三类非线性反应扩散方程的无网格配置方法 Alternative Title Meshfree collocation methods for three kinds of nonlinear reaction diffusion equations 肖艳敏 Thesis Advisor 伍渝江 2011-06-01 Degree Grantor 兰州大学 Place of Conferral 兰州 Degree Name 硕士 Keyword 无网格配置方法 广义Fisher方程 Allen-Cahn方程 奇异反应扩散方程 径向基函数插值 Abstract 自然界中许多生物, 物理, 化学问题的数学模型都可以用微分方程的形式给出, 而这些微分方程的精确解大多数是无法求出的. 随着计算机技术的迅速发展,可以利用计算机求方程的数值解或者模拟这些现象. 无网格方法是近年来迅速发展起来的一种新兴的科学计算方法, 它任意选取求解域内的坐标点构造插值基函数的组合逼近微分方程的解, 不需要或较少的需要网格剖分, 这样就可以很容易的模拟许多复杂的自然现象. 目前, 这种方法已经成为求解偏微分方程数值解的 一种重要方法. 在这篇文章中, 主要是利用基于径向基函数插值的无网格配置方法求解一些非线性反应扩散方程的数值解, 这些方程有广义Fisher方程,Allen-Cahn方程和奇异反应扩散方程. 此方法可以把非线性问题转化为线性问题, 与传统的有限元和有限差分相比不需要网格剖分, 简单灵活易推广到多维等优点. 它本身的这些优点促使它迅速发展并成为科学研究者进行科学和工程研究的重要方法. 文章最后的数值结果说明了这种方法的有效性. Other Abstract Many mathematical models of biology, physics and chemistry in nature can be represented by differential equations, but most of differential equations’ exact solutions can’t be found. With the swift development of Computer technology, We can figured out the numerical solution of differential equations by computer and simulate natural phenomenon . Meshfree method is a rapid development new numerical method in recent years. Compared with the traditional finite element method and finite difference methods, it is with an arbitrary take fixed coordinates tectonic interpolation basis function discrete partial differential equation, partially or completely mesh dissection,This makes it easy to simulate many complex natural phenomenon. At present,it has become an important approach for solving partial differential equations. This paper performs a meshfree collocation method based on the interpolation of radial basis functions(RBFs) for solving the nonlinear reaction-diffusion equations.These problems concern generalized Fisher equation, Allen-Cahn equation and Nonlinear singular reaction-diffusion equations. A nonlinear problem is changed into a linear one via this method. Compared with the traditional finite element method and finite differential method, the new algorithm in this paper doesn’t need any mesh and it is simple and vivid to extend hign-dimensional problems. Some numerical results and simulations are also presented to demonstrate the effectiveness of the method. URL 查看原文 Language 中文 Document Type 学位论文 Identifier https://ir.lzu.edu.cn/handle/262010/224749 Collection 数学与统计学院 Recommended CitationGB/T 7714 肖艳敏. 三类非线性反应扩散方程的无网格配置方法[D]. 兰州. 兰州大学,2011.
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