Numerical Methods Based on Sinc Functions

基于Sinc函数的数值方法

• 批准号: 11450038
• 负责人: SUGIURA Masaaki
• 金额: $ 7.74万
• 依托单位: Nagoya 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-11450038/
• 关键词: Sinc functions; Sinc numerical methods; spectral methods; double exponential transformation; indefinite integrals; 2-point boundary value problems; Sturm-Liouville eigenvalue problems; Poisson equations; SINC関数; 常微分方程式の混合境界値問題; Sinc-Galerkin法; ; Sinc functions; Sinc numerical methods; spectral methods; double exponential transformation; indefinite integrals; 2-point boundary value problems; Sturm-Liouville eigenvalue problems; Poisson equations; SINC関数; 常微分方程式の混合境界値問題; Sinc-Galerkin法

This project aims at developing numerical methods based on Sine functions incorporated with the double exponential transformation technique. The following results have been obtained.1. A Sinc method using the double exponential transformation technique is developed for computing indefinite integrals. Two Sinc methods using the conventional single exponential transformation technique are well-known : one is due to Kearfott, and the other due to Haber. While these well-known methods converge at the rate exp(-c-√<n>) (n : the number of function evaluations), our method converges at the rate exp(-c'n/ log n).2. A Sinc-Galerkin method incorporated with the double exponential transformation technique for two-point boundary value problems is developed. While the rate of the convergence of the original Sinc-Galerkin method due to Stenger is exp(-c-√<n>)(n: the number of basis functions), that of our method is exp(-c'n/ logn), which is a remarkable improvement.3. A Sinc-collocation method combined with the double exponential transformation technique for Sturm-Liouville eigenvalue problems is developed. Our method enjoyes the convergence rate O(exp(-c'n/ logn)) (n : the number of basis functions), whereas the original Sine-collocation method proposed by Lund et al. does the convergence rate O(exp(-c-√<n>)).4. Three spectral methods using the double exponential transformation technique are developed for solving the Poisson equation on a fan-shaped domain. One employes the Sinc functions as basis functions, another does the Legendre polynomials, and the other does the Chebyshev polynomials. All the methods converge at the rate exp(-c√<n>/logn), where n is the number of basis functions.

该项目旨在开发基于与双重指数转换技术合并的正弦函数的数值方法。已经获得以下结果。1。使用双重指数转换技术的SINC方法是为计算无限积分而开发的。使用常规的单指数转换技术的两种SINC方法是众所周知的：一种是由于Kearfott造成的，另一种是由于Haber引起的。尽管这些众所周知的方法以速率exp（-c-√<n>）（n：函数评估的数量）收敛，但我们的方法以exp（-c'n/ log n）的速率收敛。2。开发了一种与双重指数转换技术合并的SINC-Galerkin方法，用于两点边界值问题。虽然原始SINC-GALERKIN方法的收敛速率是EXP（-c-√<n>）（n：基础函数的数量），但我们方法的函数的数量是EXP（-c'n/ logn），这是一个显着的改进。3。开发了一种与Sturm-liouville特征值问题的双指数转换技术结合使用的SINC-协议方法。我们的方法享受融合率O（exp（-c'n/ logn））（n：基函数的数量），而Lund等人提出的原始正弦协议方法。收敛速率O（exp（-c-√<n>））。4。使用双重指数转换技术的三种光谱方法是为了求解扇形域上的泊松方程的三种光谱方法。一个使用SINC作为基础函数，另一个用作Legendre多项式，另一个是Chebyshev多项式。所有方法都以速率EXP（-c√<n>/logn）收敛，其中n是基本函数的数量。