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Self-validated computation of singular integral and integral equations

Research Project

Project/Area Number 15540111
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionThe University of Electro-Ccmmunications

Principal Investigator

YAMAMOTO Nobito  UEC, Department of electro-communications, Associate professor, 電気通信学部, 助教授 (30210545)

Co-Investigator(Kenkyū-buntansha) IMAMURA Toshiyuki  UEC, Department of electro-communications, Lecturer, 電気通信学部, 講師 (60361838)
Project Period (FY) 2003 – 2004
Project Status Completed (Fiscal Year 2004)
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)
Keywordssingular 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
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.

Report

(3 results)
  • 2004 Annual Research Report   Final Research Report Summary
  • 2003 Annual Research Report
  • Research Products

    (7 results)

All 2004 2003

All Journal Article (7 results)

  • [Journal Article] A numerical verification of nontrivial solutions for the heat convection problems2004

    • Author(s)
      Watanabe, Y., Yamamoto, N., Nakao, M.T., Nishida,T.
    • Journal Title

      J. Math. Fluid Mech 6

      Pages: 1-20

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] A numerical verification of nontrivial solutions for the heat convection problems2004

    • Author(s)
      Watanabe, Y., Yamamoto, N., Nakao, M.T., Nishida, T.
    • Journal Title

      J.Math.Fluid Mech. 6

      Pages: 1-20

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] A Numerical Vertification of Nontrivial Solutions for the Heat Convection Problem2004

    • Author(s)
      Y.Watanabe, N.Yamamoto, M.T.Nakao, T.Nishida
    • Journal Title

      Journal of Mathematical Fluid Mechanics 6

      Pages: 1-20

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Error estimation with guaranteed accuracy of finite element method in nonconvex polygonal domains2003

    • Author(s)
      Yamamoto, N., Hayakawa, K
    • Journal Title

      J. Comput. Appl. Math. 159

      Pages: 173-183

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Some computer assisted proofs for solutions of the heat convection problems2003

    • Author(s)
      Nakao, M.T., Watanabe, Y., Yamamoto, N., Nishida,T.
    • Journal Title

      Reliable Computing 9

      Pages: 359-372

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Error estimation with guaranteed accuracy of finite element method in nonconvex polygonal domains2003

    • Author(s)
      Yamamoto, N., Hayakawa, K.
    • Journal Title

      J.Comput.Appl.Math 159

      Pages: 173-183

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary
  • [Journal Article] Some computer assisted proofs for solutions of the heat convection problems2003

    • Author(s)
      Nakao, M.T., Watanabe, Y., Yamamoto, N., Nishida, T.
    • Journal Title

      Reliable Computing 9

      Pages: 359-372

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2004 Final Research Report Summary

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Published: 2003-04-01   Modified: 2016-04-21  

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