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Numerical analysis of the chaotic bahavior of the free boundary

Research Project

Project/Area Number 09440080
Research Category

Grant-in-Aid for Scientific Research (B)

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

Principal Investigator

IMAI Hitoshi  The University of Tokushima, Faculty of Engineering, Professor, 工学部, 教授 (80203298)

Co-Investigator(Kenkyū-buntansha) SAKAGUCHI Hideo  The University of Tokushima, Faculty of Engineering, Instructor, 工学部, 助手 (80274265)
OKAMOTO Hisashi  Kyoto University, Research Institute for Mathematical Sciences, Professor, 数理解析研究所, 教授 (40143359)
IKEDA Tsutomu  Ryukoku University, Faculty of Science and Technology, Professor, 理工学部, 教授 (50151296)
NISHIURA Yasumasa  Hokkaido University, Research Institute for Electronic Science, Professor, 電子科学研究所, 教授 (00131277)
TAKEUCHI Toshiki  The University of Tokushima, Faculty of Engineering, Associate Professor, 工学部, 助教授 (30264964)
Project Period (FY) 1997 – 1999
Project Status Completed (Fiscal Year 1999)
Budget Amount *help
¥3,300,000 (Direct Cost: ¥3,300,000)
Fiscal Year 1999: ¥1,600,000 (Direct Cost: ¥1,600,000)
Fiscal Year 1998: ¥1,700,000 (Direct Cost: ¥1,700,000)
Keywordsfree boundary / chaos / PVM / pattern / spectral method / bifurcation / phase separation / reaction-difusion / 高精度 / 任意精度 / 無限精度
Research Abstract

In the term of the project (three years) many results are obtained. Important results are shown as follows.
1. Development of numerical methods for analysis of chaotic phenomena.
The group of Tokushima Univ. developed a method for analysis of chaotic phenomena in free boundary problems.It is based on numerical methods, so it is general. Moreover, it has a surprising feature that attractors in the infinite-dimensional solution space can be approximated arbitrarily. This means the method connects theoretical analysis and numerical analysis, therefore it enhances development of theoretical and numerical researches.
2. A proposal of a free boundary problem with attractors.
The group of Tokushima Univ. proposed a one-dimensional free boundary problem with some parameters. It has exact solutions for special values of the parameters, so it is covenient for the check of the method and programs. Various attractors are found numerically.
3. Analysis of the pattern.
Nishiura solved self-replicating dynamics of the dissipative system. He also revealed the minimizer of the system which describes the pattern formation of the diblock copolymer has the fine structure with the meso-scale. Sakaguchi proposed a model to formation of colony patterns by a bacterial cell population. From numerical simulation various patterns are observed in spite of the simplicity of the model.
4. Development of fast numerical computation and its application.
The group of Tokushima Univ., Ikeda and Okamoto developed methods for fast numerical computation in the environment of parallel computing offered by PVM. These methods enabled large-scaled numerical simulation of free boundary problems or fluid mechanics. Some new phenomena were found.

Report

(4 results)
  • 1999 Annual Research Report   Final Research Report Summary
  • 1998 Annual Research Report
  • 1997 Annual Research Report
  • Research Products

    (63 results)

All Other

All Publications (63 results)

  • [Publications] Tarmizi(共著者にT.Takeuchi,H.Imai): "NUMERICAL SIMULATION OF ONE-DIMENSIONAL FREE BOUNDARY PROBLEMS IN INFINITE PRECISION"Advances in Mathematical Sciences and Applications,Gakkotosho. Vol.10,No.2(to appear). (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T.Takeuchi: "Numerical computation of attractors in two-phase Stefan problems"数理解析研究所講究録. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] H.IMAI: "Infinite Precision Numerical Simulation for PDE systems and Its Applications"数理解析研究所講究録. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] H.Imai: "On numerical computation of Lyapunov exponents of attractors in free boundary problems"NIFS-PROC. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] H.Ikeda(共著者にT.Ikeda): "Bifurcation phenomena from standing pulse solutions of bistable reaction-difhsion systems"Jornal of Dynamics and Differential Equations. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T.Ikeda: "Hopf bifurcation of traveling pulses in some bistable reaction-diffusion systems"Methods and Applications of Analysis. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M.Mimura(共著者にT.Ikeda): "Dynamics of travelling breathers arising in reaction-diffusion systems-ODEmodelling approach-"Hiroshima Mathematical Journal. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] I.Okamoto: "Some similarity solutions of the Navier-Stokes equations and related topics"Taiwanese J.Math.. (to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M.Matsushita(共著者にH.Sakaguchi): "Formation of colony patterns by a bacterial cell population"Physica A. Vol.247. 190-199 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Y.Nishiura: "A skeleton structure of self-replicating dynamics"Physica D. Vol.130. 73-104 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] I.Ohnishi(共著者にY.Nishiura): "Analytical solutions describing the phase separation driven by a free energy functional containinga Iong-range intreraction term"Chaos. Vol.9,No.2. 329-341 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T.TAKEUCHI: "On Numerical Simulation of Partial Differential Equations in Arbitrary Precision"Science International(Lahore). Vol.10,No.4(to appear).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] H.IMAI: "On Numerical Simulation of Partial Differential Equations a Infinite Precision"Advances in Mathematical Sciences and Applications. Vol.9,No.2. 1007-1016 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T.Ikeda: "Parallel computation of spots-and-stripes patterns formed by motile bacteria"Proceedings of Third China-Japan Joint Seminar on Numerical Mathematics(eds.Z.C.Shi and M.Mori),Science Press. 57-72 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Y.Nishiura: "A hidden bifurcational structure for self-replicating dynamics"ACH-Models in Chemistry. Vol.135,No.3. 343-360 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T.Sakajo(共著者にH.Okamoto): "An application of Draghicescu's fast Summation method to vortex Sheet motion"J.Phys.Soc.Japan. Vol.67. 462-470 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M.Matsushita(共著者にH.Sakaguchi): "Interface growth and pattern formation in bacterial colonies"Physica A. Vol.249. 517-524 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] H.Imai: "Numerical simulation of free boundary problems in quadruple precision arithmetic using explicit schemes"数理解析研究所講究録,京都大学. Vol.989. 18-30 (1997)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T.Ikeda: "Parallel computation of intrerfacial dynamics"Gakuto International Series. Vol.11. 52-61 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T.Sakajo(共著者にH.Okamoto): "The application of PVM to the computation of vortex sheet motion"Gakuto lnternational Series. Vol.11. 160-170 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Tarmizi (Coauthor : T. Takeuchi, H. Imai): "NUMERICAL SIMULATION OF ONE-DIMENSIONAL FREE BOUNDARY PROBLEMS IN INFINITE PRECISION"Advances in Mathematical Sciences and Applications. Gakkotosho. Vol. 10, No. 2(to appear). (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T. Takeuchi: "Numerical computation of attractors in two-phase Stefan problems"RIMS Kokyuroku, Kyoto University. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] H. Imai: "Infinite Precision Numerical Simulation for PDE systems and Its Applications"RIMS Kokyuroku, Kyoto University. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] H. Imai: "On numerical computation of Lyapunov exponents of attractors in free boundary problems"NIFS-PROC. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] H. Ikeda (Coauthor : T. Ikeda): "Bifurcation phenomena from standing pulse solutions of bistable reaction-diffusion systems"Journal of Dynamics and Differential Equations. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T. Ikeda: "Hopf bifurcation of traveling pulses in some bistable reaction-diffusion systems"Methods and Applicatons of Analysis. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M. Mimura (Coauthor : T. Ikeda): "Dynamics of travelling breathers arising in reaction-diffusion systems -ODE modelling approach-"Hirosihma Mathematical Journal. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] H. Okamoto: "Some similarity solutions of the Navier-Stokes equations and related topics"Taiwanese J. Math.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M. Matsushita (Coauthor : H. Sakaguchi): "Formation of colony patterns by a bacterial cell population"Physica A. 247. 190-199 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Y. Nishiura: "A skeleton structure of self-replicating dynamics"Physica D. 130. 73-104 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Ohnishi (Coauthor : Y. Nishiura): "Analytical solutions describing the phase separation driven by a free energy functional containing long-range interaction term"Chaos. Vol. 9, No. 2. 329-341 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T. Takeuchi: "On Numerical Simulation of Partial Differential Equations in Arbitrary Precision"Science International (Lahore). Vol. 10, No. 4(to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] H. Imai: "On numerical Simulation of Partial Differential Equations in Infinite Precision"Advances in Mathematical Sciences and Applications. Vol. 9, No. 2. 1007-1016 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T. Ikeda: "Parallel computation of spots-and-stripes patterns formed by motile bacteria"Proceedings of Third China-Japan Joint Seminar on Numerical Mathematics (eds. Z. C. Shi and M. Mori), Science Press. 57-72 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Y. Nishiura: "A hidden bifurcation structure for self-replicating dynamics"ACH-Models in Chemistry. Vol. 135, No. 3. 343-360 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T. Sakajo (Coauthor : H. Okamoto): "An application of Draghicescu's fast summation method to vortex sheet motion"J. Phys. Soc. Japan. 67. 462-470 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] M. Matsushita (Coauthor : H. Sakaguchi): "Interface growth and pattern formation in bacterial colonies"Physiica A. 249. 517-524 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] H. Imai: "Numerical simulation of free boundary problems in quadruple precision arthmetic using explicit schemes"RIMS Kokyuroku, Kyoto University. 989. 18-30 (1997)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T. Ikeda: "parallel computation of Interfacial dynamics"Gakuto International Series. 11. 52-61 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] T. Sakajo (Coauthor : H. Okamoto): "The application of PVM to the computation of vortex sheet motion"Gakuto International Series. 11. 160-170 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Tarmizi (共著者に T.Takeuchi,H.Imai): "NUMERICAL SIMULATION OF ONE-DIMENSIONAL FREE BOUNDARY PROBLEMS IN INFINITE PRECISION"Advances in Mathematical Sciences and Applications,Gakkkotosho. Vol.10,No.2(to appear). (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] T.Takeuchi (共著者に H.Imai): "Numerical computation of attractors in two-phase Stefan problems"数理解析研究所講究録. (to appear).

    • Related Report
      1999 Annual Research Report
  • [Publications] H.IMAI (共著者に T.TAKEUCHI,H.SAKAGUCHI): "Infinite Precision Numerical Simulation for PDE systems and Its Applications"数理解析研究所講究録. (to appear).

    • Related Report
      1999 Annual Research Report
  • [Publications] H.Imai (共著者に T.Takeuchi): "On numerical computation of Lyapunov exponents of attractors in free boundary problems"NIFS-PROC. (to appear).

    • Related Report
      1999 Annual Research Report
  • [Publications] H.Ikeda (共著者に T.Ikeda): "Bifurcation phenomena from standing pulse solutions of bistable reaction-diffusion systems"Journal of Dynamics and Differentual Equations. (to appear).

    • Related Report
      1999 Annual Research Report
  • [Publications] T.Ikeda: "Hopf bifurcation of traveling pulses in some bistable reaction-diffusion systems"Methods and Applications of Analysis. (to appear).

    • Related Report
      1999 Annual Research Report
  • [Publications] M.Mimura (共著者に T.Ikeda): "Dynamics of travelling breathers arising in reaction-diffusion systems-ODE modelling approach-"Hiroshima Mathematical Journal. (to appear).

    • Related Report
      1999 Annual Research Report
  • [Publications] H.Okamoto: "Some similarity solutions of the Navier-Stokes equations and related topics"Taiwanese J.Math.. (to appear).

    • Related Report
      1999 Annual Research Report
  • [Publications] M.Matsushita (共著者に H.Sakaguchi): "Formation of colony patterns by a bacterial cell population"Physica A. Vol.247. 190-199 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Y.Nishiura: "A skeleton structure of self-replicating dynamics"Physica D. Vol.130. 73-104 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] I.Ohnishi (共著者に Y.Nishiura): "Analytical solutions describing the phase separation driven by a free energy functional containinga long-range interaction term"Chaos. Vol.9,No.2. 329-341 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] T.TAKEUCHI(共著者:H.IMAI,H.SAKAGUCHI 他): "On Numerical Simulation of Partial Differential Equations in Arbitrary Precision" Science International(Lahore). (in press).

    • Related Report
      1998 Annual Research Report
  • [Publications] H.IMAI(共著者:T.TAKEUCHI他): "On Numerical Simulation of Partial Differential Equations in Infinite Precision" Advances in Mathematical Sciences and Applications. (in press).

    • Related Report
      1998 Annual Research Report
  • [Publications] T.Ikeda: "Parallel computation of spots-and-stripes patterns formed by motile bacteria" Proceedings of Third China-Japan Joint Seminar on Numerical Mathematics(eds.Z.C.Shi and M.Mori). 57-72 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Y.Nishiura: "A hidden bifurcational structure for self-replicating dynamics" ACH-Models in Chemistry. Vol.135,No.3. 343-360 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] T.Sakajo(共著者:H.Okamoto): "An application of Draghicescu's fast summation method to vortex sheet motion" J.Phys.Soc.Japan. Vol.67. 462-470 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] M.Matsushita(共著者:H.Sakaguchi 他): "Interface growth and pattern formation in bacterial colonies" Physica A. Vol.249. 517-524 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] H.Imai, Y.Shinohara, M.Natori, W.Zhou, I.Ohnishi and Y.Nishiura: "Numerical simulation of free boundary problems in quadruple precision arithmetic using explicit schemes" 数理解析研究所講究録,京都大学. Vol.989. 18-30 (1997)

    • Related Report
      1997 Annual Research Report
  • [Publications] 今井仁司, 竹内敏己, 坂口秀雄, 篠原能材, Tarmizi: "第微分方程式の無限精度数値シミュレーションについて" 第47回応用力学連合講演会講演予稿集. 435-436 (1998)

    • Related Report
      1997 Annual Research Report
  • [Publications] T.Ikeda and A.Ueda: "Parallel computation of interfacial dynamics(to appear)" Gakuto International Series. Vol.11. (1998)

    • Related Report
      1997 Annual Research Report
  • [Publications] T.Sakajo and H.Okamoto: "The application of PVM to the computation of vortex sheet motion (to appear)" Gakuto International Series. Vol.11. (1998)

    • Related Report
      1997 Annual Research Report
  • [Publications] T.Sakajo and H.Okamoto: "An application of Draghicescu's fast summation method to vortex sheet motion(to appear)" J.Phys.Soc.Japan. Vol.67. (1998)

    • Related Report
      1997 Annual Research Report
  • [Publications] Y.Nishiura and D.Ueyama: "A SKELETON STRUCTURE FOR SELF-REPLICATION DYNAMICS(to appear)" Proceeding of 2nd Tohwa University International Meeting on Statistical Physics. (1998)

    • Related Report
      1997 Annual Research Report

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Published: 1998-04-01   Modified: 2016-04-21  

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