抄録
Offer Organization: Japan Society for the Promotion of Science, System Name: Grants-in-Aid for Scientific Research, Category: Grant-in-Aid for Scientific Research (B), Fund Type: -, Overall Grant Amount: - (direct: 12100000, indirect: -)
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.