Masatake Mori

The double-exponential transformation in numerical analysis

The double-exponential transformation was first proposed by Takahasi and Mori in 1974 for the efficient evaluation of integrals of an analytic function with end-point singularity. Afterwards, this transformation was improved for the evaluation of oscillatory functions like Fourier integrals. Recently, it turned out that the double-exponential transformation is useful not only for numerical integration but also for various kinds of Sinc numerical methods. The purpose of the present paper is to review the double-exponential transformation in numerical integration and in a variety of Sinc numerical methods.

Quadrature formulas obtained by variable transformation

AbstractQuadrature formulas suitable for evaluation of improper integrals such as

Function classes for double exponential integration formulas

\int\limits_{ - 1}^1 {f(x)(1 - x)^{ - \alpha } (1 + x)^{ - \beta } dx,\alpha ,\beta< 1}

Spatially Accurate Dissipative or Conservative Finite Difference Schemes Derived by the Discrete Variational Method

are obtained by means of variable transformations κ=tanhu and κ=erfu, and subsequent use of trapezoidal quadrature rule. Error analysis is carried out by the method of contour integral, and the results are confirmed on several concrete examples. Similar formulas are also obtained to accelerate the convergence of infinite integrals

Numerical Solution of Volterra Integral Equations with Weakly Singular Kernel Based on the DE-Sinc Method*

\int\limits_\infty ^\infty {f(x)dx}

An Error Analysis of Quadrature Formulas Obtained by Variable Transformation

by means of variable transformations κ=sinhu and κ=tanu.