Application of the double exponential transform to integral transformations

双指数变换在积分变换中的应用

• 批准号: 11554002
• 负责人: OKAMOTO Hisashi
• 金额: $ 2.5万
• 依托单位: KYOTO UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-11554002/
• 关键词: singularity; FFT; solitary wave; double exponential transform; Yamada's integralequation; blow-up of solutions; interior layer; self-similar solution; 2重指数関数変換; 積分方程式; Cahn-Hilliard方程式; Navier-Stokes方程式; 反応拡散系; 差分法; Euler方程式; Nekrasov方程式

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

概述：（1）发现了Navier-Stokes方程的新解，包括具有内层的解。（2）发展了一种新的求解积分方程近似奇异解的数值方法。(3)（3）发展了一种新的振荡积分的数值方法;（4）发展了一种高精度的偏微分方程数值方法，该方法有效地利用了离散变分方法。K.小林提出了一种数值计算极小曲面的新方法，并以此获得了JSIAM的年度论文奖。冈本和他的学生K.小林将双指数变换应用于描述孤立波的积分方程。结果表明，用传统的数值方法需要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。一些积分的被积函数振荡和衰减的代数率，被称为是难以计算的高精度。他推广了Salzer变换，这是用来加速收敛的系列，以求积规则。在一些例子中，传统方法可以计算的积分只有三位数的精度，而他的新方法可以计算相同的积分的精度为8位数。他还提出了一种新的算法来计算圆比，他被授予JSIAM年度论文奖

项目成果

M.Nagayama: "Dynamics of travelling breathers arising in reaction-diffusion systems-ODE modelling approach (with M.Mimura,H.Ikeda and T.Ikeda)"Hiroshima Math.J.. 30(2). 221-256 (2000)

M.Nagayama：“反应扩散系统中产生的移动呼吸动力学 - ODE 建模方法（与 M.Mimura、H.Ikeda 和 T.Ikeda）”Hiroshima Math.J. 30(2)。

T, Matsuo, M. Sugihara, D. Furihata and M. Mori: "Spatially accurate conservative or dissipative finite difference schemes derived by the discrete variational method"Japan J. Indus. Appl. Math.. (to appear).

T、Matsuo、M. Sugihara、D. Furihata 和 M. Mori：“通过离散变分方法导出的空间精确保守或耗散有限差分格式”Japan J. Indus。

T.Matsuo, M.Sugihara, D.Furihata, M.Mori: "Spatially accurate conservative or dissipative finite difference schemes derived by the discrete variational method"to appear in Japan J. Indus. Appl. Math..

T.Matsuo、M.Sugihara、D.Furihata、M.Mori：“通过离散变分方法导出的空间精确保守或耗散有限差分格式”出现在日本 J. Indus 上。

D.Furihata: "Finite difference schemes for (∂t/∂u)=((∂x/∂))^α(δu/δG) that inherit energy conservation or dissipation property"J. Comput. Phys.. 156. 181-205 (1999)

D.Furihata：“继承能量守恒或耗散性质的 (∂t/∂u)=((∂x/∂))^α(δu/δG) 的有限差分格式”J. 计算。 -205 (1999)

D.Furihata: "Finite difference schemes for nonlinear wave equation that inherit energy conservation property"J. Comp. Appl. Math.. 134. 35-57 (2001)

D.Furihata：“继承能量守恒性质的非线性波动方程的有限差分格式”J。