本文選題：四維反應擴散方程 + 緊差分格式��； 參考：《延邊大學》2017年碩士論文

【摘要】：隨著科學技術的快速發(fā)展,微分方程在理論、實際應用中都起著不可替代的作用,比如石油的開發(fā)、圖像分析、航空航天、生物制藥以及自動控制技術等日常生產(chǎn)生活的研究都可以抽象成高維、大范圍的數(shù)學偏微分方程定解問題來解決,然而只有少數(shù)微分方程可以求解出精確解,絕大多數(shù)的方程無法正常求出其精確解,所以人們想到用近似解代替精確解來解決實際問題中的數(shù)學模型問題,但近似解的精度直接影響了實際問題的研究,所以提高微分方程近似解精度的研究一直備受學者關注。有限差分法是用于求解微分方程定解問題最常用的數(shù)值方法之一,其基本思想是用含有有限個離散未知量的差分方程組去近似代替連續(xù)變量的微分方程和定解的條件,并把微分方程組的解作為微分方程定解問題的近似解,離散型微分方程組的解與連續(xù)性微分方程的解之間誤差越小,精確度越高,對解決實際問題的影響也就越小。所以離散型微分方程組的構建對現(xiàn)實的生產(chǎn)生活有十分重要的意義。而對于微分方程的差分算法所涉及的離散后的網(wǎng)格點越少、邊界條件不需要特殊處理、更高精度的計算方法是學者們研究的熱點。同時隨著計算機產(chǎn)業(yè)的迅速發(fā)展,越來越多的人能夠熟練的應用計算機求解數(shù)學上的問題,從而就可以用計算機進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,使其更加貼近實際問題。本文主要應用基本的差分公式推導出四維常系數(shù)反應擴散方程的緊交替方向差分格式,并通過MATLAB軟件進行相應的數(shù)值實驗,驗證了格式的精確度。主要內容如下:在第一章的序言部分,簡單介紹了關于反應擴散方程研究背景和差分的基礎知識,以及近來,國內外學者對反應擴散方程求解的研究進程,并簡要說明本文的文章結構及做的主要工作。在第二章中,我們?yōu)樗木S反應擴散方程的邊值問題,建立了一種緊差分格式,并得到相應的截斷誤差表達式,然后通過格式的變形推導出緊交替方向差分格式,并用Fourier穩(wěn)定性分析法證明了該格式的穩(wěn)定性和收斂性。再通過外推算法求出格式的近似解.最后通過對一個相關的數(shù)值算例進行計算求解,驗證了該格式的有效性及精確性。
[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.
【學位授予單位】：延邊大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：O241.8

【參考文獻】

相關期刊論文 前10條

1 韓曉晨;歷君;侯陽陽;;偏微分方程在期權定價中的應用[J];遼寧工程技術大學學報(自然科學版);2016年09期

2 陳貞忠;張芳;;反應擴散方程的緊交替方向差分算法[J];天津工業(yè)大學學報;2010年06期

3 馬明書;付立志;谷淑敏;;解三維拋物型方程的一個ADI格式[J];純粹數(shù)學與應用數(shù)學;2009年01期

4 李雪玲;孫志忠;;二維變系數(shù)反應擴散方程的緊交替方向差分格式[J];高等學校計算數(shù)學學報;2006年01期

5 孫鴻烈;解高維熱傳導方程的一族高精度的顯式差分格式[J];高校應用數(shù)學學報A輯(中文版);1999年04期

6 曾文平;多維拋物型方程的分支絕對穩(wěn)定的顯式格式[J];高等學校計算數(shù)學學報;1997年02期

7 劉曉麗;偏微分方程在物理學中的應用[J];鞍山師范學院學報;1991年03期

8 鄧陽生;解一維拋物方程的差分格式[J];數(shù)值計算與計算機應用;1990年03期

9 周順興;解拋物型偏微分方程的高精度差分格式[J];計算數(shù)學;1982年02期

10 楊情民;解拋物型方程的一族顯格式[J];高等學校計算數(shù)學學報;1981年04期

相關碩士學位論文 前1條

1 李萍;拋物型偏微分方程的MWLS算法研究及應用[D];西安理工大學;2008年

，

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