- Research
- Open Access
- Published:
Numerical stability and oscillation of the Runge-Kutta methods for equation
Advances in Difference Equations volume 2012, Article number: 146 (2012)
Abstract
This paper deals with the numerical properties of Runge-Kutta methods for the alternately of retarded and advanced equation . Necessary and sufficient conditions for the stability and oscillation of the numerical solution are given. The conditions that the Runge-Kutta methods preserve the stability and oscillations of the analytic solutions are obtained. Some numerical experiments are illustrated.
1 Introduction
This paper deals with the numerical solutions of the alternately of retarded and advanced equation with piecewise continuous arguments (EPCA)
where is the greatest integer function, and M and N are positive integers such that . Differential equations of this form have simulated considerable interest and have been studied by Wiener and Aftabizadeh [11]. Cooke and Wiener [2] studied the special cases of this form with , and , . In these equations the argument derivation are piecewise linear periodic function with periodic M. Also, is negative for and positive for (n is integer). Therefore, Eq. (1.1) is of advanced type on and of retarded type on .
EPCA describe hybrid dynamics systems, combine properties of both differential and difference equations, and have applications in certain biomedical models in the work of Busenberg and Cooke [1]. For these equations of mixed type, the change of sign in the argument derivation leads not only to interesting periodic properties but also to complications in the asymptotic and oscillatory behavior of solutions. Oscillatory, stability and periodic properties of the linear EPCA of the form with alternately of retarded and advanced have been investigated in [2].
There exist some papers concerning the stability of the numerical solutions of delay differential equations with piecewise continuous arguments, such as [5, 8, 12]. Also, there have been results concerning oscillations of delay differential equations and delay difference equations, even including delay differential equations with piecewise continuous arguments [6]. But there is no paper concerning with the stability and oscillation of the numerical solution of these equations.
In this paper, we will investigate the numerical properties, including the stability and oscillation of Runge-Kutta methods for equation . In Section 2 we give some preliminary results for the analytic solutions of this equation. In Section 3 we consider the adaptation of the Runge-Kutta methods. In Section 4 we give the conditions of the stability and oscillation of the Runge-Kutta methods. In Section 5 we investigate the preservation of the stability and oscillations of the Runge-Kutta methods. In Section 6 some numerical experiments are illustrated.
2 Preliminary results
We consider the following equation
where a, are constants, M and N are positive integers such that () and is the greatest integer function.
Definition 2.1[10]
A solution of Eq. (2.1) on is a function that satisfies the conditions:
-
1.
is continuous on .
-
2.
The derivative exists at each point with the possible exception of the points (), where one-sided derivatives exist.
-
3.
Eq. (2.1) is satisfied on each interval for .
As follows, we use these notations
The following theorems give the existence and uniqueness of solutions and then provide necessary and sufficient conditions for the asymptotic stability and the oscillation of all solutions of Eq. (2.1).
Theorem 2.2[10]
Assume that, then the initial value problem (2.1) has ona unique solutiongiven by
if, where.
Theorem 2.3[10]
The solutionof Eq. (2.1) is asymptotically stable asif and only if.
Lemma 2.4[10]
For, the equation
has a unique nonzero solution with respect to a. This solution is negative ifand positive if.
Theorem 2.5[10]
Letbe the nonzero solution of (2.3). The solutionof Eq. (2.1) is asymptotically stable (as) if any one of the following hypotheses is satisfied:
() , ;
() , or;
() , .
In the following, we will give the definition of the oscillation and the non-oscillation.
Definition 2.6 A nontrivial solution of (2.1) is said to be oscillatory if there exists a sequence such that as and . Otherwise it is called non-oscillatory. We say Eq. (2.1) is oscillatory if all nontrivial solutions of Eq. (2.1) are oscillatory. We say Eq. (2.1) is non-oscillatory if all nontrivial solutions of Eq. (2.1) are non-oscillatory.
Theorem 2.7[10]
A necessary and sufficient condition for all solutions of Eq. (2.1) to be oscillatory is that either of the conditions, holds.
3 The Runge-Kutta methods
In this section we consider the adaptation of the Runge-Kutta methods . Let be a given stepsize with integer and the gridpoints be defined by ().
For the Runge-Kutta methods we always assume that and .
The adaptation of Runge-Kutta methods to Eq. (2.1) leads to a numerical process of the following type
where matrix , vectors , and is an approximation to at , and are approximation to and , respectively.
Let , for ; for . Then can be defined as according to Definition 2.1 ().
Let . Then (3.1) reduces to
where . Hence we have
where , is the stability function of the method.
We can obtain from (3.3)


4 Stability and oscillation of the Runge-Kutta methods
In this section we will discuss stability and oscillation of the Runge-Kutta methods.
4.1 Numerical stability
Definition 4.1 The Runge-Kutta method is called asymptotically stable at if there exists a constant C such that Eq. (3.1) defines () that satisfy for all () and any given .
For any given Runge-Kutta methods, , where and are polynomials. is a continuous function at the neighborhood of zero, and , so there exist , with such that
which implies , .
Remark 4.2[7]
It is known that is an increasing function in , and for , for . Hence we can take , for simplicity.
In the following, we always suppose , i.e., . It is easy to see from (3.4) and (3.5) that as if and only if as . Hence we have the following theorem.
Theorem 4.3 The Runge-Kutta method is asymptotically stable if any one of the following hypotheses is satisfied:
() , ;
() or, ;
() , ,
where, , is a nonzero solution of.
Proof Let , . From (3.5), we only need to verify the inequalities . Assume , i.e., . Then is equivalent to
Assume , i.e., . Then is equivalent to
The theorem is proved. □
4.2 Numerical oscillation
Consider the difference equation
where , , and its associated characteristic equation is
Definition 4.4 A nontrivial solution of (4.2) is said to be oscillatory if there exists a sequence such that as and . Otherwise, it is called non-oscillatory. Eq. (4.2) is said to be oscillatory if all nontrivial solutions of Eq. (4.2) are oscillatory. Eq. (4.2) is said to be non-oscillatory if all nontrivial solutions of Eq. (4.2) are non-oscillatory.
Theorem 4.5[3]
Eq. (4.2) is oscillatory if and only if the characteristic equation (4.3) has no positive roots.
We obtain the following theorem
Theorem 4.6 Supposeandare given by (3.4) and (3.5) respectively, then the following statements are equivalent:
-
1.
is oscillatory;
-
2.
is oscillatory;
-
3.
, or .
Proof is not oscillatory if and only if
i.e.,
Hence we have for .
which is equivalent to
In view of (3.4), is not oscillatory.
Moreover, is oscillatory if and only if
which is equivalent to
□
5 Preservation of stability and oscillations of the Runge-Kutta methods
In this section, we will investigate the conditions under which the numerical solution and the analytic solution are oscillatory simultaneously. We will study the stability and the oscillation of Runge-Kutta methods with the stability function which is given by the -padé approximation to .
In order to do this, the following lemmas and corollaries will be useful.
The-padé approximation tois given by
where
with error
It is the unique rational approximation to of order , such that the degree of numerator and denominator are r and s, respectively.
Ifis the-padé approximation to, then
-
1.
there are s bounded star sectors in the right-half plane, each containing a pole of ;
-
2.
there are r bounded white sectors in the left-half plane, each containing a zero of ;
-
3.
all sectors are symmetric with respect to the real axis.
We can obtain the following corollary.
Supposeis the-padé approximation to. Then
-
1.
:
for allif and only if s is even,
for allif and only if s is odd,
-
2.
:
for allif and only if r is even,
for allif and only if r is odd,
whereis a real zero ofandis a real zero of.
5.1 Preservation of the stability
In this subsection, in order to analyse preservation of stability, the following lemmas are useful.
Lemma 5.4 Let. Then
-
1.
the function has a minimum at , and is decreasing in and increasing in ;
-
2.
the function has a unique solution , which is less than one when and larger than one when ;
-
3.
when :
if,
ifor;
when:
if,
ifor.
Lemma 5.5 Let. Then
-
1.
the function has extremum at ;
-
2.
when :
is increasing inand,
is decreasing inand;
-
3.
when :
is increasing inand,
is decreasing inand.
Lemma 5.6 Assume thatis a unique solution of the function, , . Then the following hypothesis are satisfied:
-
1.
, if ;
-
2.
, if .
Proof By virtue of Bernoulli inequality, we obtain
and
By virtue of Lemma 5.4, the proof is complete. □
Theorem 5.7 Assume that Eq. (2.1) is asymptotically stable and. Then the Runge-Kutta method is asymptotically stable if one of the following conditions is satisfied.
-
(i)
();
-
(ii)
();
-
(iii)
().
Proof In view of Theorem 2.5 and Theorem 4.3 we will prove that the conditions , , are satisfied under conditions , , . It is known from () and Lemma 5.4 and Lemma 5.5 that is decreasing and is increasing. Hence and
If , then holds which together with (5.4) imply , , . If , then holds which together with (5.4) imply . If , then holds which implies . The proof is complete. □
Theorem 5.8 Assume that Eq. (2.1) is asymptotically stable and. Then the Runge-Kutta method is asymptotically stable if one of the following conditions is satisfied.
-
(i)
();
-
(ii)
( and );
-
(iii)
().
Proof In view of Theorem 2.5 and Theorem 4.3 we will prove that the conditions , , are satisfied under conditions , , . It is known from () and Lemma 5.4 and Lemma 5.5 that is increasing and is decreasing. Hence . The rest of the proof is similar to that in Theorem 5.7. □
From Theorem 5.7 and Theorem 5.8, we obtain the following corollary.
Corollary 5.9 The v-stage A-stable higher order Runge-Kutta method preserves the stability of Eq. (2.1), if one of the following conditions is satisfied.
When:
-
(i)
for Gauss-Legendre and Lobatto IIIC methods,
v is odd, if,
v is even, if;
-
(ii)
for Lobatto IIIA and Lobatto IIIC methods,
v is even, if,
v is odd, if;
-
(iii)
for Radau IA and IIA methods,
v is even, if,
v is odd, otherwise.
When:
-
(i)
for Gauss-Legendre and Lobatto IIIC methods,
v is odd, if,
v is even, if;
-
(ii)
for Lobatto IIIA and Lobatto IIIC methods,
v is even, if,
v is odd, if;
-
(iii)
for Radau IA and IIA methods,
v is odd, if,
v is even, otherwise.
5.2 Preservation of the oscillation
In the subsection we will investigate the conditions under which the Runge-Kutta method preserves the oscillation of Eq. (2.1).
We have from Theorem 2.7 and Theorem 4.6 that the Runge-Kutta method preserves the oscillation of Eq. (2.1) if and only if
Hence the following theorem is obvious.
Theorem 5.10 The Runge-Kutta method preserves the oscillation of Eq. (2.1) if and only if
Corollary 5.11 The v-stage A-stable higher order Runge-Kutta method preserves the oscillation of Eq. (2.1) if and only if
-
(i)
for Gauss-Legendre and Lobatto IIIC methods, v is odd;
-
(ii)
for Lobatto IIIA and Lobatto IIIC methods, v is even;
-
(iii)
for Radau IA and IIA methods, v is even if , v is odd if .
5.3 The explicit Runge-Kutta methods
It is known that all ν-stage explicit Runge-Kutta methods with possess the stability function (see [4])
which is the -padé approximation to .
Theorem 5.12 The ν-stage explicit Runge-Kutta methods withpreserve the asymptotic stability of Eq. (2.1) ifand
-
(i)
ν is odd for and ;
-
(ii)
ν is even for and ;
-
(iii)
ν is odd for and .
Theorem 5.13 The ν-stage explicit Runge-Kutta methods withpreserve the oscillation of Eq. (2.1) withif ν is odd.
6 Numerical examples
In this section, we will give some examples to illustrate the conclusions in the paper. In order to illustrate the stability, we consider the following two problems


In Figure 1-Figure 3, we draw the numerical solutions for Eq. (6.1) and Eq. (6.2), respectively. It is easy to see that the numerical solutions are asymptotically stable.
In order to illustrate the oscillation, we consider the following two problems


In Figure 4-Figure 6, we draw the numerical solutions for Eq. (6.3) and Eq. (6.4), respectively. It is easy to see that the numerical solutions are oscillatory.
References
Busenberg S, Cooke KL: Models of vertically transmitted diseases with sequential-continuous dynamics. In Nonlinear Phenomena in Mathematical Sciences. Edited by: Lakshmikantham V. Academic Press, New York; 1982.
Cooke KL, Wiener J: An equation alternately of retarded and advanced type. Proc. Am. Math. Soc. 1987, 99: 726–732. 10.1090/S0002-9939-1987-0877047-8
Györi I, Ladas G: Oscillation Theory of Delay Differential Equations. Clarendon Press, Oxford; 1991.
Hairer E, Wanner G Stiff and Differential Algebraic Problems. In Solving Ordinary Differential Equations II. Springer, New York; 1993.
Liu MZ, Song MH, Yang ZW:Stability of Runge-Kutta methods in the numerical solution of equation . J. Comput. Appl. Math. 2004, 166: 361–370. 10.1016/j.cam.2003.04.002
Liu MZ, Gao JF, Yang ZW: Oscillation analysis of numerical solution in the θ -methods for equation . Appl. Math. Comput. 2007, 186(1):566–578. 10.1016/j.amc.2006.07.119
Ran XJ, Liu MZ, Zhu QY: Numerical methods for impulsive differential equation. Math. Comput. Model. 2008, 48: 46–55. 10.1016/j.mcm.2007.09.010
Song MH, Yang ZW, Liu MZ: Stability of θ -methods advanced differential equations with piecewise continuous argument. Comput. Math. Appl. 2005, 49: 1295–1301. 10.1016/j.camwa.2005.02.002
Wanner G, Hairer E, Nøsett SP: Order stars and stability theorems. BIT Numer. Math. 1978, 18: 475–489. 10.1007/BF01932026
Wiener J: Generalized Solutions of Functional Differential Equations. World Scientific, Singapore; 1993.
Wiener J, Aftabizadeh AR: Differential equations alternately of retarded and advanced type. J. Math. Anal. Appl. 1988, 129: 243–255. 10.1016/0022-247X(88)90246-6
Yang ZW, Liu MZ, Song MH:Stability of Runge-Kutta methods in the numerical solution of equation . Appl. Math. Comput. 2005, 162: 37–50. 10.1016/j.amc.2003.12.081
Acknowledgements
The financial support from the National Natural Science Foundation of China (No.11071050) is gratefully acknowledged.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors have equal contributions to each part of this article. All the authors read and approved the final manuscript.
Authors’ original submitted files for images
Below are the links to the authors’ original submitted files for images.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Song, M., Liu, M. Numerical stability and oscillation of the Runge-Kutta methods for equation . Adv Differ Equ 2012, 146 (2012). https://doi.org/10.1186/1687-1847-2012-146
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1847-2012-146
Keywords
- stability
- oscillation
- differential equation
- Runge-Kutta method