A SINGULAR APPROACH TO A CLASS OF IMPULSIVE DIFFERENTIAL EQUATION

Huaxiong Chen; Mingkang Ni

doi:10.11948/2016079

2016 Volume 6 Issue 4

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Huaxiong Chen, Mingkang Ni. A SINGULAR APPROACH TO A CLASS OF IMPULSIVE DIFFERENTIAL EQUATION[J]. Journal of Applied Analysis & Computation, 2016, 6(4): 1195-1204. doi: 10.11948/2016079

Citation:

Huaxiong Chen, Mingkang Ni. A SINGULAR APPROACH TO A CLASS OF IMPULSIVE DIFFERENTIAL EQUATION[J]. Journal of Applied Analysis & Computation, 2016, 6(4): 1195-1204. doi: 10.11948/2016079

A SINGULAR APPROACH TO A CLASS OF IMPULSIVE DIFFERENTIAL EQUATION

Huaxiong Chen¹,
Mingkang Ni¹

1 Department of Mathematics, East China Normal University, Shanghai, 200241, China;
2 Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, China

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Abstract

Abstract

In this paper, a singular approach to study the solutions of an impulsive differential equation from a qualitative and quantitative point of view is proposed. In the approach, a suitable singular perturbation term is introduced and a singularly perturbed system with infinite initial values is defined, in which, the reduced problem of the singularly perturbed system is exactly the impulsive differential equation under consideration. Then the boundary layer function method is applied to construct the uniformly valid asymptotic solutions to the singularly perturbed system. Based on the continuous asymptotic solution, the discontinuous solutions of the impulsive differential equation are described and approximated. An example, namely, a classical Lotka-Volterra prey-predator model with one pulse is carried out to illustrate the main results.
- Singular perturbation /
- asymptotic solution /
- impulsive differential equation /
- boundary layer function method
MSC: 34E05;34E20

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References

[1]	D. Bainov and P. Simeonov, Impulsive Differential Equations:Periodic Solutions and Applications, Longman Scientific and Technical, Harlow, 1993. Google Scholar
[2]	H. Cheng and T. Zhang, A new predator-prey model with a profitless delay of digestion and impulsive perturbation on the prey, Applied Mathematics and Computation, 217(2011), 9198-9208. Google Scholar
[3]	T. Fang and J. Sun, Stability of complex-valued impulsive and switching system and application to the lü system, Nonlinear Analysis:Hybrid Systems, 14(2014), 38-46. Google Scholar
[4]	X. Fu, B. Yan and Y. Liu, Introduction of impulsive differential systems, Science Press, Beijing, 2005. Google Scholar
[5]	X. Fu and S. Zheng, Chatter dynamic analysis for van der pol equation with impulsive effect via the theory of flow switchability, Commun Nonlinear Sci Numer Simulat, 19(2014), 3023-3035. Google Scholar
[6]	V. Lakshmikantham, D. Bainov and P. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1998. Google Scholar
[7]	M. Ni and W. Lin, Contrast Spatial Structure for Singularly Perturbed Problems, Science Press, Beijing, 2014. Google Scholar
[8]	A. Samoilenko and N. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995. Google Scholar
[9]	A. Tikhonov, Systems of differential equations containing a small parameter, Math. Sb, 64(1948), 193-204. Google Scholar
[10]	A. Tikhonov, On the dependence of solutions of differential equations on a small parameter, Math. Sb, 72(1952), 575-586. Google Scholar
[11]	A. Vasili'eva and V. Butuzov, Asymptotic Expansion of Solutions of Singular Perturbed Equation, Higher Education Press, Beijing, 2008. Google Scholar
[12]	Z. Zhang and H. Liang, Collocation methods for impulsive differential equations, Applied Mathematics and Computation, 228(2014), 336-348. Google Scholar