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Research on the double exponential formula for indefinite integrals

不定积分双指数公式的研究

基本信息

批准号：
15607017
负责人：
MORI Masatake
金额：
$ 1.34万
依托单位：
Tokyo Denki University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2005
项目状态：
已结题

项目摘要

The double exponential (abbreviated as DE) transformation was first proposed in 1974 by Takahasi and Mori, the head investigator, in order to evaluate definite integrals efficiently. Until now it has come to be used in various fields of science and technology and also incorporated in several famous mathematical softwares such as Mathematica and Maple. On the other hand recently it has been found that the DE transformation can also be applied to efficient numerical evaluation of indefinite integrals. In fact in 2003 the head investigator and people in his group proposed a formula for indefinite integration combining the DE transformation and the Sinc method (DE-Sinc method) and they published two papers on the new formula in 2003 and 2005. As an application of the formula for indefinite integrals they proposed a new DE formula for efficient evaluation of iterated integrals and published also two papers. Also they applied the DE transformation to numerical solution of Volterra integral equations of the second kind and published a paper in 2005. Since a given ordinary differential equation can be transformed into a Volterra integral equation they proposed a new method based on the DE-Sinc method to solve numerically the initial value problem of ordinary differential equation. Furthermore they applied their idea to the solution of boundary value problems of ordinary differential equations. In September 2004 an international workshop titled "Thirty years of the double exponential transforms' was held in Research Institute for Mathematical Sciences, Kyoto University and it was a good opportunity to make the DE transformation more popular also to people from outside Japan.

双指数变换（double exponential transformation，简称DE变换）是由Takahasi和Mori于1974年提出的，目的是有效地计算定积分。到目前为止，它已被应用于各个科学技术领域，并已被纳入一些著名的数学软件，如Mathematica和Maple。另一方面，最近发现DE变换也可以应用于不定积分的有效数值计算。事实上，在2003年，首席研究员和他的小组成员提出了一个结合DE变换和Sinc方法的不定积分公式（DE-Sinc方法），他们在2003年和2005年发表了两篇关于新公式的论文。作为不定积分公式的应用，他们提出了一个新的DE公式，有效地评估迭代积分，并发表了两篇论文。他们还将DE变换应用于第二类沃尔泰拉积分方程的数值求解，并于2005年发表了一篇论文。由于给定的常微分方程可以转化为沃尔泰拉积分方程，他们提出了一种基于DE-Sinc方法数值求解常微分方程初值问题的新方法。此外，他们应用他们的想法解决边值问题的常微分方程。在2004年9月国际研讨会题为“三十年的双指数变换”举行了研究所数学科学，京都大学，这是一个很好的机会，使DE变换更受欢迎的人也来自日本以外。