华东师范大学学报(自然科学版) ›› 2015, Vol. 2015 ›› Issue (6): 59-71.doi: 10.3969/j.issn.1000-5641.2015.06.009

• 应用数学与基础数学 • 上一篇    下一篇

 Laplace分解法的推广和应用

李亨达1，柳银萍2   

  1. 1. 华东师范大学~~计算机科学技术系,上海200241;
    2. 华东师范大学~~系统科学研究所, 上海200241
  • 收稿日期:2014-08-22 出版日期:2015-11-25 发布日期:2015-12-23
  • 通讯作者: 柳银萍, 女, 教授,研究方向为符号计算、数学机械化. E-mail: ypliu2@hotmail.com. E-mail:ypliu_2@hotmail.com.
  • 作者简介:李亨达, 女, 硕士研究生,研究方向为符号计算. E-mail: gyyzlihengda@163.com.
  • 基金资助:

    国家自然科学基金重点项目~(11435005)

Extension of the Laplace decomposition method and its application

 LI  Heng-Da1, LIU  Yin-Ping2   

  • Received:2014-08-22 Online:2015-11-25 Published:2015-12-23

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摘要： 分解法思路简单且应用广泛,但单纯使用\,Adomian\,分解法所获得级数解的收敛范围往往很有限.把\,Laplace\,变换法与\,Adomian\,分解法结合起来求解非线性初边值问题的算法,即为\,Laplace\,分解法.本文将\,Laplace\,分解法推广应用到非线性偏微分方程情形,并针对直接推广得到算法的缺陷,进一步提出了适用于偏微分方程的改进\,Laplace\,分解算法.以\,1+1\,维非线性演化方程为例, 阐述了算法的思路和过程.最后通过几个实例,比较了由新算法所获得级数解与\,Adomian\,级数解的精度,由此可看出这些新级数解收敛性更好.

关键词: Laplace;分解法;Adomian分解法, 非线性演化方程

Abstract: The Adomian decomposition method was simple and widely used in solving nonlinear differential equations. The convergence region of the Adomian series solution is always very limited.Therefore the Laplace decomposition method, which is a combination of Laplace transformation method and Adomian decomposition method,is proposed to solve initial boundary value problems. In this paper,the Laplace decomposition method is extended to solve nonlinear
partial differential equations. For the flaws of the directlyextended algorithm, we further proposed a modified algorithm to solve nonlinear partial differential equations. Take, for example,1+1 dimensional nonlinear evolution equation to expound the idea and procedure of the algorithm. Finally, several examples were given to demonstrate the high precision and large convergence region of the new solutions by comparing these new solutions with those Adomian series solutions as well as other known exact solutions.

Key words: Laplace decomposition method, Adomian decomposition method, nonlinear evolution equation

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