四维常系数反应扩散方程的紧交替方向差分格式

发布时间：2018-07-02 21:38

本文选题：四维反应扩散方程 + 紧差分格式　；参考：《延边大学》2017年硕士论文

【摘要】：随着科学技术的快速发展,微分方程在理论、实际应用中都起着不可替代的作用,比如石油的开发、图像分析、航空航天、生物制药以及自动控制技术等日常生产生活的研究都可以抽象成高维、大范围的数学偏微分方程定解问题来解决,然而只有少数微分方程可以求解出精确解,绝大多数的方程无法正常求出其精确解,所以人们想到用近似解代替精确解来解决实际问题中的数学模型问题,但近似解的精度直接影响了实际问题的研究,所以提高微分方程近似解精度的研究一直备受学者关注。有限差分法是用于求解微分方程定解问题最常用的数值方法之一,其基本思想是用含有有限个离散未知量的差分方程组去近似代替连续变量的微分方程和定解的条件,并把微分方程组的解作为微分方程定解问题的近似解,离散型微分方程组的解与连续性微分方程的解之间误差越小,精确度越高,对解决实际问题的影响也就越小。所以离散型微分方程组的构建对现实的生产生活有十分重要的意义。而对于微分方程的差分算法所涉及的离散后的网格点越少、边界条件不需要特殊处理、更高精度的计算方法是学者们研究的热点。同时随着计算机产业的迅速发展,越来越多的人能够熟练的应用计算机求解数学上的问题,从而就可以用计算机进行高精度的求解微分方程的近似解,大大提高了微分方程的近似解的精度,使其更加贴近实际问题。本文主要应用基本的差分公式推导出四维常系数反应扩散方程的紧交替方向差分格式,并通过MATLAB软件进行相应的数值实验,验证了格式的精确度。主要内容如下:在第一章的序言部分,简单介绍了关于反应扩散方程研究背景和差分的基础知识,以及近来,国内外学者对反应扩散方程求解的研究进程,并简要说明本文的文章结构及做的主要工作。在第二章中,我们为四维反应扩散方程的边值问题,建立了一种紧差分格式,并得到相应的截断误差表达式,然后通过格式的变形推导出紧交替方向差分格式,并用Fourier稳定性分析法证明了该格式的稳定性和收敛性。再通过外推算法求出格式的近似解.最后通过对一个相关的数值算例进行计算求解,验证了该格式的有效性及精确性。
[Abstract]:With the rapid development of science and technology, differential equations play an irreplaceable role in theory and practical applications, such as oil development, image analysis, aerospace, The study of daily production life, such as biopharmaceutical technology and automatic control technology, can be solved by abstracting into high-dimensional, large-scale mathematical partial differential equations, but only a few differential equations can solve the exact solutions. Most equations can not find their exact solutions normally, so people think of using approximate solutions instead of exact solutions to solve mathematical model problems in practical problems, but the accuracy of approximate solutions directly affects the study of practical problems. Therefore, the research of improving the accuracy of approximate solution of differential equations has been paid much attention by scholars. The finite difference method is one of the most commonly used numerical methods for solving the definite solutions of differential equations. Its basic idea is to approximate the conditions of the differential equations and definite solutions with finite discrete unknown variables instead of the differential equations and solutions of continuous variables. The solution of the system of differential equations is regarded as the approximate solution of the definite solution of the differential equation. The smaller the error between the solution of the discrete differential equation system and the solution of the continuous differential equation, the higher the accuracy and the smaller the influence on the solution of the practical problem. Therefore, the construction of discrete differential equations is of great significance to practical production and life. The difference algorithm for differential equations involves less discrete grid points, and the boundary conditions do not need special treatment. Therefore, more accurate calculation methods are the focus of scholars' research. At the same time, with the rapid development of the computer industry, more and more people can skillfully use the computer to solve mathematical problems, so that the computer can be used to solve the approximate solution of differential equations with high accuracy. The accuracy of approximate solution of differential equation is greatly improved, and it is closer to the practical problem. In this paper, the compact alternating direction difference scheme of the four-dimensional constant coefficient reaction diffusion equation is derived by using the basic difference formula, and the accuracy of the scheme is verified by the corresponding numerical experiments with MATLAB software. The main contents are as follows: in the preface of the first chapter, the basic knowledge of the research background and difference of the reaction diffusion equation is briefly introduced, as well as the recent research progress of the reaction diffusion equation solved by domestic and foreign scholars. And briefly describes the structure of this article and the main work done. In the second chapter, we establish a compact difference scheme for the boundary value problem of the four-dimensional reaction-diffusion equation and obtain the corresponding truncation error expression. Then we derive the compact alternating direction difference scheme through the deformation of the scheme. The stability and convergence of the scheme are proved by Fourier stability analysis. The approximate solution of the scheme is obtained by extrapolation method. Finally, the validity and accuracy of the scheme are verified by a numerical example.
【学位授予单位】：延边大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.8

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