Numerical oscillation and non-oscillation for differential equation with piecewise continuous arguments of mixed type

Applied Mathematics and Computation 299 (2017) 16–27

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Numerical oscillation and non-oscillation for differential equation with piecewise continuous arguments of mixed typeR Jianfang Gao School of Mathematical Sciences, Harbin Normal University, Harbin 150025, China

a r t i c l e

i n f o

Keywords: Oscillation Numerical solution Runge–Kutta method Delay differential equation

a b s t r a c t The paper deals with the oscillation and non-oscillation of the Runge–Kutta methods for a differential equation with piecewise continuous arguments of mixed type. The conditions of the oscillation and non-oscillation for the Runge–Kutta method are obtained. It is proved that oscillation of the analytic solution is not preserved by the Runge–Kutta method under any conditions. The conditions under which the non-oscillation of analytic solutions is preserved by the Runge–Kutta method are obtained. Some numerical experiments are given. © 2016 Elsevier Inc. All rights reserved.

1. Introduction Oscillatory behavior is one of the main considerations in the qualitative study of delay differential equations (DDEs) and is the subject of many investigations. In the last few decades the oscillatory theory of DDEs has been extensively developed. We refer to [1] and [2] for the general theory of oscillation. In recent years, there has been much interest in researching the differential equations with piecewise constant argument (EPCA) [3–12]. The strong interest in such equations is motivated by the fact that they represent a hybrid of continuous and discrete dynamical systems and combine the properties of both differential and difference equations. However, all of the researchers mostly pay attention to the oscillation of analytic solutions not numerical solutions, such as [13–21] and so on. As is known to all, various models in biology, mechanics, and electronics are developed by the numerical solutions of EPCA, such as [22]. Hence the study on the numerical solution of DDEs is needful. In 2007, professor Liu, Gao and Yang (see [23]) firstly investigated oscillation of numerical solution in the θ -methods for a kind of differential equations with piecewise constant argument, which is delay type, and the preservation of oscillation for θ -methods was studied. In 2009, Liu, Gao and Yang (see [24]) discussed the numerical oscillation of the same equation for Runge–Kutta methods, and also the preservation of oscillation and non-oscillation for Runge–Kutta methods are investigated. For a kind of nonlinear DDEs of population dynamics and a linear neutral DDEs, Gao and Liu (see [25,26]) investigated the numerical oscillations in 2011. In the same year, Wang et al. (see [27]) discussed stability and oscillation of EPCA of alternately advanced and retarded type and stability and oscillation of another type EPCA were studied by Song and Liu (see [28,29]) in 2012. In [30], Wang and Zhu investigated stability of EPCA of mixed type, the numerical oscillation is not discussed. In this paper, we will investigate the numerical oscillation for this kind of equations.

R This work is supported by the Natural Science Foundation of Heilongjiang Province of China (A201411, F2015012) and the National Natural Science Foundation of China (11401145). E-mail addresses: [email protected], [email protected]

http://dx.doi.org/10.1016/j.amc.2016.11.031 0 096-30 03/© 2016 Elsevier Inc. All rights reserved.

J. Gao / Applied Mathematics and Computation 299 (2017) 16–27

17

The general form of EPCA is

x (t ) = f (t, x(t ), x(α1 (t )), x(α2 (t )), x(α3 (t ))),

t ≥ 0,

x(−1 ) = x−1 , x(0 ) = x0 ,

(1.1)

where the arguments α i (t) (i = 1, 2, 3 ) have intervals of constancy. In this paper, we consider the following differential equation with piecewise continuous argument (EPCA) of mixed type:

x (t ) = px(t ) + p−1 x([t − 1] ) + p0 x([t] ) + p1 x([t + 1] ),

t ≥ 0,

x(−1 ) = x−1 , x(0 ) = x0 ,

(1.2)

where p, p−1 , p0 , p1 , x−1 , x0 are real constants, [·] denotes the greatest integer function and p−1 = 0, p1 = 0. Definition 1.1 [31]. A solution of Eq. (1.2) on [0, ∞) is a function x(t) satisfying the conditions: (1) x(t) is continuous on [0, ∞); (2) The derivative x (t) exists at each point t ∈ [0, ∞), with the possible exception of the points [t] ∈ [0, ∞), where one-sided derivatives exist; (3) Eq. (1.2) is satisfied on each interval [n, n + 1 ) ⊂ [0, ∞ ) with integral end-points. The following theorem gives the solution of Eq. (1.2). Theorem 1.2 [31]. If p = 0, p−1 = 0 and q1 = 1, then Eq. (1.2) has on [0, ∞) a unique solution

x(t ) = m−1 ({t } )c[t−1] + m0 ({t } )c[t] + m1 ({t } )c[t+1] ,

(1.3)

where {t} is the fractional part of t and

λ[1t+1] (x0 − λ2 x−1 ) + (λ1 x−1 − x0 )λ[2t+1] , λ1 − λ2 m−1 (t ) = (e pt − 1 ) p−1 p−1 , m0 (t ) = e pt + (e pt − 1 ) p−1 p0 , m1 (t ) = (e pt − 1 ) p−1 p1 , c[t] =

q−1 = m−1 (1 ),

q0 = m0 ( 1 ),

q1 = m1 ( 1 ),

(1.4) (1.5) (1.6)

λ1 and λ2 are the roots of equation

(1 − q1 )λ2 − q0 λ − q−1 = 0. Proof. Let xn (t) be the solution of Eq. (1.2) on the interval [n, n + 1 ). If we let cn = x(n ) for integer n, then we have the equation

xn (t ) = pxn (t ) + p−1 cn−1 + p0 cn + p1 cn+1

(1.7)

with the solution

xn (t ) = e p(t−n ) cn + p−1 (e p(t−n ) − 1 )( p−1 cn−1 + p0 cn + p1 cn+1 ), which can be written, by virtue of (1.5), as

xn (t ) = m−1 (t − n )cn−1 + m0 (t − n )cn + m1 (t − n )cn+1 .

(1.8)

From (1.8) we can see that it suffices to know the constants cn in order to determine x(t). Taking into account

xn (n + 1 ) = xn+1 (n + 1 ) = cn+1 , we obtain

cn+1 = m−1 (1 )cn−1 + m0 (1 )cn + m1 (1 )cn+1 , n ≥ 0. With the notations (1.6), this equation takes the form

(1 − q1 )cn+1 − q0 cn − q−1 cn−1 = 0. Its particular solution is sought as cn =

λn .

(1.9) Then

(1 − q1 )λ − q0 λ − q−1 = 0. 2

(1.10)

Eq. (1.10) has two nontrivial solutions because of p = 0 , p−1 = 0 and q1 = 1. If the roots λ1 and λ2 of Eq. (1.10) are different, the general solution of Eq. (1.9) is

cn = k1 λn1 + k2 λn2 , with arbitrary constants k1 and k2 . For n = −1 and n = 0, in view of x(−1 ) = x−1 and x(0 ) = x0 , we have −1 k1 λ−1 1 + k2 λ2 = x−1 ,

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J. Gao / Applied Mathematics and Computation 299 (2017) 16–27

k1 + k2 = x0 , then

k1 =

λ1 (x0 − λ2 x−1 ) , λ1 − λ2

k2 =

λ2 (λ1 x−1 − x0 ) . λ1 − λ2

These results establish (1.4). If λ1 = λ2 = λ, then general solution of Eq. (1.9) is

cn = k1 λn + nk2 λn , in view of x(−1 ) = x−1 and x(0 ) = x0 , we get

cn = λn [x0 (n + 1 ) − λx−1 n], which is the limiting case of (1.4) as λ1 → λ2 . The uniqueness of solution follows from its continuity and from the uniqueness of the problem xn (n ) = cn for (1.7) on each interval [n, n + 1].  Remark 1.3. If p = 0, then

xn (t ) = cn + ( p−1 cn−1 + p0 cn + p1 cn+1 )(t − n ), which is the limiting case of (1.3) as p → 0. In the following we will give the definition of the oscillation and non-oscillation. Definition 1.4. A non-trivial solution of Eq. (1.2) is said to be oscillatory if there exists a sequence {tk }∞ such that tk k=1 → ∞ as k → ∞ and x(tk )x(tk−1 ) ≤ 0; otherwise it is called non-oscillatory. We say Eq. (1.2) is oscillatory if all the nontrivial solution of Eq. (1.2) are oscillatory; we say Eq. (1.2) is non-oscillatory if all the non-trivial solutions of Eq. (1.2) are non-oscillatory. Theorem 1.5 [1]. Eq. (1.2) is oscillatory if and only if the following conditions are satisfied:

(i ) p1 ≥ 1, p0 > −1, p−1 > 0 (ii ) p1 ≤ 1, p0 < −1, p−1 < 0 where p = 0, p p , p0 > −p , p−1 > 0 1 − ep e −1 p p p1 ≤ − , p0 < −p , p−1 < 0 where p = 0. 1 − ep e −1

(iii ) p1 ≥ − ( iv )

Eq. (1.2) is non-oscillatory if and only if any of the following conditions is satisfied:

(1 ) (2 ) (3 ) (4 ) (5 ) (6 )

p1 > 1, p0 ≤ −1, p−1 ≤ 0 p1 > 1, p0 > −1, p−1 ≤ 0 p1 > 1, p0 < −1, p−1 ≥ 0

where p = 0,

p1 < 1, p0 ≥ −1, p−1 ≥ 0 p1 < 1, p0 < −1, p−1 ≥ 0 p1 < 1, p0 > −1, p−1 ≤ 0

(7 ) (8 ) (9 ) (10 ) (11 ) (12 )

p p1 > − 1−e p , p0 ≥

p1 > p1 > p1 < p1 < p1 <

p − 1−e p, p − 1−e p , p − 1−e p, p − 1−e p, p − 1−e p ,

p0 < p0 < p0 ≥ p0 < p0 >

p , e−p −1 p , e−p −1 p , e−p −1 p , −p e −1 p , e−p −1 p , e−p −1

p−1 ≤ 0 p−1 ≤ 0 p−1 ≥ 0 p−1 ≥ 0

where p = 0.

p−1 ≥ 0 p−1 ≤ 0

Now we consider a scalar difference equation in the form

xn+ j + d1 xn+ j−1 + · · · + d j xn = 0, where n, j ∈

N+

and di ∈ R for i = 1, . . . , j.

(1.11)

J. Gao / Applied Mathematics and Computation 299 (2017) 16–27

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Definition 1.6. A nontrivial solution xn of Eq. (1.11) is said to be oscillatory if there exists a sequence {nk } such that nk → ∞ as k → ∞ and x(nk )x(nk−1 ) ≤ 0; otherwise it is called non-oscillatory. We say Eq. (1.11) is oscillatory if all the nontrivial solutions of Eq. (1.11) are oscillatory; we say Eq. (1.11) is non-oscillatory if all the nontrivial solutions of Eq. (1.11) are nonoscillatory. Remark 1.7. If a solution x(t) of Eq. (1.2) is continuous and non-oscillatory, then it must be eventually positive or negative. That is, there exists a T ∈ R such that x(t) is positive for t ≥ T or negative for t ≥ T. Similarly, if xn of Eq. (1.11) is nonoscillatory, then xn is eventually positive or negative. Theorem 1.8 [1]. Assume that j ∈ N + and di ∈ R for i = 1, . . . , j. Then Eq. (1.11) oscillates if and only if the characteristic equation

λ j + λ j−1 d1 + · · · + λd j−1 + d j = 0 has no positive real roots. Remark 1.9. From Theorems 1.5 and 1.8, we can see that the oscillatory property of the solution is independent of the initial conditions. 2. Analysis of oscillation and non-oscillation 2.1. Runge–Kutta method In this subsection, we consider a ν -stage Runge–Kutta method, which is completely specified by its Butcher array c

A bT

. We always suppose b1 + b2 + · · · + bν = 1 and 0 ≤ c1 ≤ c2 ≤  ≤ cν ≤ 1. The general ν -stage Runge–Kutta

methods (A, b, c) are defined by

xn+1 = xn + h



ν 

bi ki ,

i=1

ki = f tn + ci h, xn + h

ν 

 ai j k j , i = 1 , 2 , . . . , ν .

(2.1)

i=1

1 be a given stepsize, m be a positive integer and gridpoints tn be defined by tn = nh(n = 1, 2, 3, . . . ). According to m [24], the adaptation of the Runge–Kutta methods to Eq. (1.1) leads to a numerical process of the following type, generating approximations x1 , x2 , . . . to the exact solution x(t) at the gridpoints tn (n = 1, 2, . . . ), Let h =

xn+1 = xn + h

ν 

bi f (tn + ci h, yi(n ) , ui(n ) , zi(n ) , vi(n ) ),

i=1

yi(n ) = xn + h

ν 

ai j f (tn + c j h, y(jn ) , u(jn ) , z(jn ) , v(jn ) ),

(2.2)

j=1

the arguments ui(n ) , zi(n ) and vi(n ) denote the given approximations to x(α1 (tn + ci h )), x(α2 (tn + ci h )) and x(α3 (tn + ci h )) (i = 1, 2, . . . , ν ; n = 0, 1, 2, . . .), respectively. The application of the process (2.2) to Eq. (1.2) yields

xn+1 = xn + h

ν 

bi ( pyi(n ) + p−1 zi(n−m ) + p0 zi(n ) + p1 zi(n+m ) ),

i=1

yi(n ) = xn + h

ν 

ai j ( py(jn ) + p−1 z(jn−m ) + p0 z(jn ) + p1 z(jn+m ) ).

(2.3)

j=1

If we denote n = km + l (l = 0, 1, . . . , m − 1 ), then zi(km+l ) can be defined as xkm . Let Y (n ) = (y1(n ) , y2(n ) , . . . , yν(n ) )T . Then Eq. (2.3) reduces to

xkm+l+1 = xkm+l + hpbT Y (km+l ) + hp−1 x(k−1)m + hp0 xkm + hp1 x(k+1)m , Y (km+l ) = xkm+l e + hpAY (km+l ) + hp−1 Aex(k−1)m + hp0 Aexkm + hp1 Aex(k+1)m , where l = 0, 1, . . . , m − 1 and e =

( 1, 1, . . . , 1 ) T ,

therefore we have

(2.4)

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J. Gao / Applied Mathematics and Computation 299 (2017) 16–27

p−1 (R(s ) − 1 )x(k−1)m p p0 p + (R(s ) − 1 )xkm + 1 (R(s ) − 1 )x(k+1)m if p = 0, p p = xkm+l + hp−1 x(k−1)m + hp0 xkm + hp1 x(k+1)m if p = 0,

xkm+l+1 = R(s )xkm+l +

xkm+l+1

where s = hp, l = 0, 1, . . . , m − 1 and R(s ) = 1 +

x(k+1)m =

1 + p0 p−1 x + x 1 − p1 km 1 − p1 (k−1)m

sbT (I

− sA )−1 e

is the stability function of the method. Therefore

for p = 0,

xn = xkm+l = (1 + lhp0 )xkm + lhp−1 x(k−1)m + lhp1 x(k+1)m x(k+1)m =

+

(2.6) for p = 0,

Rm (s ) + ( p0 /p)(Rm (s ) − 1 ) ( p−1 /p)(Rm (s ) − 1 ) xkm + x 1 − ( p1 /p)(Rm (s ) − 1 ) 1 − ( p1 /p)(Rm (s ) − 1 ) (k−1)m

xkm+l = [Rl (s ) +

(2.5)

(2.7)

for p = 0,

(2.8)

p0 l p (R (s ) − 1 )]xkm + −1 (Rl (s ) − 1 )x(k−1)m p p

p1 l (R (s ) − 1 )x(k+1)m p

for p = 0.

(2.9)

2.2. Analysis of oscillation and non-oscillation Definition 2.1. We say a Runge–Kutta method preserves oscillations of Eq. (1.2), if Eq. (1.2) oscillates then there is an h0 such that Eq. (2.7) or Eq. (2.9) oscillates for h < h0 . Similarly, we say a Runge–Kutta method preserves non-oscillations of Eq. (1.2), if Eq. (1.2) is non-oscillatory then there is an h0 such that Eq. (2.7) or Eq. (2.9) is non-oscillatory for h < h0 . One would be interested in the study of special methods for oscillatory problems (see [32] and [33]). However, many of them are based on Runge–Kutta schemes but are not directly the Runge–Kutta methods studied in this paper. Perhaps study of such methods can be made later. First, we assume that p = 0. Theorem 2.2. Eq. (2.6) is non-oscillatory if and only if any of the following conditions is satisfied:

p1 > 1, p0 ≤ −1, p−1 ≤ 0; p1 > 1, p0 > −1, p−1 ≤ 0; p1 > 1, p0 < −1, p−1 ≥ 0; p1 < 1, p0 ≥ −1, p−1 ≥ 0; p1 < 1, p0 < −1, p−1 ≥ 0; p1 < 1, p0 > −1, p−1 ≤ 0. Eq. (2.6) is oscillatory if and only if either of the following conditions is satisfied:

p1 ≥ 1, p0 > −1, p−1 > 0; p1 ≤ 1, p0 < −1, p−1 < 0. Proof. According to Theorem 1.8, Eq. (2.6) is non-oscillatory if and only if

λ2 +

1 + p0 p λ + −1 = 0 p1 − 1 p1 − 1

has at least one positive root, which is equivalent to

λ1 > 0, λ2 ≤ 0, or

λ1 > 0, λ2 ≥ 0. We know λ1 > 0, λ2 ≥ 0, applying the relation between the roots and coefficients to the above equation , which is equivalent to

 p0 λ1 + λ2 = − 1+ > 0, p1 −1 λ1 λ2 = pp1−1 ≥ 0, −1

J. Gao / Applied Mathematics and Computation 299 (2017) 16–27

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that is to say

p1 > 1, p0 < −1, p−1 ≥ 0; p1 < 1, p0 > −1, p−1 ≤ 0.

λ1 > 0, λ2 ≤ 0 is equivalent to λ1 λ2 ≤ 0 that is to say



or



p−1 < 0, p1 > 1, p−1 > 0, p1 < 1.

 The following theorem gives the relationship of non-oscillations and oscillations between {xn } and {xkm }. Theorem 2.3. Suppose {xkm } and {xn } are given by (2.6) and (2.7) respectively, then (1) {xn } is non-oscillatory if and only if {xkm } is non-oscillatory; (2) {xn } is oscillatory if and only if {xkm } is oscillatory. Proof. We only prove (1). It is easy to see that {xn } is non-oscillatory, then {xkm } is non-oscillatory. Now we assume {xkm } is non-oscillatory, without loss of generality, {xkm } is eventually positive, that is, there exists an integer k0 , such that xkm > 0 for k > k0 . We will prove xkm+l > 0 for all k with k > k0 + 1 and l = 0, 1, 2, . . . , m − 1, we know 1 − lh > 0. According to (2.6), we have

x(k−1)m =

1 + p0 1 − p1 x − x , p−1 (k+1)m p−1 km

we have

xkm+l = (1 + lhp0 )xkm + lhp1 x(k+1)m + lhp−1

1 − p

1

x(k+1)m −

1 + p0 x p−1 km



p−1 = [(1 + lhp0 ) − lh(1 + p0 )]xkm + [lhp1 + lh(1 − p1 )]x(k+1)m = (1 − lh )xkm + lhx(k+1)m > 0,

Hence xkm+l > 0 .



Remark 2.4. According to Theorems 1.5, 2.2 and non-oscillation of Eq. (1.2) for p = 0.

2.3, we know all Runge–Kutta methods can preserve oscillations and

In the rest of the paper, we assume that p = 0.

P (s ) , where P(s), Q (s )  Q(s) are polynomials, P(s) is a continuous function at the neighborhood of 0, and R(0 ) = R (0 ) = 1, Q (0 ) = 1, there exists a δ1 = inf{1,R(s)−1 ∪ 1,Q (s) ∪ {+∞}} ∈ (0, +∞], δ2 = sup{2,R(s)−1 ∪ 2,R(s)+1 ∪ {−∞}} ∈ [−∞, 0 ), such that For a continuous function f(s), let 1, f (s ) = {s < 0 : f (s ) = 0}, 2, f (s ) = {s > 0 : f (s ) = 0}. Since R(s ) =

1 < R (s ) < ∞ 0 < R (s ) < 1

for for

0 < s < δ1 ,

δ2 < s < 0.

In the following, we always suppose h <

δ , where δ = min{|δ1 |, |δ2 |}. | p|

The following theorem gives the relationship of non-oscillations and oscillations between {xn } and {xkm }. Theorem 2.5. Suppose {xkm } and {xn } are given by (2.8) and (2.9) respectively, then (1) {xn } is non-oscillatory if and only if {xkm } is non-oscillatory; (2) {xn } is oscillatory if and only if {xkm } is oscillatory. Proof. We only prove (1). It is easy to see that {xn } is non-oscillatory, then {xkm } is non-oscillatory. Now we assume {xkm } is non-oscillatory, without loss of generality, {xkm } is eventually positive, that is, there exists an integer k0 , such that xkm > 0 for k > k0 . We will prove xkm+l > 0 for all k with k > k0 + 1 and l = 0, 1, 2, . . . , m − 1. According to Eq. (2.8), we have

x(k−1)m =

Rm (s ) + pp0 (Rm (s ) − 1 ) ( Rm ( s ) − 1 ) x − xkm , ( k +1 ) m p−1 p−1 ( Rm ( s ) − 1 ) ( Rm ( s ) − 1 ) p p

1−

p1 p

22

J. Gao / Applied Mathematics and Computation 299 (2017) 16–27

We can combine Eq. (2.8) with (2.9), we have

p0 l p (R (s ) − 1 )]xkm + 1 (Rl (s ) − 1 )x(k+1)m p p

xkm+l = [Rl (s ) + +



Rl ( s ) +

=

 + =



Rm (s ) + pp0 (Rm (s ) − 1 ) 1 − p1 ( Rm ( s ) − 1 ) p−1 l (R (s ) − 1 ) p−1 p m x(k+1)m − xkm p−1 p (R (s ) − 1 ) ( Rm ( s ) − 1 ) p p



Rl ( s ) − 1 m p0 l p0 m (R (s ) − 1 ) − m R (s ) + (R (s ) − 1 ) p R (s ) − 1 p



p1 m p1 l Rl ( s ) − 1 (R (s ) − 1 ) + m 1− (R (s ) − 1 ) p R (s ) − 1 p





xkm



x(k+1)m

Rm ( s ) − Rl ( s ) Rl ( s ) − 1 xkm + m x . m R (s ) − 1 R (s ) − 1 (k+1)m

If p > 0, then R(s) > 1 and Rl (s) < Rm (s), we have

Rl ( s ) − 1 >0 Rm ( s ) − 1

and

Rm ( s ) − Rl ( s ) > 0, Rm ( s ) − 1

If p < 0, then 0 < R(s) < 1 and Rm (s) < Rl (s), we have

Rl ( s ) − 1 >0 Rm ( s ) − 1 Hence xkm+l > 0.

and

Rm ( s ) − Rl ( s ) > 0. Rm ( s ) − 1



Theorem 2.6. Eq. (2.8) is non-oscillatory if and only if any of the following conditions is satisfied:

p , p0 1 − Rm ( s ) p >− , p0 1 − Rm ( s ) p >− , p0 1 − Rm ( s ) p <− , p0 1 − Rm ( s ) p <− , p0 1 − Rm ( s ) p <− , p0 1 − Rm ( s )

p1 > −

≥

p1

<

p1 p1 p1 p1

< ≥ < >

p , p−1 R−m (s ) − 1 p , p−1 R−m (s ) − 1 p , p−1 R−m (s ) − 1 p , p−1 R−m (s ) − 1 p , p−1 R−m (s ) − 1 p , p−1 R−m (s ) − 1

≤ 0; ≤ 0; ≥ 0; ≥ 0; ≥ 0; ≤ 0.

Proof. According to Theorem 1.8, Eq. (2.8) is non-oscillatory if and only if

λ2 +

( p0 /p)(1 − Rm (s )) − Rm (s ) ( p−1 /p)(1 − Rm (s )) λ + =0 1 − ( p1 /p)(Rm (s ) − 1 ) 1 − ( p1 /p)(Rm (s ) − 1 )

has at least one positive root, which is equivalent to

λ1 > 0, λ2 ≤ 0, or λ1 > 0, λ2 ≥ 0. If λ1 > 0, λ2 ≥ 0, applying the relation between the roots and coefficients to the above equation , we have

⎧ ( p /p)(1 − Rm (s )) − Rm (s ) ⎪ > 0, ⎨λ1 + λ2 = − 0 1 + ( p1 /p)(1 − Rm (s )) ( p−1 /p)(1 − Rm (s )) ⎪ ⎩ λ1 λ2 = ≥ 0, 1 + ( p1 /p)(1 − Rm (s ))

that is to say

p p , p0 < −m , p−1 ≥ 0; 1 − Rm ( s ) R (s ) − 1 p p p1 < − , p0 > −m , p−1 ≤ 0. 1 − Rm ( s ) R (s ) − 1

p1 > −

If λ1 > 0, λ2 ≤ 0, then



p−1 ≥ 0, p1 < Rm (sp)−1 ,

J. Gao / Applied Mathematics and Computation 299 (2017) 16–27

or



23

p−1 ≤ 0, p1 > Rm (sp)−1 .

 Theorem 2.7. Eq. (2.8) is oscillatory if and only if the following conditions are satisfied:

p p , p0 < −m , p−1 < 0; 1 − Rm ( s ) R (s ) − 1 p p or p1 ≥ − , p0 > −m , p−1 > 0. 1 − Rm ( s ) R (s ) − 1 p1 ≤ −

p p Let B = − 1−e p , B (m ) = − 1−Rm (s ) and C =

p , e−p −1

C (m ) =

p . R−m (s )−1

Theorem 2.8. B ≥ B(m), C ≤ C(m) (B ≤ B(m), C ≥ C(m) ) if any of the following conditions is satisfied: • p > 0 and es ≥ R(s) (es ≤ R(s)); • p < 0 and es ≤ R(s) (es ≥ R(s)). Corollary 2.9. The Runge–Kutta method preserves non-oscillation of Eq. (1.2) if and only of either B ≥ B(m), C ≤ C(m); B ≤ B(m), C ≥ C(m) or B ≥ B(m), C ≥ C(m). Corollary 2.10. The Runge–Kutta method preserves oscillation of Eq. (1.2) if and only of either B ≥ B(m), C ≥ C(m) or B ≤ B(m), C ≤ C(m). By Theorem 2.8 and Corollary 2.10, we have the following result. Theorem 2.11. Runge–Kutta methods cannot preserve oscillation of Eq. (1.2) under any conditions. The following result is given by Theorem 2.8 and Corollary 2.9. Theorem 2.12. Runge–Kutta methods can preserve non-oscillation of Eq. (1.2) under any conditions. Remark 2.13. In 2011, Wang and Zhu (see [27]) considered numerical stability of Eq. (1.2) in Runge–Kutta method and gave the conditions under which Runge–Kutta method can preserve stability of Eq. (1.2). According to Theorems 2.11 and 2.12 we know that no matter how es and R(s) have any relationship, Runge–Kutta method cannot preserve oscillation and can preserve non-oscillation of Eq. (1.2) under any conditions, which is different from stability. Hence we can see that numerical methods which are convergent and preserve stability not necessarily preserve oscillation. It is necessary to study the conditions under which the numerical solution and the analytic solution have the same oscillatory and non-oscillatory properties. 3. Numerical examples Example 1. Consider the equation

x (t ) = 0.5x([t − 1] ) + 0.7x([t] ) + 2x([t + 1] ), x(−1 ) = 1, x(0 ) = 1,

(3.1)

where p−1 = 0.5, p0 = 0.7, p1 = 2, (i) in Theorem 1.5 is satisfied. In Figs. 1 and 2, we draw the analytical solution and the numerical solution in 2-stage Lobatto IIIC method with h = 0.02. From the two figures, we can see the analytical solution and the numerical solution are oscillatory, which is in agreement with Remark 2.4. Example 2. Consider the equation

x (t ) = −0.05x([t − 1] ) − 1.87x([t] ) + 2x([t + 1] ), x(−1 ) = 1, x(0 ) = 1,

(3.2)

where p−1 = −0.05, p0 = −1.87, p1 = 2, (1) in Theorem 1.5 is satisfied. In Figs. 3 and 4, we draw the analytical solution and the numerical solution in 1-stage Radau IIA method with h = 0.01. From the two figures, we can see the analytical solution and the numerical solution are non-oscillatory, which is in agreement with Remark 2.4. Example 3. Consider the equation

x (t ) = −x(t ) − x([t − 1] ) − 0.5x([t] ) + 1.6x([t + 1] ), x(−1 ) = 1, x(0 ) = 1,

(3.3)

24

J. Gao / Applied Mathematics and Computation 299 (2017) 16–27

9

x 10

10 8 6 4

x(t)

2 0 −2 −4 −6 −8

0

10

20

30

40 t

50

60

70

80

70

80

70

80

Fig. 1. The analytical solution of Eq. (3.1).

9

10

x 10

8 6

2

n

x (t)

4

0 −2 −4 −6 −8

0

10

20

30

40 t

50

60

Fig. 2. The numerical solution of Eq. (3.1).

1 0.9 0.8 0.7

x(t)

0.6 0.5 0.4 0.3 0.2 0.1 0

0

10

20

30

40 t

50

60

Fig. 3. The analytical solution of Eq. (3.2).

J. Gao / Applied Mathematics and Computation 299 (2017) 16–27

25

1 0.9 0.8 0.7

xn(t)

0.6 0.5 0.4 0.3 0.2 0.1 0

0

10

20

30

40 t

50

60

70

80

70

80

70

80

Fig. 4. The numerical solution of Eq. (3.2).

95

14

x 10

12

10

x(t)

8

6

4

2

0

0

10

20

30

40 t

50

60

Fig. 5. The analytical solution of Eq. (3.3).

95

14

x 10

12

10

n

x (t)

8

6

4

2

0

0

10

20

30

40 t

50

60

Fig. 6. The numerical solution of Eq. (3.3).

26

J. Gao / Applied Mathematics and Computation 299 (2017) 16–27

a

b

x(t) 2

x

n

2

1.5

1.5

1 1 0.5 0.5

0 −0.5

0

50

c

100

150

0

200

2

0.5

1.5

0

1

−0.5

0.5

50

100

150

200

100

150

200

150

200

sign(xn)

1

0

50

d

sign(x(t))

−1

0

0

0

50

100

Fig. 7. The analytical solution(a) and the numerical solution(b) of Eq. (3.4).

where p = −1, p−1 = −1, p0 = −0.5, p1 = 1.6, (7) in Theorem 1.5 is satisfied. In Figs. 5 and 6, we draw the analytical solution and the numerical solution in 1-stage Radau IA method with h = 0.02. From the two figures, we can see the analytical solution and the numerical solution are non-oscillatory, which is in agreement with Theorem 2.12. Example 4. Consider the equation

x (t ) = 2x(t ) + x([t − 1] ) − 2x([t] ) + 0.4x([t + 1] ), x(−1 ) = 1, x(0 ) = 1,

(3.4)

where p = 2, p−1 = 1, p0 = −2, p1 = 0.4, (iii) in Theorem 1.5 is satisfied. In Fig. 7, we draw the analytical solution and the numerical solution in 2-stage Radau IIA method with h = 0.01 (we also draw the sign function of the analytic solution and the numerical solution for convenience). We can see the analytical solution is oscillatory and the numerical solution is non-oscillatory, which is in agreement with Theorem 2.11. References [1] I. Györi, G. Ladas, Oscillation Theory of Delay Differential Equations, Clarendon Press, Oxford, 1991. [2] R. Agarwal, L. Berezansky, E. Braverman, A. Domoshnitsky, Nonoscillation Theory of Functional Differential Equations with Applications, Springer, New York, 2012. [3] M.U. Akhmet, On the reduction principle for differential equations with piecewise constant argument of generalized type, J. Math. Anal. Appl. 336 (2007) 646–663. [4] G.Q. Wang, periodic solutions of a neutral differential equation with piecewise constant arguments, J. Math. Anal. Appl. 326 (2007) 736–803. [5] M.U. Akhmet, Asymptotic behavior of solutions of differential equations with piecewise constant arguments, Appl. Math. Lett. 21 (2008) 951–956. [6] H. Li, Y. Muroya, R. Yuan, A sufficient condition for the global asymptotic stability of a class of logistic equations with piecewise constant delay, Nonlinear Anal. RWA 10 (2009) 244–253. [7] I. Györi, F. Hartung, ON numerical approximation using differential equations with piecewise constant arguments, Period. Math. Hungar. 56 (2008) 55–69. [8] M.U. Akhmet, Stability of differential equations with piecewise constant arguments of generalized type, Nonlinear Anal. 68 (2008) 794–803.

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