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.

双指数公式最初由 H. Takahasi 和 M. Mori 为了高效地计算定积分而提出，并已被应用于科学技术的各个领域。该公式的基本思想来自于双指数变换。在之前的一个研究项目中，我们通过将双指数变换与 Slue 函数相结合，开发了除数值积分之外的几种强大的数值方法。这种方法称为 DE-Sine 方法。具体来说，我们将DE-Sine方法应用于不定积分的数值计算，得到了迭代积分和Voloterra积分方程数值求解的方法。在本研究项目中，我们扩展了双指数变换的思想，开发了边值问题和初值问题的数值方法，通过将 DE-Sine 方法与 Galerkin 方法或搭配方法相结合，给出了非常高精度的结果。得到了二阶常微分方程边值问题数值格林函数法、基于伽辽金法的四阶常微分方程边值问题数值求解方法、积分方程数值求解方法弱奇异核和微分代数方程的数值求解方法。特别是通过本方法对于奇异摄动问题，我们可以获得非常高精度的数值解，而无需特别注意方程是奇异摄动的事实。