Pursuit of explicit representation formula using elliptic and analysis of the global bifurcation structure
| Project/Area Number |
15K04972
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Mathematical analysis
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| Research Institution | Ryukoku University |
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Principal Investigator |
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| Co-Investigator(Kenkyū-buntansha) |
森田 善久 龍谷大学, 理工学部, 教授 (10192783)
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| Project Period (FY) |
2015-04-01 – 2019-03-31
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| Project Status |
Completed (Fiscal Year 2018)
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| Budget Amount *help |
¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000)
Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2016: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2015: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
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| Keywords | 非線形境界値問題 / 完全楕円積分 / 楕円関数 / 交差拡散方程式 / 反応拡散方程式 / 極限方程式 / 非局所 / 線形化固有値問題 / 弾性曲線 |
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| Outline of Final Research Achievements |
We have deepened our original method and extended its scope in order to solve various problems for differential equations. We obtain all the candidate solutions and derive transcendental equations which are equivalent to problems to be solved. For instance, as for a cross-diffusion equation,we obtained the stability of stationary solutions in multidimensional case, which is inspired by 1 dimensional case results obtained by our method. As for linearized eigen-value problems for reaction diffusion equations, we have obtained all exact values of eigen-values and representation formulas of eigen-functions for Allen-Cahn type reaction diffusion equations. Moreover, we have asymptotic formulas of eigen-values as diffusion coefficients converge to zero.
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| Academic Significance and Societal Importance of the Research Achievements |
従来,微分方程式の解の存在のための条件を求めたり,局所的な解の分岐構造に対して,現代的関数解析的な手法により多くの数学的研究がなされ現在も発展を続けている.さらに詳しく精密に解の形状を知ること,解の大域的な分岐構造の解明はより困難な問題である.しかし,生物学の発生や生命現象等にあらわれる数理モデルに対して,数学的な結果を利用できるようにするためには,是非とも克服すべき問題である.
我々は,基本的で典型的ないくつかの微分方程式で記述される数理モデルに対し,特異摂動問題の解の精密な陽的表示,さまざまな極限形状の精密表示,２次分岐等も含めた解の大域的分岐構造等の数学的解明を行なった.
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Report
(5 results)
Research Products
(68 results)
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[Presentation] Existence, nonexistence, multiplicity, and numerical stability of solutions for the SKT cross-diffusion stationary limiting equation2018
Author(s)
T. Mori, Y. Lou, H. Matsubara, W.-M. Ni, S. Sukekuni and S. Yotsutani
Organizer
The 12th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Taipei, Taiwan
Related Report
Int'l Joint Research / Invited
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[Presentation] Cross diffusion models in population biology2016
Author(s)
Y. Lou, W.-M. Ni, M. Winkler and S. Yotsutani
Organizer
The 11th AIMS Conference on Dynamical Systems, Differential Equations and Applications
Place of Presentation
Hyatt Regency Orlando,Florida, USA
Year and Date
2016-07-01
Related Report
Invited
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