Self-validated computation of singular integral and integral equations
| Project/Area Number |
15540111
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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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 | The University of Electro-Ccmmunications |
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Principal Investigator |
YAMAMOTO Nobito UEC, Department of electro-communications, Associate professor, 電気通信学部, 助教授 (30210545)
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| Co-Investigator(Kenkyū-buntansha) |
IMAMURA Toshiyuki UEC, Department of electro-communications, Lecturer, 電気通信学部, 講師 (60361838)
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| Project Period (FY) |
2003 – 2004
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| Project Status |
Completed (Fiscal Year 2004)
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| Budget Amount *help |
¥2,600,000 (Direct Cost: ¥2,600,000)
Fiscal Year 2004: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2003: ¥1,600,000 (Direct Cost: ¥1,600,000)
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| Keywords | singular integral / self-validated computation / integral equations / numerical verification of local uniqueness / 精度保証法 / 数値的検証法 / 関数方程式 / 常微分方程式 / 初期値問題 / 局所一意性 / 固有値 |
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| Research Abstract |
The present research has two purposes.
1. Establishment of self-validated computation for singular integral using DE transformation, which is one of most effective methods for calculating approximate values of singular integrals.
2. Development of numerical verification methods for the existence and the local uniqueness of solutions to integral equations.
On the self-validated computation of DE transformation, we constructed a set of basic techniques for a computer program library of computation with guaranteed accuracy of singular integrals. For the arguments of the programs, we suppose integrands composed of elementary functions. The class of the DE transformation is chosen corresponding to the singularity of the integrand at the ends of the interval of integral. In order to estimate the error bounds, we need to verify the regularity of the integrand on an expanded area in a complex region. For this purpose, the method by Sugiura et al. is adopted. These results are described in detail in the report of the research results.
On the numerical verification of integral equations, we carried out our study as follows.
1. Establish a numerical verification for the local uniqueness of solutions to function equations including integral equations.
2. Develop a numerical verification methods for the systems of ordinal differential equations with initial values which are derived from integral equations
3. Implementation of 1 to 2.
For 1, we have got an important result which will be useful for a wide range of self-validation. For 2 and 3, we have developed a new method and are now improving it for practical use.
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Report
(3 results)
Research Products
(7 results)
