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Developments of applications of the double exponential transformation

双指数变换的应用进展

基本信息

批准号：
18560063
负责人：
MORI Masatake
金额：
$ 0.8万
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：
2006
资助国家：
日本
起止时间：
2006 至 2007
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-18560063/
关键词：
double exponential transformation double exponential formula DE transformation DE formula Sine method Galerkin method collocation method singular perturbation 積分方程式常微分方程式数値解法

项目摘要

The double exponential formula was first proposed by H. Takahasi and M. Mori in order to evaluate definite integrals in high efficiency and has come to be used in various fields of science and technology. The basic idea for this formula comes from the double exponential transformation. In a former research project we developed several powerful numerical methods other than numerical integration by incorporating the double exponential transformation with the Slue function. This kind of methods is called the DE-Sine method. Specifically we applied the DE-Sine method to numerical computation of indefinite integrals and obtained methods of iterated integration and numerical solution of Voloterra integral equation. In the present research project we extended the idea of the double exponential transformation to develop numerical methods for boundary value problems and initial value problems which give results with very high precision by incorporating the DE-Sine method with the Galerkin method or with the collocation method. As a result we obtained a numerical Green's function method for boundary value problem of 2nd order ordinary differential equation, a method for numerical solution of boundary value problem of 4th order ordinary differential equation based on the Galerkin method, a method for numerical solution of integral equation with weakly singular kernel and a method for numerical solution of differential-algebraic equation. In particular by the present method for singularly perturbed problem we can obtain a numerical solution with very high accuracy without paying special care for the fact that the equation is of singular perturbation.

双指数公式是由H. Takahasi和M. Mori算法是为了高效率地计算定积分而提出的，并已被应用于科学技术的各个领域。这个公式的基本思想来自双指数变换。在以前的一个研究项目中，我们开发了几个强大的数值方法，而不是数值积分，将双指数变换与Slue函数。这种方法称为DE-Sine方法。具体地说，我们将DE-Sine方法应用到不定积分的数值计算中，得到了Voloterra积分方程的迭代积分和数值解的方法。在本研究项目中，我们扩展了双指数变换的思想，发展了数值方法的边界值问题和初值问题，给出了非常高的精度，通过将DE正弦方法与Galerkin方法或配点法。得到了二阶常微分方程边值问题的数值绿色函数方法、基于Galerkin方法的四阶常微分方程边值问题的数值解法、具有弱奇异核的积分方程的数值解法和微分代数方程的数值解法。特别是对于奇摄动问题，本文的方法可以得到很高精度的数值解，而不必特别注意方程是奇摄动的事实。