| 使用解析方法获得FitzHugh-Nagumo方程新的peakon解 |
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| 引用本文： | 邹丽,王振,梁辉,宗智,邹昊. 使用解析方法获得FitzHugh-Nagumo方程新的peakon解[J]. 应用数学和力学, 2013, 34(11): 1141-1149. DOI: 10.3879/j.issn.1000-0887.2013.11.003 |
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| 作者姓名： | 邹丽 王振 梁辉 宗智 邹昊 |
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| 作者单位： | 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 |
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| 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） |
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| 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．
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| 关 键 词： | FitzHugh-Nagumo方程 尖峰解 微分变换法 Padé逼近 收敛区域和速度 |
| 收稿时间： | 2013-06-14 |
Finding New Types of Peakon Solutions for FitzHugh-Nagumo Equation by an Analytical Technique |
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| Affiliation: | 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 |
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| 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. |
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