Developments of applications of the double exponential transformation

双指数公式最早是由H.Takahasi和M.Mori为了高效地计算定积分而提出的，并已被广泛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，我们通过将双指数变换与Slue函数相结合，开发了除了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。具体地，我们将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和数值求解方法。在本研究项目中，我们将双指数变换的思想推广到边值问题和初值问题的数值方法中，通过将DE-Sine方法与Galerkin方法或配置法相结合，得到了具有很高精度的结果。由此得到了求解二阶常微分方程边值问题的数值格林函数方法、基于Galerkin方法的四阶常微分方程边值问题的数值求解方法、具有弱奇异核的积分方程数值求解方法以及微分-代数方程的数值求解方法。特别地，对于奇摄动问题，不需要特别注意方程具有奇异摄动的事实，用本方法可以得到非常高精度的数值解。

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.