Project/Area Number 
11554002

Research Category 
GrantinAid for Scientific Research (B)

Allocation Type  Singleyear Grants 
Section  展開研究 
Research Field 
General mathematics (including Probability theory/Statistical mathematics)

Research Institution  KYOTO UNIVERSITY 
Principal Investigator 
OKAMOTO Hisashi Research Institute for Mathematical Sciences, Professor, 数理解析研究所, 教授 (40143359)

CoInvestigator(Kenkyūbuntansha) 
FURIHARA Daisuke Cybermedia Center, Osaka Univ., Lecturer, サイバーメディアセンター, 講師 (80242014)
MUROTA Kazuo Research Institute for Mathematical Sciences, Professor, 数理解析研究所, 教授 (50134466)
MORI Masataka Dept. of Mathematical Sciences, Tokyo Denki Univ., Professor, 理工学部, 教授 (20010936)
OOURA Takuya Research Institute for Mathematical Sciences, Research Associate, 数理解析研究所, 助手 (50324710)
NAGAYAMA Masaharu Research Institute for Mathematical Sciences, Research Associate, 数理解析研究所, 助手 (20314289)

Project Period (FY) 
1999 – 2001

Project Status 
Completed (Fiscal Year 2001)

Budget Amount *help 
¥3,900,000 (Direct Cost: ¥3,900,000)
Fiscal Year 2001: ¥1,600,000 (Direct Cost: ¥1,600,000)
Fiscal Year 2000: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 1999: ¥1,400,000 (Direct Cost: ¥1,400,000)

Keywords  singularity / FFT / solitary wave / double exponential transform / Yamada's integralequation / blowup of solutions / interior layer / selfsimilar solution / 2重指数関数変換 / 積分方程式 / CahnHilliard方程式 / NavierStokes方程式 / 反応拡散系 / 差分法 / Euler方程式 / Nekrasov方程式 / 保存型スキーム / CahnHiliard方程式 / 燃焼合成 / 極小曲面 
Research Abstract 
Overview : (1) New solutions of the NavierStokes 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 120degree singularity, (3) a new numerical method for oscillatory integral was developed (4) a highaccurate 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 NavierStokes flows which have interior layers. T. Ooura discovered a new method of onedimensional 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
