1999 Fiscal Year Final Research Report Summary
Numerical Amalysis and Glolal Behavioz of Chemotactic Equations
| Project/Area Number |
10640205
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Global analysis
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| Research Institution | Osaka University |
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Principal Investigator |
YAGI Atsushi Grcaduate School of Engineering, Osaka University. Professor, 大学院・工学研究科, 教授 (70116119)
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| Co-Investigator(Kenkyū-buntansha) |
NAKAGUCHI Etsushi Grcaduate School of Engineering, Osaka University. Assistant, 大学院・工学研究科, 助手 (70304011)
YAMAMOTO Yoshitaka Grcaduate School of Engineering, Osaka University. Lectunen, 大学院・工学研究科, 講師 (30259915)
OHNAKA Kohzabuww Grcaduate School of Engineering, Osaka University. Professor, 大学院・工学研究科, 助教授 (60127199)
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| Project Period (FY) |
1998 – 1999
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| Keywords | Chemotactic Equations / Non Lineur Diffusion / Numerical Calculation / Finite Element Methocl / Runge-Kutta Methocl / Stalility / Convergence |
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| Research Abstract |
In 1998 we devised a discretization scheme for the chemotactic equations which is based on the finite element methods and the Runge-Kutta methods, and proved theoretically stability of the scheme and convergence of the approximate solution. In the proof, notions of the discrete semigroup and discrete evolution operator were newly introduced to describe the approximate solutions precisely.
In 1999 we set up algorithm for calculations by the scheme devised. If one uses usual algorithm for some finite element method, enormous memories of machine are needed. So in this research we made some device that we exchange components of the matrix in a suitable way in order to condense a size of the band of the matrix. By this the spatial variable can be divided into 8192 in the one dimensional case, and into 256 in the two dimensional case. Using this algorithm we performed numerical calculations for the chemotactic equations. As the results, the following two things were clarified mainly on the global behavior of solutions. With the forms of the sensitive function included in the equations the pattern of cellular mold obtained from the solution changes substantially. For the chemotactic equations having the growing term the desired types of patterns of mold, that is concentric circles and ramifications, are really observed in the solutions of the equations.
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