| Project/Area Number |
13440035
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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 | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Kyushu University |
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Principal Investigator |
NAKAO Mitsuhiro Faculty of Mathematics, Professor, 大学院・数理学研究院, 教授 (10136418)
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| Co-Investigator(Kenkyū-buntansha) |
OISHI Shin'ichi Waseda University, Informatics, Professor, 理工学部, 教授 (20139512)
IMAI Hitoshi Tokushima University, Faculty of Engineering, Professor, 工学部, 教授 (80203298)
ISO Yusuke Kyoto University, Graduate School of Informatics, Professor, 大学院・情報学研究科, 教授 (70203065)
YAMAMOTO Tetsurou Waseda University, Informatics, Professor, 理工学部, 教授 (80034560)
NISHIDA Takaaki Kyoto University, Mathematics, Professor, 大学院・理学研究科, 教授 (70026110)
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| Project Period (FY) |
2001 – 2002
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| Keywords | Numerical analysis / Validated computation / Numerical verification / Computer assisted proof |
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| Research Abstract |
In this research, we newly developed the self-validating numerical methods which can be applied to wide mathematical and analytical problems as well as extended or improved the existing techniques.
And we actually applied these methods to particular problems such as equations in the mathematical fluid mechanics and oscillation problems. The important research results obtained by investigators and co-investigators are as follows :
1. Nakao, N.Yamamoto, Watanabe established several refinements and extensions for the numerical verification methods of solutions for elliptic problems. Namely, they succeeded the numerical computation with guaranteed error bounds for the inverse eigenvalue problems of second order elliptic operator. They also obtained some results for enclosing the solutions for elliptic variational inequlities. Moreover, they computed an optimal constant with guaranteed accuracy appearing in the a priori error estimates for the finite element projection of the Poisson problem, which is an important contribution for the numerical verification for nonlinear elliptic problems.
2. Nagatou and Minamoto obtained interesting computer assisted proofs for the Kolmogorov problem and for the perturbed Gelfand equation, respectively.
3. Oishi established some fast algorithms for the fundamental validated computations for the solutions of linear equations.
4. Nishida et al. computed with guaranteed error bounds for the non-trivial solution of heat convection problems, which is an important result for a computer assisted proof in the fluid mechanics.
5. T. Yamamoto obtained some convergence results of the finite difference scheme for the singular solutions of two point boundary value problems.
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