Mathematical analysis and reaction diffusion approximation for pattern formations with nonlocal interactions
| Project/Area Number |
17K14228
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| Research Category |
Grant-in-Aid for Young Scientists (B)
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| Allocation Type | Multi-year Fund |
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| Research Field |
Foundations of mathematics/Applied mathematics
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| Research Institution | Future University-Hakodate (2018-2019)
Hokkaido University (2017) |
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Principal Investigator |
Yoshitaro Tanaka 公立はこだて未来大学, システム情報科学部, 准教授 (80783977)
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| Project Period (FY) |
2017-04-01 – 2020-03-31
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| Project Status |
Completed (Fiscal Year 2019)
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| Budget Amount *help |
¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000)
Fiscal Year 2019: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2018: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2017: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
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| Keywords | 反応拡散系 / 非局所相互作用 / 非局所発展方程式 / 反応拡散近似 / 応用数学 / 解析学 / パターン形成 |
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| Outline of Final Research Achievements |
Recently the existence of the spatially nonlocal interactions which can change the influence on the objects depending on the distance has been reported in fish cells. As this interaction influences spatially globally, it can be modeled by the convolutions with suitable kernel, and various nonlocal evolution equations were proposed. Although the nonlocal evolution equations can reproduce various patterns and many applications are expected, the technique of analyzing a nonlocal evolution equations were not rich. Motivated by these backgrounds, we proposed a method to approximate the nonlocal evolution equations into reaction diffusion system in which the various theories have already been established. We revealed that the nonlocal evolutions equations with any even kernels can be approximated by a reaction diffusion system in one-dimensional spaces.
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| Academic Significance and Societal Importance of the Research Achievements |
動物や魚の表皮等に観察されるパターン形成や，昆虫の脳における神経形成，また細胞接着現象など，積分付きの相互作用をもつ発展方程式はさまざまな現象を記述することができる．広い分野の現象に応用が期待できるため，解析手法を整備することが求められるが，現状発展途上である．そこで我々は，すでに多くの理論が整備されている反応拡散系という方程式に，積分つきの相互作用をもつ発展方程式を近似する方法を提案し，理論的に近似できることを示した．このことから，積分つきの相互作用をもつ発展方程式を解析することが期待でき，さらなる応用例や理論的な知見を生み出すことができると考えている．
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Report
(4 results)
Research Products
(15 results)
