which are utilized in these two parts, respectively.
Theorem (I)Let k≥be even.Then()is asymptotically stable if and only if one of the following conditions holds:
|bh| ≤, |b|+a< , ()
< bh< –ah, τ<τ*(h). ()
(II)Let k≥be odd and m= (k– )/.Then()is asymptotically stable if and only if one of the following conditions holds:
a≤b< –a, |bh|< , ()
|b|+a< , (–)mbh= , ()
b+|a|< , bh> –, τ<τ*(h), ()
(–)mb+a< , (–)mbh> , τ<τ*(h), ()
(–)mb+a> , (–)m+bh> , τ<τ*(h). ()
Proof The proof is based on the application of Theorem to () and the ensuing anal-ysis of the obtained conditions. In particular, if we consider () in the form of (), the coefficientsαandβof () are given by
α= – +ah
–ah, β= –
bh
–ah ()
and the indicesandkare in the relation=k– .
Case(I): Investigating the case ofkeven, we utilize parts (iii) and (iv) of Theorem . Firstly, we focus on condition (iii): considering the coefficients (), the assumptionα< implies|ah|< . Thus, is equivalent to|b|+a< . Therefore, condition (iii) coincides with|bh|< –ah< .
Now, we analyze condition (iv): analogously, assumptionα> implies|ah|> . Hence, () gives|bh| ≤ providingah< –, while for the caseah> relation () cannot occur. We now turn to (). Relation () can be read as < bh< –ah. Furthermore, the
restriction ()becomes
k– < arcsin a+b
h |( +ah)b|
arccos +a
Since|ah|> andk=τ/h, it can be written asτ<τ*
(h). Therefore, condition (iv) is satisfied
if and only if either|bh| ≤,ah< – or ().
Finally, Theorem does not cover the case ofαβ= (i.e.,ah= – orb= ). In our case we do not consider the eventualityb= with respect to the fact that we deal with the discretization of (). Accordingly, forah= –, equation () turns to
Y(n+ ) –bhY(n–k+ ) = , n= , , . . .
and the necessary and sufficient condition for its asymptotic stability is given by Theo-rem as|bh|< . Summarizing the above discussion, we conclude that ifkis even, () is asymptotically stable if either () or () holds.
Case(II): Forkodd, we consider conditions (i) and (ii) of Theorem . Condition (i) can be rewritten as
by use of (). With respect to the parity of power in the first relation, we obtain by a sign discussion of terms in the other relation a set of conditions equivalent to () as
|ah|< , b> , a< –b,
In the Case (ii) of Theorem , condition () can be reformulated as
–bh
An analogous analysis to that above shows that formodd, () is equivalent to
|ah|< , b< , a≤b,
ah< –, b< , –bh≤.
()
In the casemeven, condition () is satisfied if and only if () or
ah< –, b> , bh≤
holds. The above discussion of the Case (i), the part of (ii) considering () and including the caseα= (i.e.,ah= –,|bh|< , seeCase(I)) gives ()-().
Now it remains to analyze condition () adapted for equation () by (),i.e.,
under the assumption –bh–ah(+ah–ah)+m> . In the same manner as above, we get the
equiva-lency to the following set of conditions:
|ah|< , b< , +ah≤–bh, –bh< , b<a, τ<τ*(h), ()
|ah|< , b< , +ah> –bh, b< –|a|, τ<τ*(h), ()
ah> , b> , +ah≤bh, b<a, τ<τ*(h), ()
ah> , b> , +ah> bh, <bh, τ<τ*(h), ()
ah< –, b< , +ah< bh, –bh> , τ<τ*(h), ()
ah< –, b< , +ah≥bh, –bh> , a<b, τ<τ*(h) ()
formodd and (), (),
ah> , b< , +ah≤–bh, –b<a, τ<τ*(h), ()
ah> , b< , +ah> –bh, < –bh, τ<τ*(h), ()
ah< –, b> , +ah< –bh, bh> , τ<τ*(h), ()
ah< –, b> , +ah≥–bh, bh> , a< –b, τ<τ*(h) ()
formeven. These conditions are jointly expressed by ()-(). In fact, () coincides with (), (). Condition () is equivalent to (), () and (), () formodd andmeven, respectively. Finally, () is the same as (), () formodd and (), () formeven. The
proof is complete.
3 Asymptotic stability discussion
In this section we introduce some remarks and comments to the obtained result formu-lated in Theorem . First, we focus on the connection with the asymptotic stability proper-ties of (). Particularly, we investigate a limit form of Theorem consideringh→. In the Case (I) ofkeven, the asymptotic stability region of () becomes|b|+a< . Let us note that with the exception of the boundary, this region corresponds to (). In the Case (II) ofkodd, it may be shown (by the L’Hospital rule) that the asymptotic stability conditions turn to
a≤b< –a,
|a|+b< , τ<arccos(–a/b) (b–a)/
ash→. These are equivalent to the conditions defining the asymptotic stability region of ().
Now we present the necessary and sufficient conditions for the asymptotic stability of the midpoint method discretization of () in the form
Y(n+ ) –Y(n) – bhY(n–k+ ) = , n= , , . . . , ()
corresponding differential equation, as well as with the stability intervals for the forward Euler method discretization of ().
Corollary Equation()is asymptotically stable if and only if
k is odd, >b> – hsin
πh τ.
Proof The assertion is an immediate consequence of Theorem . Settinga= , we realize that conditions () and () cannot occur. Therefore, () is unstable for anyb∈Rin the case ofkeven.
Consideringkodd, we investigate conditions ()-(). Fora= , there arises a contra-diction in all conditions except for (), which becomes
b< , bh> –, τ<h+ harccos(–bh)
arccos( – bh). ()
We omit ()because it is imposed by the domain of the last relation (). Our next aim
is to simplify the delay restriction ()to the form more convenient for further analysis.
We use the formula
arccos – x=π– arccosx, x>
and taking into account that functioncosxis decreasing forx∈(,π/), we arrive at
cosπ(τ–h) τ > –bh.
Sincecos(π/ –x) =sinx, we get
sinπh τ > –bh.
Finally, we rewrite () as
>b> – hsin
πh τ,
which is the necessary and sufficient condition for asymptotic stability of () providing
kis odd.
We emphasize that the stability regions are captured just by stability intervals for val-ues of parameterb. We denoteIM
τ (h) = (–hsin πh
τ, ) stability intervals of () derived in
Corollary . Next assertion describes the relation between stability intervalsIM τ (h) with
respect to stepsizeh.
Theorem Let≤k<kbe arbitrary positive odd integers and let h=τ/k>τ/k=h
be corresponding stepsizes.Then
Proof Let us define a function
f(h) = – hsin
πh
τ, h∈(,τ/,
which represents the dependence of the left endpoint ofIM
τ (h) on stepsizeh. Since the
right endpoint is zero for anyh, we discuss only the monotony off(h). Doing this, we drop the constraint h=τ/k and we considerf(h) as a function of a continuous argumenth.
τ, the last relation becomes
tanx>x, x∈(,π/ . ()
Obviously,tan() = and (tan(x))=cos–(x) > =xforx∈(,π/ . Therefore () holds
for anyx∈(,π/ . Thus, we have proved thatf(h) > forh∈(,τ/ and consequently IM
τ (h)⊃IτM(h).
Next, we compare stability intervalsIM
τ (h) with the stability interval of (), which we
denoteI*
τ= (–π/(τ), ).
Remark An important property is the behavior ofIM
τ (h) ash→. Using the L’Hospital
h=τ/k, wherekis odd. Note that the midpoint method discretization of () is not asymp-totically stable.
Finally, we discuss a relation betweenIτM(h) and asymptotic stability intervals for the
forward Euler discretization of (). They are derived in [], and we denote them asIE τ(h) =
(–hcosτ+hπ τ , ).
Theorem Let k≥be an arbitrary positive odd integer and let h=τ/k be the corre-sponding stepsize.Then
Proof Since the right endpoints ofIM
τ (h) andIτE(h) are zero for anyh, we investigate only
the behavior of the left endpoints with respect to changing stepsizeh. We define a function
g(h) = – hcos
π τ
τ+h, h∈(,τ ,
which expresses the dependence of the left endpoint ofIE
τ(h) onh. In the further analysis,
To do this, we introduce the following proposition.
Lemma Let F ∈Ca,b be a function such that F(a) =F(b) = ,F(a)≤,F(b) > ,
since each term in the sum is positive for alls∈(, ). Then by the previous lemma, we have thatG(s) < for alls∈(, ) and consequentlyf(h) <g(h) forh∈(,τ), which concludes
the proof.
4 Conclusions
form of the delay differential equation (). The authors believe that analogous investigation is possible also for more complicated numerical formulae (applied to ()) as far as there are known stability criteria for corresponding difference equations. Such analysis may be done,e.g., for the-method.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors declare that the research was realized in collaboration with the same contribution. Both authors read and approved the final manuscript.
Acknowledgements
The first author was supported by the project FSI-S-11-3 of Brno University of Technology. The second author was supported by the grant P201/11/0768Qualitative properties of solutions of differential equations and their applicationsof the Czech Science Foundation.
Received: 1 March 2013 Accepted: 28 May 2013 Published: 18 June 2013 References
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doi:10.1186/1687-1847-2013-177