Application of the double exponential transform to integral transformations

概述：(1)发现了Navier-Stokes方程的新解，包括具有内层的解。(2)提出了一种求解积分方程近似奇异解的新方法。(3)成功地将该技术应用于具有120度奇点的孤立波；(3)建立了一种新的振荡积分数值方法；(4)建立了一种有效利用离散变量法求解偏微分方程的高精度数值方法。K. Kobayashi提出了一种新的数值计算最小曲面的方法，并因此获得了JSIAM的年度论文奖。冈本和他的学生K. Kobayashi将双指数变换应用于描述孤波的积分方程。他们表明，在传统的数值方法中，计算需要1000多个网格点的近似奇异解，只需128个网格点就可以很好地计算出来。Okamoto和M. Nagayama发现了具有内层的Navier-Stokes流。欧拉发现了一种新的一维数值积分方法。有些积分的被积项以代数速率振荡和衰减，因此很难精确计算。他将萨尔泽变换推广到一个求积分规则，这个变换被用来加速级数的收敛。在一些例子中，传统方法计算积分的精度只有三位数，而他的新方法计算同样的积分的精度可以达到8位数。他还提出了一种计算圆比的新算法，并因此获得了JSIAM的年度论文奖

Overview : (1) New solutions of the Navier-Stokes equations, including those solutions having interior layers were found. (2) a new numerical technique for nearly singular solutions of integral equations was developed. (3) That technique was successfully applied to solitary waves with 120-degree singularity, (3) a new numerical method for oscillatory integral was developed (4) a high-accurate numerical method for partial differential equations, which effectively use the discrete vanational method, was developed. K. Kobayashi proposed a new method for numerically computing minimal surfaces, by which an annual award of papers by JSIAM was awarded on him.H. Okamoto and his student K. Kobayashi applied the double exponential transform to the integral equations which describes the solitary waves. They showed that a nearly singular solution, whose computation required more'than 1000 mesh points in conventional numerical methods, can be computed very well with only 128 mesh points.H. Okamoto and M. Nagayama found Navier-Stokes flows which have interior layers.T. Ooura discovered a new method of one-dimensional numerical integration. Some integrals whose integrands oscillate and decay with an algebraic rate, were known to be difficult to compute with high accuracy. He generalized a Salzer transformation, which was used to accelerate the convergence of series, to a quadrature rule. In some examples, conventional methods can compute the integrals with an accuracy of only three digits, while his new method can compute the same integrals with an accuracy of 8 digits. He also proposed a new algorithm to compute the circle ratio, by which he was awarded an annual award of papers by JSIAM