2001 Fiscal Year Final Research Report Summary
Application of the double exponential transform to integral transformations
| Project/Area Number |
11554002
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 展開研究 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | KYOTO UNIVERSITY |
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Principal Investigator |
OKAMOTO Hisashi Research Institute for Mathematical Sciences, Professor, 数理解析研究所, 教授 (40143359)
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| Co-Investigator(Kenkyū-buntansha) |
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)
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| Project Period (FY) |
1999 – 2001
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| Keywords | singularity / FFT / solitary wave / double exponential transform / Yamada's integralequation / blow-up of solutions / interior layer / self-similar solution |
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| Research Abstract |
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
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