非齐次电报方程的无网格数值解法
发布时间：2019-03-02 17:54
【摘要】：非齐次电报方程作为一类特殊的非线性发展型偏微分方程,在电学、光学、声学以及微波技术等领域中得到了广泛应用.除极少数情况外,该方程的解析解是难以求得的,只能通过数值方法求其近似解,因此对于非齐次电报方程的数值方法研究具有极其重要的理论意义和应用价值.本文的创新之处是本文主要研究非齐次电报方程定解问题的无网格特解方法.首先,通过有限差分方法离散方程中的时间导数；其次,适当选取径向基函数对未知函数及其空间导数进行逼近；最后通过逐层求解配置矩阵得到方程的数值解.本文结构安排如下：绪论中主要介绍本文的研究背景、研究方法、所考虑问题的研究现状及本文的主要研究工作；第二章,在给出径向基函数的相关概念后详细论述径向基函数的插值理论；第三章,介绍无网格方法中,基于径向基函数的三种基本数值方法；第四章,利用无网格特解方法数值求解非齐次电报方程并通过数值算例说明该方法对其方程的有效率和稳定性.第五章,本文工作总结及后续研究展望.
[Abstract]:As a kind of special nonlinear evolution partial differential equation, the inhomogeneous telegram equation has been widely used in the fields of electricity, optics, acoustics and microwave technology. With the exception of a few cases, the analytical solution of the equation is difficult to obtain, and the approximate solution can only be obtained by numerical method. Therefore, it is of great theoretical significance and practical value to study the numerical method of inhomogeneous Telegraph equation. The innovation of this paper is that the meshless special solution method for inhomogeneous telegram equation is studied in this paper. Firstly, the time derivative of the equation is discretized by the finite difference method; secondly, the radial basis function is selected to approximate the unknown function and its spatial derivative; finally, the numerical solution of the equation is obtained by solving the configuration matrix layer by layer. The structure of this paper is arranged as follows: the introduction mainly introduces the research background, research methods, the status quo of the problems considered and the main research work of this paper; In chapter 2, the interpolation theory of radial basis function is discussed in detail after the concept of radial basis function is given, and in chapter 3, three basic numerical methods based on radial basis function are introduced in meshless method. In chapter 4, the meshless special solution method is used to solve the inhomogeneous telegram equation, and the numerical examples are given to illustrate the efficiency and stability of the method to the equation. The fifth chapter summarizes the work of this paper and looks forward to the follow-up research.
【学位授予单位】：山东师范大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O241.82
本文编号：2433311
[Abstract]:As a kind of special nonlinear evolution partial differential equation, the inhomogeneous telegram equation has been widely used in the fields of electricity, optics, acoustics and microwave technology. With the exception of a few cases, the analytical solution of the equation is difficult to obtain, and the approximate solution can only be obtained by numerical method. Therefore, it is of great theoretical significance and practical value to study the numerical method of inhomogeneous Telegraph equation. The innovation of this paper is that the meshless special solution method for inhomogeneous telegram equation is studied in this paper. Firstly, the time derivative of the equation is discretized by the finite difference method; secondly, the radial basis function is selected to approximate the unknown function and its spatial derivative; finally, the numerical solution of the equation is obtained by solving the configuration matrix layer by layer. The structure of this paper is arranged as follows: the introduction mainly introduces the research background, research methods, the status quo of the problems considered and the main research work of this paper; In chapter 2, the interpolation theory of radial basis function is discussed in detail after the concept of radial basis function is given, and in chapter 3, three basic numerical methods based on radial basis function are introduced in meshless method. In chapter 4, the meshless special solution method is used to solve the inhomogeneous telegram equation, and the numerical examples are given to illustrate the efficiency and stability of the method to the equation. The fifth chapter summarizes the work of this paper and looks forward to the follow-up research.
【学位授予单位】：山东师范大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O241.82
【参考文献】
相关期刊论文 前10条
1 廖勇;徐刚;谢平;石小燕;陆巍;丁恩燕;杨周炳;孟凡宝;;非线性传输线数值模拟方法[J];强激光与粒子束;2015年08期
2 谢平;徐刚;廖勇;石小燕;杨周炳;;非线性传输线产生射频脉冲原理研究[J];强激光与粒子束;2014年04期
3 郭鹏;陈宗广;孙小伟;;非线性电报方程的简洁解法[J];大学物理;2014年04期
4 褚洪学;姜同松;;求解一类线性反应扩散方程的特解方法[J];聊城大学学报(自然科学版);2013年02期
5 王晓蕾;姜同松;;两种求解一类二维热传导方程的无网格数值算法的比较[J];临沂大学学报;2012年06期
6 张能伟;郭欣;;带有非线性阻尼项的非线性电报方程的cauchy问题[J];安阳师范学院学报;2012年05期
7 戚祖敏;张军;钟辉煌;张点;;利用非线性传输线产生高功率射频场[J];强激光与粒子束;2012年04期
8 杨中兵,徐万达;非线性电报方程解的存在性[J];黄金学报;1999年02期
9 范恩贵,张鸿庆;非线性波动方程的孤波解[J];物理学报;1997年07期
10 王海明;非线性电报方程的周期解[J];应用数学;1997年02期
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