【作者】 李运利；

【导师】 郭清伟；

【作者基本信息】 合肥工业大学 ， 计算数学， 2010， 硕士

【摘要】 科学和工程技术中的许多实际问题都可以转化为微分方程的求解问题,而大部分的微分方程很难求出其解析解,因此,微分方程的数值解法的研究就具有重要的意义。由于样条函数具有许多好的性质而被广泛用于数值逼近、计算几何和微分方程数值解等领域。本文主要任务是利用几类样条函数来构造常微分方程的数值解。本文章节内容安排如下:第一、二两章介绍了本文的研究背景、几类样条函数和几种经典的微分方程数值解方法。第三章在总结了近年来Bernstein多项式在微分方程数值解中的应用的基础上,用Bernstein多项式构造了一类两点边值问题方程组和奇异边值问题的数值解法,并用数值实例说明方法的有效性。第四章总结了已有的两点边值问题的数值解法,在此基础上,给出了一类带有小参数ξ的两点边值问题的三次多项式样条解法,并分析了截断误差,用所给的方法计算了相关参考文献中的数值例子,与其他方法的结果进行了比较。第五章讨论了高阶微分方程的数值解法,主要是对已有的方法特别近几年的新方法作了总结,在此基础上,给出了任意高阶微分方程的一种三次B样条解法,用所给的方法计算了参考文献中的数值例子,与已有的结果进行了比较,说明本文方法的有效性。第六章总结了全文并进行了展望。更多还原

【Abstract】 There are practical problems occurring in many areas of science and engineering, which can be transformed into solve differential equations. Most of differential equations are difficult to obtain their analytical solutions. Therefore, the research of numerical solution of differential equations is very important and necessary. The spline has many useful properties, such as the continuity, unity partition. So they are applied widely in function approximation, computational geometry, and numerical solution of differential equations and so on. In this paper, we study the numerical solution methods based on various kinds of spline function for initial and boundary value problems of ordinary differential equations.This paper is organized as follows:Chapter1and 2: Investigative background of numerical solution of differential equations, and introduce various types of spline functions and several classical numerical methods for solving differential equations.Chapter 3: Summarize the application of Bernstein polynomial to solve differential equations, especially the recent existent numerical methods. Based on the result, we constructed the numerical method of a linear system of second-order boundary value problems and a kind of singular boundary value problem. Several numerical examples are presented to illustrate the efficiency of the methods.Chapter 4: Summarize the existing numerical solution of second-order boundary value problems, based on the result, we developed a numerical technique of a class of two-point boundary value problem with parametricξusing cubic polynomial spline function, analysis the error and compute the numerical examples of the relevant references using the given method, and compare with other articles.Chapter 5: Discuss the solution of high-order boundary value problems, sum up the recent existent numerical methods, based on the result, we constructed the numerical method of any high-order boundary value problems using third B-spline, we applied the new method to compute several numerical examples of the reference literature, and compared the result with other articles, it is shown that the new method is efficient.Chapter 6: Summarizes the whole dissertation and give some expect ion.更多还原