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Dynamic analysis of a class of neutral delay model based on the Runge-Kutta algorithm

EURASIP Journal on Wireless Communications and Networking20182018:100

https://doi.org/10.1186/s13638-018-1125-2

  • Received: 20 March 2018
  • Accepted: 24 April 2018
  • Published:

Abstract

In this paper, we study the dynamics of a class of second-order neutral delay nonlinear models. This study is applicable to many fields, such as engineering, cybernetics, and physics. We use the Runge-Kutta algorithm and the Riccati transform method. First, we give a neutral delay nonlinear model based on the Runge-Kutta algorithm. Then, we study the dynamic characteristics of the neutral delay model and establish some new sufficient conditions for the oscillation. The results of our research are new, and these results promote and improve the results already available. The results are also verified by numerical experiments. The neutral delay nonlinear model has an important application in engineering, cybernetics, and physics. Therefore, the study of this paper has great help and promotion to engineering, cybernetics, and physics.

Keywords

  • Runge-Kutta algorithm
  • Neutral delay model
  • Dynamic analysis
  • Oscillation

1 Introduction

The Runge-Kutta algorithm is a more practical algorithm built on the basis of mathematical support [1]. This algorithm is an important implicit or explicit iterative method for solving the solutions of nonlinear ordinary differential equations [2]. Because of the high precision of the algorithm, it is a kind of high-precision single-step algorithm widely used in engineering [3]. However, some measures need to be taken to suppress the deviation, so the implementation principle is more complex [4]. In recent years, due to the widespread application of neutral delay differential equations in engineering cybernetics and physics fields, a wide range of attention has been drawn from scholars both at home and abroad [58]. With the further improvement of the Runge-Kutta algorithm and the further development of the neutral delay differential equation theory, many scholars have studied the delay differential equations and get some related results about oscillation [914]. People use a series of techniques and methods, such as calculation and reasoning, to study these equations and to obtain the oscillation conditions of the solution of the equation [10, 1519]. How to get the oscillation criterion of the neutral delay differential equation model becomes the key and the difficult problem [20, 21]. The Runge-Kutta algorithm and Riccati transform provide an effective and practical method for us to study the two-order neutral time-delay model.

In this paper, we study the dynamic characteristics of a class of second-order neutral delay models by using the Runge-Kutta algorithm. We have obtained some new oscillation criteria for a class of second-order neutral delay nonlinear differential equation models. These results promote and improve the known results in the literature.

2 Model and methodology

The vibration problems in engineering, cybernetics, communication technology, physics, and other fields can be represented by the neutral delay model differential equation model. For a long time, the problem of dynamics has been the concern of experts and scholars at home and abroad. To this end, the experts also set up some neutral delay model to study the vibration of engineering, automatic control, communication technology, and other practical problems. On the basis of the existing literature, this paper studies a class of engineering control problems, that is, a class of second-order neutral delay differential equations with the expression equation model:

$$ {\left(r(t){\phi}_{\alpha}\left({z}^{\prime }(t)\right)\right)}^{\prime }+p(t){\phi}_{\alpha}\left({z}^{\prime }(t)\right)+f\left(t,x\left(\sigma (t)\right)\right)=0,t\ge {t}_0. $$
(1)

where z(t) = x(t) + c(t)x(τ(t)), φ α (s) = |s|α − 1s and the following conditions are satisfied:

(h1) q(t) C[t0, ∞), f(t, x) sgn x ≥ q(t)|x| β , α and β are constants.
$$ \left({h}_2\right)p(t),r(t)\in C\left(\left[{t}_0,\infty \right]\right),p(t)\ge 0,r(t)>0,-1<c(t)<0 $$
$$ \left({h}_3\right)\sigma (t)\in {C}^1\left(\left[{t}_0,\infty \right],R\right),\sigma (t)>0,{\sigma}^{\prime }(t)>0,\sigma (t)\le t,\underset{t\to \infty }{\lim}\sigma (t)=\infty . $$
$$ \left({h}_4\right){\int}_{t_0}^{\infty }{R}^{-\frac{1}{\alpha }}(t) dt=+\infty, R(t)=E(t)r(t),E(t)=\exp {\int}_{t_0}^t\frac{p(s)}{r(s)} ds. $$

This model (1) has been widely used in engineering, automatic control, communication technology, physics, and other systems. By using Riccati transformation and computational reasoning, some new vibration criteria for two-order neutral delay differential Eq. (1) are obtained. These results promote and improve some of the well-known results.

3 Results and discussion

In this paper, in order to study the vibration of the system (1), we will use the generalized Riccati transform method to study Eq. (1).

Lemma 1 Assume that (h1)~(h4) holds, and x(t) is an eventually positive solution of Eq. (2), then z(t) > 0, z(t) > 0 , or x(t) → 0.

Proof We Suppose x(t) is an eventually positive solution of Eq. (2). If z(t) > 0, we have
$$ {\left(R(t){\phi}_{\alpha}\left({z}^{\prime }(t)\right)\right)}^{\prime }+E(t)f\left(t,x\left(\sigma (t)\right)\right)=0. $$
Then,
$$ {\left(R(t){\phi}_{\alpha}\left({z}^{\prime }(t)\right)\right)}^{\prime}\le 0, $$
that is,

(R(t)|z(t)|α − 1z(t)) ≤ 0.

z(t) is eventually of one sign, that is, z(t) > 0 or z(t) < 0. Otherwise, if there exists T, such that z(t) < 0 for t ≥ T, then for arbitrary positive K, we have
$$ R(t){\left|{z}^{\prime }(t)\right|}^{\alpha -1}{z}^{\prime }(t)\le -R(T)\left(-{z}^{\prime }(T)\right)=-K<0. $$
$$ -{z}^{\prime }(t)\ge {\left(\frac{k}{R(t)}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\alpha $}\right.}. $$
$$ 0<z(t)\le z(T)-{K}^{\frac{1}{\alpha }}{\int}_T^t\ {R}^{-\frac{1}{\alpha }}(s) ds\to -\infty . $$

Therefore, z(t) > 0.

If z(t) < 0, then x(t) is bounded. Otherwise, if x(t) is unbounded, \( \exists {\left\{{t}_n\right\}}_{n=1}^{\infty } \), such that \( \underset{n\to \infty }{\lim }{t}_n=\infty \), let \( x\left({t}_n\right)=\underset{s\in \left[T,{x}_n\right]}{\max}\left\{x(s)\right\} \); thus, t n  ≥ τ(t n ) ≥ T.

\( x\left(\tau \left({t}_n\right)\right)\le \underset{s\in \left[T,{x}_n\right]}{\max}\kern0.5em \left\{x(s)\right\}=x\left({t}_n\right) \)

< − c(t n )x(τ(t n )) < x(τ(t n )).

Therefore, x(t) is bounded.

\( \underset{t\to \infty }{0\ge \lim \kern0.5em \sup \kern0.5em \sup \kern0.5em z(t)} \)

\( \underset{t\to \infty }{\ge \lim \kern0.5em \sup \kern0.5em x(t)}+\underset{t\to \infty }{\lim \kern0.5em \operatorname{inf}\kern0.5em c(t)x(t)\Big)} \)

\( \underset{t\to \infty }{\ge \left(1-c\right)\kern0.5em \lim \kern0.5em \operatorname{inf}\kern0.5em x{(t)}_{\ge 0}} \).

Thus, \( \underset{t\to \infty }{\lim }x(t)=0 \).

Lemma 2 We suppose x(t) is an eventually positive solution of Eq. (2), then

(1)z(t) > tz(t);

(2)\( \frac{z(t)}{t} \) is strictly decreasing eventually.

Proof Since (R(t)(z(t)) α ) ≤ 0, then z′′(t) ≤ 0. Let g(t) = z(t) − tz(t); we get g(t) =  − tz′′(t) > 0 and we assert that g(t) > 0 eventually. Otherwise, g(t) < 0, so
$$ {\left(\frac{z(t)}{t}\right)}^{\prime }=-\frac{g(t)}{t^2}>0. $$

Thus, \( \frac{z(t)}{t} \) is strictly increasing.

\( \frac{z\left(\sigma (t)\right)}{\sigma (t)}\ge \frac{z\left(\sigma (T)\right)}{\sigma (T)}=b>0 \), t ≥ T.

We have z(σ(t)) ≥ (t); thus,

0 < R(t)(z(t))

\( \le R(T){\left({z}^{\hbox{'}}(T)\right)}^{\alpha }-{\int}_T^tQ(s){z}^{\beta}\left(\sigma (s)\right) ds \)
$$ \le R(T){\left({z}^{\prime }(T)\right)}^{\alpha }-{b}^{\beta }{\int}_T^t\ Q(s){\sigma}^{\beta }(s) ds\to -\infty . $$

Then, z(t) > tz(t), and \( \frac{z(t)}{t} \) is strictly decreasing eventually.

Theorem 1 Assume that \( {\int}_T^t\ \left[\rho (s)Q(s){\left(\frac{\sigma (s)}{s}\right)}^{\beta }-\frac{R(s){\left({\rho}^{\prime }(s)\right)}^{\lambda +1}}{{\left(\lambda +1\right)}^{\lambda +1}{\left( m\rho (s)\right)}^{\lambda }}\right] ds=\infty \), then Eq. (2) is almost oscillatory.

Proof We suppose x(t) is an eventually positive solution of Eq. (2); from Lemma 1, we have

z(t) > 0, z(t) > 0, or x(t) → 0.

We define the function
$$ w(t)=\frac{R(t){\left({z}^{\prime }(t)\right)}^{\alpha }}{z^{\beta }(t)}. $$
If β ≥ α, we have
$$ {w}^{\prime }(t)=\frac{{\left(R(t){\left({z}^{\prime }(t)\right)}^{\alpha}\right)}^{\prime }}{z^{\beta }(t)}-\frac{\beta }{R^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\alpha $}\right.}(t)}{\left[z(t)\right]}^{\frac{\beta -\alpha }{\alpha }}{w}^{\frac{\alpha +1}{\alpha }}(t) $$
$$ \le -Q(t){\left(\frac{\sigma (t)}{t}\right)}^{\beta }-\frac{\alpha {m}_1}{R^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\alpha $}\right.}(t)}{w}^{\frac{\alpha +1}{\alpha }}(t). $$
where \( {m}_1=\min \left\{1,{\left[z(T)\right]}^{\frac{\beta -\alpha }{\alpha }}\right\} \).
If β < α, we have
$$ {w}^{\prime }(t)\le -Q(t){\left(\frac{\sigma (t)}{t}\right)}^{\beta }-\frac{\beta }{R^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beta $}\right.}(t)}{\left[{z}^{\prime }(t)\right]}^{\frac{\beta -\alpha }{\beta }}{w}^{\frac{\beta +1}{\beta }}(t) $$
$$ \le -Q(t){\left(\frac{\sigma (t)}{t}\right)}^{\beta }-\frac{\beta {m}_2}{R^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\beta $}\right.}(t)}{w}^{\frac{\beta +1}{\beta }}(t). $$
where \( {m}_2=\min \left\{1,{\left[{z}^{\hbox{'}}(T)\right]}^{\frac{\beta -\alpha }{\beta }}\right\} \).
Therefore, if β < α or β < α, we have
$$ {w}^{\prime }(t)\le -Q(t){\left(\frac{\sigma (t)}{t}\right)}^{\beta }-\frac{\lambda m}{R^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\lambda $}\right.}(t)}{w}^{\frac{\lambda +1}{\lambda }}(t) $$
where λ = min {α, β}. Let \( A(t)=\frac{\lambda m}{R^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\lambda $}\right.}(t)} \); we have
$$ {\int}_T^t\ \rho (s)Q(s){\left(\frac{\sigma (s)}{s}\right)}^{\beta } ds\le -{\int}_T^t\ \rho (s){w}^{\prime }(s) ds-{\int}_T^t\ \rho (s)A(s){w}^{\frac{\lambda +1}{\lambda }}(s) ds $$
$$ \le \rho (T)w(T)-\rho (t)w(t)+{\int}_T^t\ \left[{\rho}^{\prime }(s)w(s) ds-\rho (s)A(s){w}^{\frac{\lambda +1}{\lambda }}(s)\right] ds $$
$$ \le \rho (T)w(T)+{\int}_T^t\ \frac{\lambda^{\lambda }R(s){\left({\rho}^{\prime }(s)\right)}^{\lambda +1}}{{\left(\lambda +1\right)}^{\lambda +1}{\left(\lambda m\rho (s)\right)}^{\lambda }} ds $$
$$ =\rho (T)w(T)+{\int}_T^t\ \frac{R(s){\left({\rho}^{\prime }(s)\right)}^{\lambda +1}}{{\left(\lambda +1\right)}^{\lambda +1}{\left( m\rho (s)\right)}^{\lambda }} ds. $$
We have
$$ {\int}_T^t\ \left[\rho (s)Q(s){\left(\frac{\sigma (s)}{s}\right)}^{\beta }-\frac{R(s){\left({\rho}^{\prime }(s)\right)}^{\lambda +1}}{{\left(\lambda +1\right)}^{\lambda +1}{\left( m\rho (s)\right)}^{\lambda }}\right] ds\le \rho (T)w(T)<\infty . $$

By the Lemma 1 and the Lemma 2 and the related theory of equation oscillatory, we get Eq. (2) is almost oscillatory.

4 Conclusions

In this paper, the second-order neutral delay nonlinear model is studied by combining the Runge-Kutta algorithm and the Riccati transformation method. We have obtained the oscillation criterion of the second-order neutral delay nonlinear differential equation model. Most of the literature mainly studied the situation α ≥ β [58, 1321]. We not only studied the situation α ≥ β but also studied the situation α < β. We generalize the existing results and get the new oscillation criterion.

This second-order neutral delay differential equation describes the oscillation phenomena in the fields of engineering, control, communication, physics, and other fields. This indicates that oscillation in engineering, control and communication technologies will cause internal damage. We can predict the oscillation by the Runge-Kutta algorithm and the Riccati transform, in order to avoid the occurrence of oscillation in actual conditions such as engineering, control, communication technology and so on.

Abbreviation

Eq: 

Equation

Declarations

Acknowledgements

The research presented in this paper was supported by the China National Natural Science Foundation, Yunnan Science and Technology Department of China, and Qujing Normal University, China.

Funding

The authors acknowledge the National Natural Science Foundation of China (grant 11361048), Yunnan Natural Science Foundation of China (grant 2017FH001-014), and Qujing Normal University Science Foundation of China (grant ZDKC2016002).

Availability of data and materials

The simulation code can be downloaded at literature [11], and it is applicable.

Author’s contributions

HL is the only author of this article. By using the Runge-Kutta algorithm and Riccati transformation method, we study the dynamical properties of a class of two-order neutral delay nonlinear models and establish some new sufficient conditions. The author read and approved the final manuscript.

Competing interests

The author declares that he/she has no competing interests.

Publisher’s Note

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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
College of Mathematics and Statistics, Qujing Normal University, Qujing, China

References

  1. MH Carpenter, D Gottlieb, S Abarbanel, WS Don, The theoretical accuracy of Runge-Kutta discretization for the initial-boundary value problem: a study of the boundary error. SIAM J.Sci.Comput 16, 1241–1252 (1995)MathSciNetView ArticleMATHGoogle Scholar
  2. Q Zhang, Third order explicit Runge-Kutta discontinuous Galerkin method for linear conservation law with inflow boundary condition. J. Sci. Comput. 46(2), 294–313 (2011)MathSciNetView ArticleMATHGoogle Scholar
  3. Q Zhang, CW Shu, Error estimates to smooth solution of Runge-Kutta discontinuous Galerkin methods for scalar conservation laws. SIAM J. Numer. Anal. 42, 641–666 (2004)MathSciNetView ArticleMATHGoogle Scholar
  4. B Cockburn, CW Shu, Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J.Sci. Comput 16, 173–261 (2001)MathSciNetView ArticleMATHGoogle Scholar
  5. ME Elmetwally, SH Taher, HS Samir, Oscillation of nonlinear neural delay differential equations. J. Appl. Math. & Computing 21(1), 99–118 (2006)MathSciNetMATHGoogle Scholar
  6. P Hu, CM Huang, Analytical and numerical stability of nonlinear neural delay integro-differential equations. Journal of the Franklin Institutc 348, 1082–1100 (2011)View ArticleMATHGoogle Scholar
  7. S Zhang, Q Wang, Oscillation of second-order nonlinear neutral dynamic equations on time scales. AppliedMathematics and Computation 216, 2837–2848 (2010)MathSciNetMATHGoogle Scholar
  8. Q Li, R Wang, F Chen, TX Li, Oscillation of second-order nonlinear delay differential equations with nonpositive neutral coefficients. Advances in Difference Equations 35, 1–7 (2015)MathSciNetMATHGoogle Scholar
  9. J Liu, HY Luo, X Liu, Oscillation criteria for half-linear functional differential equation with damping. Therm. Sci. 18(5), 1537–1542 (2014)View ArticleGoogle Scholar
  10. T Candan, Oscillatory behavior of second order nonlinear neutral differential equations with distributed deviating arguments. Appl. Math. Comput. 262, 199–203 (2015)MathSciNetGoogle Scholar
  11. HY Luo, J Liu, X Liu, Oscillation behavior of a class of new generalized Emden-Fowler equations. Therm. Sci. 18(5), 1567–1572 (2014)View ArticleGoogle Scholar
  12. RP Agarwal, CH Zhang, TX Li, Some remarks o nonlinear neutral n of second order neutral differential equations. Appl. Math. Comput. 274, 178–181 (2016)MathSciNetGoogle Scholar
  13. J Dzurina, LP Stavroulakis, Oscillation criteria for second-order delay differential equation. Appl. Math. Comput. 174, 1636–1641 (2003)MathSciNetMATHGoogle Scholar
  14. FW Meng, R Xu, Oscillation criteria for certain even order quasi-linear neutral differential equations with deviating arguments. Appl. Math. Comput. 190, 458–464 (2007)MathSciNetMATHGoogle Scholar
  15. ZW Zheng, X Wang, HM Han, Oscillation criteria for forced second order differential equations with mixed nonlinearities. Appl. Math. Lett. 22, 1096–1101 (2009)MathSciNetView ArticleMATHGoogle Scholar
  16. QX Zhang, SH Liu, L Gao, Oscillation criteria for even-order half-linear functional differential equation. Appl. Math. Lett. 24, 1709–1715 (2011)MathSciNetView ArticleMATHGoogle Scholar
  17. ZL Han, TX Li, CH Zhang, Y Sun, Oscillation criteria for certain second-order nonlinear neutral differential equations of mixed type. Abstr. Appl. Anal. 387483, 1–9 (2011)MathSciNetMATHGoogle Scholar
  18. HD Liu, FW Meng, PC Liu, Oscillation and asymptotic analysis on a new generalized Emden-Fowler equation. Appl. Math. Comput. 219, 2739–2748 (2015)MathSciNetMATHGoogle Scholar
  19. SH Liu, QX Zhang, YH Yu, Oscillation of even-order half-linear differential equation with damping. Computers and Mathematics with Applications 61, 2191–2196 (2011)MathSciNetView ArticleMATHGoogle Scholar
  20. CJ Zhang, TT Qin, J Jin, An improvement of the numerical stability results for nonlinear neutral delay-integro-differential equations. Appl. Math.Comput 215, 548–556 (2009)MathSciNetMATHGoogle Scholar
  21. JJ Zhao, Y Cao, Y Xu, Legendre spectral collocation methods for volterra delay-integro-differential equations. J. Sci. Comput. 67(3), 1110–1133 (2016)MathSciNetView ArticleMATHGoogle Scholar

Copyright

© The Author(s). 2018

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