2005 Fiscal Year Final Research Report Summary
Solution structure around bifurcation points of co-dimension 2
| Project/Area Number |
15340038
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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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 | Ryukoku University |
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Principal Investigator |
IKEDA Tsutomu Ryukoku University, Faculty of Science and Technology, Professor, 理工学部, 教授 (50151296)
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| Co-Investigator(Kenkyū-buntansha) |
NISHIDA Takaaki Waseda University, Faculty of Science and Engineering, Professor, 理工学術院, 教授 (70026110)
IKEDA Hideo Toyama University, Faculty of Science, Professor, 理学部, 教授 (60115128)
MORITA Yoshihisa Ryukoku University, Faculty of Science and Technology, Professor, 理工学部, 教授 (10192783)
NINOMIYA Hirokazu Ryukoku University, Faculty of Science and Technology, Associated Professor, 理工学部, 助教授 (90251610)
NAGAYAMA Masaharu Kanazawa University, Graduate School of Natural Science and Technology, Associated Professor, 大学院・自然科学研究科, 助教授 (20314289)
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| Project Period (FY) |
2003 – 2005
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| Keywords | mono-stable system / bifurcation of pulses / phase condition / flow on slope / Hopf bifurcation / combustion equations / combustion pulses / collision of pulses |
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| Research Abstract |
Masaharu Nagayama (one of investigators of the present research project) has devised a computer code that can analyze bifurcation structures in a neighborhood of double bifurcation points. This code deals with bifurcation phenomena of pulse solutions to mono-stable reaction-diffusion systems, and is equipped with the following two functions : (1)It can find a critical point and construct its bifurcation branch, (2)It can extend existing bifurcation branches. In order to devise the code, we consider the reaction-diffusion system on a finite interval (0,L) subject to the periodic boundary condition where L is a fixed large positive number. From the phase condition we obtain the equation that determines the propagating velocity of traveling pulse, and by the Keller method we express the dependency on a parameter p included in the equation systems. The problem formularized as in the above is numerically solved by the Newton method in the computer code. We note that a solution is a set of {solutions to reaction-diffusion systems, c, p}. When a traveling pulse bifurcates from a standing pulse, there appear two zero-eigenvalues, one of which is a trivial one trivial one corresponding to parallel translation. Our code applies to not only this case but also the cases where two crucial zero-eigenvalues exist except the trivial one.
The head investigator have dealt with standing and traveling combustion pulses of a mathematical model for self-propagating high-temperature syntheses including both the cooling effect and raw material supply system. Employing a piece-wise constant function for the reaction term, we have studied the existence of pulse solutions in a mathematically rigorous way, and also the collision dynamics of combustion pulses on a circle domain.
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Research Products
(44 results)
