Advances in Difference Equations

, 2013:177 | Cite as

On stability regions of the modified midpoint method for a linear delay differential equation

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  1. Progress in Functional Differential and Difference Equations

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.

Keywords

Difference Equation Asymptotic Stability Stability Region Delay Differential Equation Left Endpoint 

1 Introduction

Qualitative investigation of difference equations plays a key role in the numerical analysis of differential equations. Particularly, the study of asymptotic stability of numerical schemes (including construction of stability regions) is based on the results for asymptotic stability of difference equations. In this paper we deal with the necessary and sufficient asymptotic stability conditions for a certain discretization applied to the linear delay differential equation
y ( t ) = a y ( t ) + b y ( t τ ) , t > 0 , Open image in new window
(1)
where a , b , τ R Open image in new window, τ > 0 Open image in new window. In particular, we compare the obtained conditions with results known for another numerical discretization applied to (1) and also with asymptotic stability conditions for delay differential equation (1) itself. For this purpose, we mention several works, which deal with the asymptotic properties of (1) and some of its discrete counterparts. First, we recall the papers of Andronov and Mayer [1], and of Hayes [2], where the necessary and sufficient conditions for asymptotic stability of (1) were derived. These conditions can be captured as follows:
a b < a for all  τ > 0 Open image in new window
(2)
and, in addition to the previous,
| a | + b < 0 for  τ < arccos ( a / b ) ( b 2 a 2 ) 1 / 2 . Open image in new window
Particularly, considering a = 0 Open image in new window, equation (1) turns to
y ( t ) = b y ( t τ ) , t > 0 Open image in new window
(3)

and it is asymptotically stable if and only if π / 2 < b τ < 0 Open image in new window.

The first result for a discrete case (related to (3)) that we mention is the paper of Levin and May [3], where the difference equation
y ( n + 1 ) y ( n ) + β y ( n ) = 0 , n = 0 , 1 , 2 , , Open image in new window
(4)
1 Open image in new window, Z Open image in new window was investigated. The necessary and sufficient condition for asymptotic stability of (4) is 0 < β < 2 cos ( π / ( 2 + 1 ) ) Open image in new window. A more general case of (4) in the form
y ( n + 1 ) + α y ( n ) + β y ( n ) = 0 , n = 0 , 1 , 2 , , Open image in new window
(5)

1 Open image in new window, Z Open image in new window was discussed by Kuruklis [4].

Theorem 1 Let α 0 Open image in new window, β be arbitrary reals. Equation (5) is asymptotically stable if and only if | α | < ( + 1 ) / Open image in new window, and
| α | 1 < β < ( α 2 + 1 2 | α | cos ϕ ) 1 / 2 for odd, | α + β | < 1 and | β | < ( α 2 + 1 2 | α | cos ϕ ) 1 / 2 for even, Open image in new window

where ϕ ( 0 , π / ( + 1 ) ) Open image in new window is a solution of the auxiliary equation sin ( x ) / sin ( ( + 1 ) x ) = 1 / | α | Open image in new window.

We note that for α = 0 Open image in new window the necessary and sufficient condition for asymptotic stability of (5) becomes | β | < 1 Open image in new window. We remark that the conditions in this famous result have an implicit form with respect to . Another equivalent set of conditions in an explicit form with respect to is introduced in [5].

The analysis of this paper is based on the assertion by Čermák and Tomášek [6], which formulates the necessary and sufficient asymptotic stability conditions for the difference equation
y ( n + 2 ) + α y ( n ) + β y ( n ) = 0 , n = 0 , 1 , 2 , , Open image in new window
(6)

where α , β R Open image in new window and 1 Open image in new window, Z Open image in new window.

Theorem 2 Let α, β be arbitrary reals such that α β 0 Open image in new window.
  1. (i)
    Let be even and β ( α ) / 2 + 1 < 0 Open image in new window. Then (6) is asymptotically stable if and only if
    | α | + | β | < 1 . Open image in new window
    (7)
     
  2. (ii)
    Let be even and β ( α ) / 2 + 1 > 0 Open image in new window. Then (6) is asymptotically stable if and only if either
    | α | + | β | 1 , Open image in new window
    (8)
     
or
| | α | | β | | < 1 < | α | + | β | , < 2 arccos α 2 + β 2 1 2 | α β | / arccos α 2 β 2 + 1 2 | α | Open image in new window
(9)
holds.
  1. (iii)

    Let be odd and α < 0 Open image in new window. Then (6) is asymptotically stable if and only if (7) holds.

     
  2. (iv)
    Let be odd and α > 0 Open image in new window. Then (6) is asymptotically stable if and only if either (8), or
    β 2 < 1 α < | β | , < 2 arcsin 1 α 2 β 2 2 | α β | / arccos α 2 β 2 + 1 2 | α | Open image in new window
    (10)
     

holds.

Recently, Ren [7] also gave an equivalent system of necessary and sufficient conditions for asymptotic stability of (6), 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 (6) in terms of some straight lines and certain parametric curves can be found in Kipnis and Nigmatullin [8].

We close the survey of the results for various linear difference equations with Dannan [9], where a general form of a three-term difference equation
y ( n + m ) + α y ( n ) + β y ( n ) = 0 , n = 0 , 1 , 2 , , Open image in new window

with positive integers m, was investigated.

The above mentioned results can be utilized to describe stability regions (i.e., sets of pairs ( a , b ) R × R Open image in new window, for which the given discretization is asymptotically stable considering given stepsize) for various numerical schemes, which solve an initial value problem for (1). For more details about numerical background, methods and their stability theory, see, e.g., Bellen and Zennaro [10] and in’ t Hout [11].

The paper is focused on the asymptotic properties of a numerical scheme, which arises from (1) 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 (1) are approximated via the trapezoidal rule and the midpoint rule, respectively. The analysis is realized on the equidistant mesh t n = n h Open image in new window, n = 0 , 1 , Open image in new window with stepsize h = τ / k Open image in new window, where k 2 Open image in new window 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 [10]). First, we apply the modified midpoint method to equation (1) to obtain a linear difference equation
Y ( n + 2 ) 1 + a h 1 a h Y ( n ) 2 b h 1 a h Y ( n k + 1 ) = 0 , n = 0 , 1 , , Open image in new window
(11)

where the stepsize h satisfies a h 1 Open image in new window. The value Y ( n ) Open image in new window then represents a numerical approximation of solution y of delay differential equation (1) at the nodal point t n Open image in new window.

The paper is organized as follows. Section 2 presents the set of necessary and sufficient conditions for asymptotic stability of (11). In Section 3 we discuss some important properties of obtained results and compare them with the results known for another discretization as well as with the asymptotic stability conditions for the corresponding differential equation. Section 4 concludes the paper by final remarks.

2 Main result

In this section we state the necessary and sufficient conditions for asymptotic stability of (11). The analysis of (11) falls naturally into two parts according to the parity of k. For an effective and clear formulation of the main result, we introduce the symbols
τ 1 ( h ) = h + 2 h arcsin a + b 2 h ( 1 + a h ) | b | / arccos 1 + a 2 h 2 2 b 2 h 2 a 2 h 2 1 , τ 2 ( h ) = h + 2 h arccos a + b 2 h | ( 1 + a h ) b | / arccos 1 + a 2 h 2 2 b 2 h 2 | a 2 h 2 1 | , Open image in new window

which are utilized in these two parts, respectively.

Theorem 3 (I) Let k 2 Open image in new window be even. Then (11) is asymptotically stable if and only if one of the following conditions holds:
| b h | 1 , | b | + a < 0 , Open image in new window
(12)
2 < 2 b 2 h 2 < 1 a h , τ < τ 1 ( h ) . Open image in new window
(13)
  1. (II)
    Let k 3 Open image in new window be odd and m = ( k 1 ) / 2 Open image in new window. Then (11) is asymptotically stable if and only if one of the following conditions holds:
    a b < a , | b h | < 1 , Open image in new window
    (14)
    | b | + a < 0 , ( 1 ) m b h = 1 , Open image in new window
    (15)
    b + | a | < 0 , b h > 1 , τ < τ 2 ( h ) , Open image in new window
    (16)
    ( 1 ) m b + a < 0 , ( 1 ) m b h > 1 , τ < τ 2 ( h ) , Open image in new window
    (17)
    ( 1 ) m b + a > 0 , ( 1 ) m + 1 b h > 1 , τ < τ 2 ( h ) . Open image in new window
    (18)
     
Proof The proof is based on the application of Theorem 2 to (11) and the ensuing analysis of the obtained conditions. In particular, if we consider (11) in the form of (6), the coefficients α and β of (6) are given by
α = 1 + a h 1 a h , β = 2 b h 1 a h Open image in new window
(19)

and the indices and k are in the relation = k 1 Open image in new window.

Case (I): Investigating the case of k even, we utilize parts (iii) and (iv) of Theorem 2. Firstly, we focus on condition (iii): considering the coefficients (19), the assumption α < 0 Open image in new window implies | a h | < 1 Open image in new window. Thus, 7 is equivalent to | b | + a < 0 Open image in new window. Therefore, condition (iii) coincides with | b h | < a h < 1 Open image in new window.

Now, we analyze condition (iv): analogously, assumption α > 0 Open image in new window implies | a h | > 1 Open image in new window. Hence, (8) gives | b h | 1 Open image in new window providing a h < 1 Open image in new window, while for the case a h > 1 Open image in new window relation (8) cannot occur. We now turn to (10). Relation (10)1 can be read as 2 < 2 b 2 h 2 < 1 a h Open image in new window. Furthermore, the restriction (10)2 becomes
k 1 < 2 arcsin a + b 2 h | ( 1 + a h ) b | / arccos 1 + a 2 h 2 2 b 2 h 2 | a 2 h 2 1 | . Open image in new window

Since | a h | > 1 Open image in new window and k = τ / h Open image in new window, it can be written as τ < τ 1 ( h ) Open image in new window. Therefore, condition (iv) is satisfied if and only if either | b h | 1 Open image in new window, a h < 1 Open image in new window or (13).

Finally, Theorem 2 does not cover the case of α β = 0 Open image in new window (i.e., a h = 1 Open image in new window or b = 0 Open image in new window). In our case we do not consider the eventuality b = 0 Open image in new window with respect to the fact that we deal with the discretization of (1). Accordingly, for a h = 1 Open image in new window, equation (11) turns to
Y ( n + 1 ) b h Y ( n k + 1 ) = 0 , n = 0 , 1 , Open image in new window

and the necessary and sufficient condition for its asymptotic stability is given by Theorem 1 as | b h | < 1 Open image in new window. Summarizing the above discussion, we conclude that if k is even, (11) is asymptotically stable if either (12) or (13) holds.

Case (II): For k odd, we consider conditions (i) and (ii) of Theorem 2. Condition (i) can be rewritten as
2 b h 1 a h ( 1 + a h 1 a h ) 1 + m < 0 , | 1 + a h 1 a h | + | 2 b h 1 a h | < 1 Open image in new window
(20)
by use of (19). 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 (20) as
| a h | < 1 , b > 0 , a < b , a h < 1 , b > 0 , b h < 1 Open image in new window
(21)
for m odd and (21),
a h < 1 , b < 0 , b h < 1 Open image in new window

for m even.

In the Case (ii) of Theorem 2, condition (8) can be reformulated as
2 b h 1 a h ( 1 + a h 1 a h ) 1 + m > 0 , | 1 + a h 1 a h | + | 2 b h 1 a h | 1 . Open image in new window
(22)
An analogous analysis to that above shows that for m odd, (22) is equivalent to
| a h | < 1 , b < 0 , a b , a h < 1 , b < 0 , b h 1 . Open image in new window
(23)
In the case m even, condition (22) is satisfied if and only if (23) or
a h < 1 , b > 0 , b h 1 Open image in new window

holds. The above discussion of the Case (i), the part of (ii) considering (8) and including the case α = 0 Open image in new window (i.e., a h = 1 Open image in new window, | b h | < 0 Open image in new window, see Case (I)) gives (14)-(15).

Now it remains to analyze condition (9) adapted for equation (11) by (19), i.e.,
| | 1 + a h 1 a h | | 2 b h 1 a h | | < 1 < | 1 + a h 1 a h | + | 2 b h 1 a h | , τ < τ 2 ( h ) Open image in new window
under the assumption 2 b h 1 a h ( 1 + a h 1 a h ) 1 + m > 0 Open image in new window. In the same manner as above, we get the equivalency to the following set of conditions:
| a h | < 1 , b < 0 , 1 + a h 2 b h , b h < 1 , b < a , τ < τ 2 ( h ) , Open image in new window
(24)
| a h | < 1 , b < 0 , 1 + a h > 2 b h , b < | a | , τ < τ 2 ( h ) , Open image in new window
(25)
a h > 1 , b > 0 , 1 + a h 2 b h , b < a , τ < τ 2 ( h ) , Open image in new window
(26)
a h > 1 , b > 0 , 1 + a h > 2 b h , 1 < b h , τ < τ 2 ( h ) , Open image in new window
(27)
a h < 1 , b < 0 , 1 + a h < 2 b h , b h > 1 , τ < τ 2 ( h ) , Open image in new window
(28)
a h < 1 , b < 0 , 1 + a h 2 b h , b h > 1 , a < b , τ < τ 2 ( h ) Open image in new window
(29)
for m odd and (24), (25),
a h > 1 , b < 0 , 1 + a h 2 b h , b < a , τ < τ 2 ( h ) , Open image in new window
(30)
a h > 1 , b < 0 , 1 + a h > 2 b h , 1 < b h , τ < τ 2 ( h ) , Open image in new window
(31)
a h < 1 , b > 0 , 1 + a h < 2 b h , b h > 1 , τ < τ 2 ( h ) , Open image in new window
(32)
a h < 1 , b > 0 , 1 + a h 2 b h , b h > 1 , a < b , τ < τ 2 ( h ) Open image in new window
(33)

for m even. These conditions are jointly expressed by (16)-(18). In fact, (16) coincides with (24), (25). Condition (17) is equivalent to (28), (29) and (32), (33) for m odd and m even, respectively. Finally, (18) is the same as (26), (27) for m odd and (30), (31) for m even. The proof is complete. □

3 Asymptotic stability discussion

In this section we introduce some remarks and comments to the obtained result formulated in Theorem 3. First, we focus on the connection with the asymptotic stability properties of (1). Particularly, we investigate a limit form of Theorem 3 considering h 0 Open image in new window. In the Case (I) of k even, the asymptotic stability region of (11) becomes | b | + a < 0 Open image in new window. Let us note that with the exception of the boundary, this region corresponds to (2). In the Case (II) of k odd, it may be shown (by the L’Hospital rule) that the asymptotic stability conditions turn to
a b < a , | a | + b < 0 , τ < arccos ( a / b ) ( b 2 a 2 ) 1 / 2 Open image in new window

as h 0 Open image in new window. These are equivalent to the conditions defining the asymptotic stability region of (1).

Now we present the necessary and sufficient conditions for the asymptotic stability of the midpoint method discretization of (3) in the form
Y ( n + 2 ) Y ( n ) 2 b h Y ( n k + 1 ) = 0 , n = 0 , 1 , , Open image in new window
(34)

and then we focus on some of their monotony properties with respect to changing stepsize h. Finally, we compare the obtained stability intervals with the stability interval of the corresponding differential equation, as well as with the stability intervals for the forward Euler method discretization of (3).

Corollary 4 Equation (34) is asymptotically stable if and only if
k is odd, 0 > b > 1 h sin π h 2 τ . Open image in new window

Proof The assertion is an immediate consequence of Theorem 3. Setting a = 0 Open image in new window, we realize that conditions (12) and (13) cannot occur. Therefore, (34) is unstable for any b R Open image in new window in the case of k even.

Considering k odd, we investigate conditions (14)-(18). For a = 0 Open image in new window, there arises a contradiction in all conditions except for (16), which becomes
b < 0 , b h > 1 , τ < h + 2 h arccos ( b h ) arccos ( 1 2 b 2 h 2 ) . Open image in new window
(35)
We omit (35)2 because it is imposed by the domain of the last relation (35)3. Our next aim is to simplify the delay restriction (35)3 to the form more convenient for further analysis. We use the formula
arccos ( 1 2 x 2 ) = π 2 arccos x , x > 0 Open image in new window
and taking into account that function cosx is decreasing for x ( 0 , π / 2 ) Open image in new window, we arrive at
cos π ( τ h ) 2 τ > b h . Open image in new window
Since cos ( π / 2 x ) = sin x Open image in new window, we get
sin π h 2 τ > b h . Open image in new window
Finally, we rewrite (35) as
0 > b > 1 h sin π h 2 τ , Open image in new window

which is the necessary and sufficient condition for asymptotic stability of (34) providing k is odd. □

We emphasize that the stability regions are captured just by stability intervals for values of parameter b. We denote I τ M ( h ) = ( 1 h sin π h 2 τ , 0 ) Open image in new window stability intervals of (34) derived in Corollary 4. Next assertion describes the relation between stability intervals I τ M ( h ) Open image in new window with respect to stepsize h.

Theorem 5 Let 3 k 1 < k 2 Open image in new window be arbitrary positive odd integers and let h 1 = τ / k 1 > τ / k 2 = h 2 Open image in new window be corresponding stepsizes. Then
I τ M ( h 2 ) I τ M ( h 1 ) . Open image in new window
Proof Let us define a function
f ( h ) = 1 h sin π h 2 τ , h ( 0 , τ / 3 , Open image in new window
which represents the dependence of the left endpoint of I τ M ( h ) Open image in new window on stepsize h. Since the right endpoint is zero for any h, we discuss only the monotony of f ( h ) Open image in new window. Doing this, we drop the constraint h = τ / k Open image in new window and we consider f ( h ) Open image in new window as a function of a continuous argument h. Then
f ( h ) = 1 h 2 sin π h 2 τ π 2 τ h cos π h 2 τ , h ( 0 , τ / 3 . Open image in new window
Since sin π h 2 τ > 0 Open image in new window for h ( 0 , τ / 3 Open image in new window, f ( h ) > 0 Open image in new window when
cot π h 2 τ < 2 τ π h , h ( 0 , τ / 3 . Open image in new window
If we substitute x = π h 2 τ Open image in new window, the last relation becomes
tan x > x , x ( 0 , π / 6 . Open image in new window
(36)

Obviously, tan ( 0 ) = 0 Open image in new window and ( tan ( x ) ) = cos 2 ( x ) > 1 = x Open image in new window for x ( 0 , π / 6 Open image in new window. Therefore (36) holds for any x ( 0 , π / 6 Open image in new window. Thus, we have proved that f ( h ) > 0 Open image in new window for h ( 0 , τ / 3 Open image in new window and consequently I τ M ( h 2 ) I τ M ( h 1 ) Open image in new window. □

Next, we compare stability intervals I τ M ( h ) Open image in new window with the stability interval of (3), which we denote I τ = ( π / ( 2 τ ) , 0 ) Open image in new window.

Remark 6 An important property is the behavior of I τ M ( h ) Open image in new window as h 0 Open image in new window. Using the L’Hospital rule, we may see that
lim h 0 1 h sin π h 2 τ = lim h 0 π 2 τ cos π h 2 τ = π 2 τ . Open image in new window

Therefore, I τ M ( h ) Open image in new window is approaching I τ Open image in new window as h 0 Open image in new window.

Remark 7 In the proof of Theorem 5 we have shown that 1 h sin π h 2 τ Open image in new window is an increasing function on h ( 0 , τ / 3 Open image in new window. Considering also Remark 6, we conclude that I τ I τ M ( h ) Open image in new window for any h = τ / k Open image in new window, where k is odd. Note that the midpoint method discretization of (3) is not asymptotically stable.

Finally, we discuss a relation between I τ M ( h ) Open image in new window and asymptotic stability intervals for the forward Euler discretization of (3). They are derived in [12], and we denote them as I τ E ( h ) = ( 2 h cos π τ 2 τ + h , 0 ) Open image in new window.

Theorem 8 Let k 3 Open image in new window be an arbitrary positive odd integer and let h = τ / k Open image in new window be the corresponding stepsize. Then
I τ M ( h ) I τ E ( h ) . Open image in new window
Proof Since the right endpoints of I τ M ( h ) Open image in new window and I τ E ( h ) Open image in new window are zero for any h, we investigate only the behavior of the left endpoints with respect to changing stepsize h. We define a function
g ( h ) = 2 h cos π τ 2 τ + h , h ( 0 , τ , Open image in new window
which expresses the dependence of the left endpoint of I τ E ( h ) Open image in new window on h. In the further analysis, we drop the constraint h = τ / k Open image in new window and consider both functions f ( h ) Open image in new window and g ( h ) Open image in new window to be functions with a continuous argument for h ( 0 , τ Open image in new window (we extend the domain of f ( h ) Open image in new window to simplify the proof). Thus our aim is to show that f ( h ) g ( h ) < 0 Open image in new window for any h ( 0 , τ ) Open image in new window, i.e.,
sin π h 2 τ + 2 cos π τ 2 τ + h < 0 , h ( 0 , τ ) . Open image in new window
(37)

To do this, we introduce the following proposition.

Lemma Let F C 3 a , b Open image in new window be a function such that F ( a ) = F ( b ) = 0 Open image in new window, F ( a ) 0 Open image in new window, F ( b ) > 0 Open image in new window, F ( a ) < 0 Open image in new window, F ( b ) > 0 Open image in new window and F ( t ) > 0 Open image in new window for all t ( a , b ) Open image in new window. Then F ( t ) < 0 Open image in new window for all t ( a , b ) Open image in new window.

Proof Since F ( t ) > 0 Open image in new window for all t ( a , b ) Open image in new window, the function F ( t ) Open image in new window is increasing. Since F ( a ) < 0 < F ( b ) Open image in new window, there is a unique point t 1 ( a , b ) Open image in new window such that F ( t 1 ) = 0 Open image in new window. Thus, the function F ( t ) Open image in new window is decreasing in ( a , t 1 ) Open image in new window and increasing in ( t 1 , b ) Open image in new window. Further, since F ( t 1 ) < F ( a ) 0 Open image in new window and F ( b ) > 0 Open image in new window, there is a unique point t 2 ( a , b ) Open image in new window such that F ( t 2 ) = 0 Open image in new window. Therefore, F ( t ) Open image in new window is decreasing in ( a , t 2 ) Open image in new window and increasing in ( t 2 , b ) Open image in new window. Taking into account F ( a ) = F ( b ) = 0 Open image in new window, we obtain that F ( t ) < 0 Open image in new window for t ( a , b ) Open image in new window. □

Next, we denote s = 2 + h / τ Open image in new window. Then we define
G ( s ) = sin π s 2 + 2 cos π s , s ( 2 , 3 ) , Open image in new window
which is equivalent to the left-hand side of (37). It holds that G ( 2 ) = G ( 3 ) = 0 Open image in new window, G ( 2 ) = 0 Open image in new window, G ( 3 ) = 3 π 9 > 0 Open image in new window, G ( 2 ) = π 2 < 0 Open image in new window and G ( 3 ) = π 2 4 π 2 81 2 3 π 27 > 0 Open image in new window. Further
G ( s ) = 12 π 2 s 5 cos π s π 3 8 cos π s 2 + 2 π s 6 ( 6 s 2 π 2 ) sin π s > 0 , Open image in new window

since each term in the sum is positive for all s ( 2 , 3 ) Open image in new window. Then by the previous lemma, we have that G ( s ) < 0 Open image in new window for all s ( 2 , 3 ) Open image in new window and consequently f ( h ) < g ( h ) Open image in new window for h ( 0 , τ ) Open image in new window, which concludes the proof.  □

4 Conclusions

To summarize the previous, the main result formulated in Theorem 3 describes the asymptotic stability regions of difference equation (11). This equation actually represents a discretization of delay differential equation (1) by a modified midpoint rule. It was shown that the asymptotic stability regions depend not only on the value of stepsize h, but also on the parity of k. In the case a = 0 Open image in new window, the obtained result was given to the connection with the results known for the Euler discretization of (3). Moreover, the connection with asymptotic stability properties of delay differential equation (3) was also mentioned. This discussion points out some interesting properties of the stability regions for the discrete form of the delay differential equation (1). The authors believe that analogous investigation is possible also for more complicated numerical formulae (applied to (1)) as far as there are known stability criteria for corresponding difference equations. Such analysis may be done, e.g., for the Θ-method.

Notes

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 P201/11/0768 Qualitative properties of solutions of differential equations and their applications of the Czech Science Foundation.

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Authors and Affiliations

  1. 1.Institute of Mathematics, Brno University of TechnologyBrnoCzech Republic

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