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.

双重指数公式首先是由H. takahasi和M. M. M. M. M. M.提出的，以评估高效率的确定积分，并已在科学技术的各个领域中使用。该公式的基本思想来自双重指数转换。在一个以前的研究项目中，我们通过将双重指数转换与SLUE函数结合在一起，开发了几种功能强大的数值方法。这种方法称为De-Sine方法。具体而言，我们将De-Sine方法应用于不确定积分的数值计算，并获得了迭代集成和Voloterra积分方程的数值解决方案的方法。在本研究项目中，我们扩展了双重指数转换的概念，以开发数值方法，以解决边界价值问题和初始值问题，这些方法通过将De-Sine方法与Galerkin方法或搭配方法合并为高精度。结果，我们获得了一种数值绿色的函数方法，用于第二阶普通微分方程的边界值问题，这是一种基于Galerkin方法的第4阶普通微分方程的数值解决方案的数值解决方案的方法，一种用于弱呈奇异核的积分方程的数值解决方案的方法，以及用于差异元素差异的方法的方法。特别是，通过目前的奇异扰动问题的方法，我们可以以非常高的精度获得一个数值解决方案，而无需对方程式具有奇异扰动的事实。