1991 Fiscal Year Final Research Report Summary
Co-Operative Research of Numerical Analysis of Nonlinear Problems
| Project/Area Number |
02302012
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| Research Category |
Grant-in-Aid for Co-operative Research (A)
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| Allocation Type | Single-year Grants |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Tokushima University |
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Principal Investigator |
SHINOHARA Yoshitane Tokushima University・Engineering, Professor, 工学部, 教授 (40035803)
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| Co-Investigator(Kenkyū-buntansha) |
MIMURA Masayasu Hiroshima University・Science, Professor, 理学部, 教授 (50068128)
MITSUI Taketomo Nagoya University・Engineering, Assistant Professor, 工学部, 助教授 (50027380)
KIKUCHI Fumio University of Tokyo・General Education, Professor, 教養学部, 教授 (40013734)
OKAMOTO Hisashi Kyoto University・Rims, Assistant Professor, 数理解析研究所, 助教授 (40143359)
USHIJIMA Teruo University of Elector-Communications, Professor, 電気通信学部, 教授 (10012410)
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| Project Period (FY) |
1990 – 1991
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| Keywords | quasiperiodic phenomena / arc-length method / bifurcation / optimum shape design / finite element analysis / electromagnetism / parallel computation / dynamics of phase and surface |
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| Research Abstract |
1. Head Investigator's results : (1) Existence and uniqueness theorem with an error estimation of Galerkin approximation has been established. This theorem is very useful for numerical analysis of quasiperiodic phenomena which appear in nonlinear oscillation, modulation and detection of communication. The theorem is applied to compute the quasiperiodic solutions to Van der Pol type and Duffing type equations. (2) The concept of numerically ill conditioning of solutions of the initial value problem of ordinary differential equations is given. The concept is novel and important in numerical analysis of ordinary differential equations. (3) The arc-length method (or geometric method) becomes very powerful for the bifurcation analysis in nonlinear problems.
2. Investigators' main results : (1) T. Ushijima investigates the eigenvalue problems of water waves and gives some important paterns of global behaviour of relative erros in eigenvalues. (2) H. Okamoto examines the stabilities of steady
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axisymmetric flows and determines how these stabilities change upon varying the geometric parameters in 2-dimensional Navier-Stokes equations. (3) , H. Kawarada studies on Stefan problems with free boundary and computes the solidification problems with change of volume. The short time existence of the unsteady free boundary appearing in the porous media is studied by use of Nash-Moser's implicit function theorem. Nash-Moser's implicit theorem has been modified in an applicable form. (4) F. Kikuchi studies the mixed formulations for finite element analysis in electromagnetism and solid-state mechanics. He developed some weak formulations for finite element analysis of magnetostatic and electrostatic problems by means of the Hilbert space method. He received the ISHIKAWA award. (5) T. Mitsui made a new A-stable 3-stage fourth order Runge-Kutta formula which can be calculated in parallel. Specifically, forcus is given to general solution for formulae parameters of IRK under the symplectic and the order conditions. Examples of such formulae are constructed and linear orders are given for up to three stages. (6) M. Mimura studies the dynamics of phase and surface of chemical waves and solidification by means of the moving pictures. A size-space distribution model of biological individuals including two effects of density-dependent growth rates for size and chemotactic aggregation for space have been proposed. Assuming that the spatial movement is rapid in comparison with the growth process, he uses time-scaling arguments to reduce the model to an approximating system of only size distribution. He attended and presented a lecture on ICIAM (1991, USA) about his work on pattern formation arising from systems of reaction diffusion equations in space two and higher. Less
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