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2003 Fiscal Year Final Research Report Summary

Analysis of nonlinear phenomena describing hysteresis operators

Research Project

Project/Area Number 14540169
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionGifu University

Principal Investigator

AIKI Toyohiko  Gifu University, Faculty of Education, Associate Professor, 教育学部, 助教授 (90231745)

Co-Investigator(Kenkyū-buntansha) YAMADA Masahiro  Gifu University, Faculty of Education, Associate Professor, 教育学部, 助教授 (00263666)
SATO Naoki  Nagaoka College of Technology, Liberal Arts, Associate Professor, 一般教育科, 助教授 (90280370)
TAKEUCHI Shigeru  Gifu University, Faculty of Education, Professor, 教育学部, 教授 (30021330)
ITO Akio  Kinki University, Faculty of Engineering, Lecturer, 工学部, 講師 (30303506)
ISHIWATA Tetsuya  Gifu University, Faculty of Education, Associate Professor, 教育学部, 助教授 (50334917)
Project Period (FY) 2002 – 2003
Keywordshysteresis operator / shape memory alloy / nonlinear PDE / ferromagnetic / weak solution of PDE
Research Abstract

[Shape memory alloy problems] : In the dynamics of shape memory alloy materials the relationship between the strain and the stress can not be described by a usual function, and is represented by a hysteresis operator. Here, the main idea of this research project is to describe the relationship by using a general stop operator. Moreover, we consider the ordinary differential equation, which is equivalent to the generalized stop operator, and propose a system describing the dynamics of shape memory alloys. The system consists of partial differential equations and the ordinary differential equation and is studied in this project. We consider the one-dimensional problem, in which we dealt the ordinary differential equation without an approximation. In this case the regularity of a solution to the ordinary differential equation is not enough in space so that it is impossible to prove the existence of a classical solution. Then we showed the existence and the uniqueness of a weak solution. Also, we considered a shape memory alloy problem in three dimensions. In 3-d case the regularity of a solution is not good. Hence, we proved the well-posed ness for only approximated problem. Here, we note that we also have a plan to analyze our mathematical model, numerically. Now, we do not success to get a numerical solution of a complete problem and have an algorithm for solving a kinetic equation, which is most difficult to compute in our process.
[magnetization in ferromagnetic] : By using a generalized Duhem model we propose a mathematical model for dynamics of ferromagnetic and prove the well-posedness.

  • Research Products

    (12 results)

All Other

All Publications (12 results)

  • [Publications] AIKI, Toyohiko: "Uniqueness for multi-dimensional Stefan problems with nonlinear boundary conditions described by maximal monotone operators"Differential and Integral Equations. 15. 973-1008 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] AIKI, Toyohiko, IMAI, Hitoshi, ISHIMURA Naoyuki: "Well-posedness of one-phase Stefan problems for sublinear heat equations"Journal of Nonlinear Analysis : Theory, Methods, and Applications. 51. 587-606 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] AIKI, Toyohiko, KENMOCHI, Nobuyuki: "Models for shape memory alloys described by subdifferentials of indicator functions"Elliptic and Parabolic Problems, Rolduc and Gaeta 2001, World Scientific Publishing. 1-10 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] AIKI, Toyohiko: "One-dimensional shape memory alloy problems including a hysteresis operator"Funkcialaj Ekvaccioj. 46. 441-469 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Ishiwata, S.Yazaki: "On the blow-up rate for fast blow-up solutions arising in an anisotropic crystalline motion"Journal of Computational and Applied Mathematics. 159. 55-64 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] AIKI, Toyohiko, IMAI, Hitoshi, ISHIMURA Naoyuki: "One-phase Stefan problems for sublinear heat equations : Asymptotic behavior of solutions"Communications in Applied Analysis. 8. 1-15 (2004)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Toyohiko Aiki: "Uniqueness for multi-dimensional Stefan problems with nonlinear boundary conditions described by maximal monotone operators."Differential and Integral Equations. 15. 973-1008 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] AIKI, Toyohiko, IMAI, Hitoshi, ISHIMURA, Naoyuki: "Well-posedness of one-phase Stefan problems for sublinear heat equations."Journal of Nonlinear Analysis : Theory, Methods, and Applications. 51. 587-606 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] AIKI, Toyohiko, KENMOCHI, Nobuyuki: "Models for shape memory alloys described by subdifferentials of indicator functions."Elliptic and Parabolic Problems, Rolduc and Gaeta 2001. 1-10 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] AIKI, Toyohiko: "One-dimensional shape memory alloy problems including a hysteresis operator."Funkcialaj Ekvaccioj. 46. 441-469 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T.Ishiwata, S.Yazaki: "On the blow-up rate for fast blow-up solutions arising in an anisotropic crystalline motion."Journal of Computational Applied and Mathematics. 159. 55-64 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] AIKI, Toyohiko, IMAI, Hitoshi, ISHIMURA Naoyuki: "One-phase Stefan problems for sublinear heat equations : Asymptotic behavior of solutions."Communications in Applied Analysis. 8. 1-15 (2004)

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2005-04-19  

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