使用解析方法获得FitzHugh-Nagumo方程新的peakon解

邹丽; 王振; 梁辉; 宗智; 邹昊

doi:10.3879/j.issn.1000-0887.2013.11.003

使用解析方法获得FitzHugh-Nagumo方程新的peakon解

doi: 10.3879/j.issn.1000-0887.2013.11.003

邹丽^1,2,
王振³,
梁辉^1,4,
宗智^1,4,
邹昊^1,2

¹工业装备结构分析国家重点实验室, 辽宁大连 116085;²大连理工大学航空航天学院, 辽宁大连 116085;³大连理工大学数学科学学院, 辽宁大连 116085;⁴大连理工大学船舶工程学院, 辽宁大连116085

基金项目: 国家重点基础研究发展计划（973计划）资助项目（2010CB832704；2013CB036101）；国家自然科学基金资助项目（51109031；11072053；51009022；51221961；51239002）；中国教育部博士点基金资助项目（20100041120037）；中央高校基本科研业务费专项资金资助项目（DUTBJS01；DUT12LK52；DUT12LK34）

详细信息

作者简介:
邹丽(1981—),女,辽宁盘锦人,副教授,博士,硕士生导师(通讯作者. Tel: +86-411-84706521; E-mail: lizou@dlut.edu.cn).

中图分类号: O242.1;O302
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出版历程
- 收稿日期: 2013-06-14
- 修回日期: 2013-08-19
- 刊出日期: 2013-11-15

Finding New Types of Peakon Solutions for FitzHugh-Nagumo Equation by an Analytical Technique

ZOU Li^1,2,
WANG Zhen³,
LIANG Hui^1,4,
ZONG Zhi^1,4,
ZOU Hao^1,2

¹State Key Laboratory of Structural Analysis for Industrial Equipment,Dalian, Liaoning 116085, P.R.China;²School of Aeronautics and Astronautics, Dalian University of Technology,Dalian, Liaoning 116085, P.R.China;³School of Mathematics, Dalian University of Technology, Dalian, Liaoning 116085, P.R.China;⁴School of Naval Architecture, Dalian University of Technology, Dalian, Liaoning 116085, P.R.China

Funds: The National Basic Research Program of China (973 Program)（2010CB832704； 2013CB036101）;The National Natural Science Foundation of China（51109031； 11072053； 51009022； 51221961； 51239002）

摘要

摘要: 依据对FitzHugh-Nagumo方程的研究，通过微分变化法近似分析出FitzHugh-Nagumo方程，获得了这个方程的尖峰孤立波（peakon soliton）的解，从而获得了更多形式的peakon解，同时也分析了微分变换法（differential transform method, DTM）收敛区域和收敛速度．构建的微分变换法，结合帕德(Padé)逼近，构建一个明确的，完全解析，对FitzHugh-Nagumo方程全部有意义的尖波解．其主要思想是限制边界条件而令导数在孤立波不存在峰值，但导数的孤立波在两侧存在．结果表明，微分变换法在参数很小的情况下可以避免摄动的限制．表明这种方法提供了一种强大而有效地获得FitzHugh-Nagumo方程新的peakon解的数学方法．
- FitzHugh-Nagumo方程 /
- 尖峰解 /
- 微分变换法 /
- Padé逼近 /
- 收敛区域和速度
Abstract: The FitzHugh-Nagumo equation was studied with an approximate analytical method: the differential transform method. Peakon soliton solutions to this equation were presented. As a result, more new types of peakon solutions were obtained. The convergence region and rate of the differential transform method were also analyzed. The differential transform method was successfully combined with the Padé approximation technique, to construct an explicit, totally analytical and uniformly valid peakon soliton solution to FitzHugh-Nagumo equation. The main idea was to limit the boundary conditions while let the derivative at the crest of the solitary wave not exist but the solitary waves of the derivative exist at both sides. The obtained results show that the differential transform method can avoid the limitation of perturbation under conditions of very small parameters. The present method provides a powerful and effective mathematical tool to obtain new types of precise peakon solutions for FitzHugh-Nagumo equation.
- FitzHugh-Nagumo equation /
- peakon solution /
- differential transform method /
- Padé approximation /
- convergence region and rate

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参考文献(18)

[1]	FitzHugh R. Impulses and physiological states in theoretical models of nerve membrane[J].Biophysical Journal,1961, 1(6): 445-466.
[2]	Nagumo J, Arimoto S, Yoshizawa S. An active pulse transmission line simulating nerve axon[J].Proceedings of the Institute of Radio Engineers,1962, 50(10): 2061-2070.
[3]	Abbasbandy S, Amirfakrian M. A new approach to universal approximations of fuzzy functions on a discrete set of points[J].Applied Mathematical Modelling,2006, 30(12): 1525-1534.
[4]	Li H Y, Guo Y C. New exact solutions to the FitzHugh-Nagumo equation[J].Applied Mathematics and Computation,2006, 180(2): 524-528.
[5]	Nucci M C, Clarkson P A. The nonclassical method is more general than the direct method for symmetry reductions. an example of the FitzHughNagumo equation[J].Physics Letters A,1992, 164(1): 49-56.
[6]	Kawahara T, Tanaka M. Interactions of traveling fronts: an exact solution of a nonlinear diffusion equation[J].Physics Letters A,1983, 97(8): 311-314.
[7]	Dancer E N, Yan S. Interior peak solutions for an elliptic system of FitzHugh-Nagumo type[J].Journal of Differential Equations,2006, 229(2): 654679.
[8]	赵家奎. 微分变换及其在电路中的应用[M]. 武汉: 华中理工大学出版社, 1986.(ZHAO Jia-kui.Differential Transformation and Its Applications for Electrical Circuits [M]. Wuhan: Huazhong University Press, 1986.(in Chinese))[9]Chen C K, Ho S H. Solving partial differential equations by twodimensional differential transform method [J].Applied Mathematics and Computation,1999, 106(2/3): 171-179.
[9]	JANG MingJyi, CHEN ChiehLi, LIU YungChin. Two-dimensional differential transform for partial differential equations[J].Applied Mathematics and Computation,2001, 121(2/3): 261-270.
[10]	AbdelHalim Hassan I H. Different applications for the differential transformation in the differential equations[J].Applied Mathematics and Computation,2002, 129(2/3): 183-201.
[11]	Ayaz F. On the twodimensional differential transform method[J].Applied Mathematics and Computation,2003,143(2/3): 361-374.
[12]	Kurnaz A, Oturnaz G, Kiris M E.n-Dimensional differential transformation method for solving PDEs[J].International Journal of Computer Mathematics,2005, 82(3): 369-380.
[13]	AdbelHalim Hassan I H. Comparison differential transformation technique with Adomian decomposition method for linear and nonlinear initial value problems[J].Chaos, Solitons & Fractals,2008, 36(1): 53-65.
[14]	Ravi Kanth A S V, Aruna K. Differential transform method for solving linear and non-linear systems of partial differential equations[J].Physics Letters A,2008, 372(46): 6896-6898.
[15]	邹丽, 王振, 宗智, 邹东阳, 张朔. 改进的微分变换法对不连续冲击波的求解[J]. 应用数学和力学, 2012, 33(12): 1465-1476.(ZOU Li, WANG Zhen, ZONG Zhi, ZOU Dong-yang, ZHANG Shuo. Solving shock wave with discontinuity by enhanced differential transform method[J].Applied Mathematics and Mechanics,2012, 33(12): 1465-1476.(in Chinese))
[16]	ZOU Li, ZOU Dong-yang, WANG Zhen, ZONG Zhi. Finding discontinuous solution to the differential-difference equation by homotopy analysis method[J].Chinese Physics Letters,2013, 30(2): 020204(5pp).
[17]	ZOU Li, WANG Zhen, ZONG Zhi. Generalized differential transform method to differential-difference equation[J].Physics Letters A,2009, 373(45): 4142-4151.
[18]	ZOU Li, ZONG Zhi, WANG Zhen, TIAN Shou-fu. Differential transform method for solving solitary wave with discontinuity[J].Physics Letters A, 2010, 374(34): 3451-3454.

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使用解析方法获得FitzHugh-Nagumo方程新的peakon解

doi: 10.3879/j.issn.1000-0887.2013.11.003

作者简介:
邹丽(1981—),女,辽宁盘锦人,副教授,博士,硕士生导师(通讯作者. Tel: +86-411-84706521; E-mail: lizou@dlut.edu.cn).

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Finding New Types of Peakon Solutions for FitzHugh-Nagumo Equation by an Analytical Technique

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使用解析方法获得FitzHugh-Nagumo方程新的peakon解

doi: 10.3879/j.issn.1000-0887.2013.11.003

作者简介: 邹丽(1981—),女,辽宁盘锦人,副教授,博士,硕士生导师(通讯作者. Tel: +86-411-84706521; E-mail: lizou@dlut.edu.cn).

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出版历程

Finding New Types of Peakon Solutions for FitzHugh-Nagumo Equation by an Analytical Technique

计量

出版历程

目录

作者简介:
邹丽(1981—),女,辽宁盘锦人,副教授,博士,硕士生导师(通讯作者. Tel: +86-411-84706521; E-mail: lizou@dlut.edu.cn).