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）找到Navier-Stokes方程的新解决方案，包括具有内部层的解决方案。 （2）开发了一种用于积分方程的几乎单数解的新数值技术。 （3）该技术成功地应用于具有120度奇异性的孤立波，（3）开发了一种用于振荡积分的新数值方法（4）一种用于部分微分方程的高精确数值方法，开发。 K. Kobayashi提出了一种新的方法，用于计算最小的表面，通过该方法，JSiam授予了JSiam的年度奖励。 Okamoto和他的学生K. Kobayashi将双重指数转换应用于描述孤独波的积分方程。他们表明，一个几乎奇异的解决方案，其计算在常规数值方法中需要更多的1000个网格点，只能使用128个网格点来计算得很好。 Okamoto和M. Nagayama发现了具有内部层的Navier-Stokes流。 Ooura发现了一种一维数值集成的新方法。一些积分的积分振荡和衰减以代数速率振荡和衰减，以高精度很难计算。他概括了一种萨尔策转变，该转化用于将串联的融合加速到正交规则。在某些示例中，常规方法可以仅准确地计算出三位数的积分，而他的新方法可以以8位精度计算相同的积分。他还提出了一种新算法来计算圈子比率，他被JSiam授予年度论文奖

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。

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)

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。

M, Nagayama, T. Ikeda, T. Ishiwata, N. Tamura and M.Ohyanagi: "Three-dimensional numerical simulation of helically propagating combustion waves"J. Mater. Synthesis Proces.. 9. 153-163 (2001)

M，Nagayama，T. Ikeda，T. Ishiwata，N. Tamura 和 M.Ohyanagi：“螺旋传播燃烧波的三维数值模拟”J。