Developments of applications of the double exponential transformation

点击翻译按钮获取中文摘要

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.