Developments of applications of the double exponential transformation

重指数公式是由高桥（H. Takahasi）和森（M. Mori）为了高效地求定积分而首先提出的，并已在科学技术的各个领域得到应用。这个公式的基本思想来自于二重指数变换。在以前的一个研究项目中，我们通过将双指数变换与Slue函数结合，开发了除数值积分之外的几种强大的数值方法。这种方法被称为DE-Sine方法。将DE-Sine方法应用于不定积分的数值计算，得到了Voloterra积分方程的迭代积分方法和数值解。在本研究项目中，我们扩展了双指数变换的思想，通过将de - sin方法与伽辽金方法或配点法相结合，发展了求解边值问题和初值问题的数值方法，得到了精度很高的结果。得到了二阶常微分方程边值问题的格林函数数值解法、基于伽辽金方法的四阶常微分方程边值问题的数值解法、弱奇异核积分方程的数值解法和微分代数方程的数值解法。特别是对于奇异摄动问题，用这种方法可以得到精度很高的数值解，而不必特别注意方程是奇异摄动的。

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.