The development of the wavelet theory for partial differential equations and its numerical analysis application

2015 Fiscal Year Final Research Report

• Project/Area Number: 24540161
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: University of Tsukuba
• Principal Investigator: Kinoshita Tamotu 筑波大学, 数理物質系, 准教授 (90301077)
• Co-Investigator(Kenkyū-buntansha): KAJITANI Kunihiko 筑波大学, 名誉教授 (00026262) ; ISHIWATA Satoshi 山形大学, 理学部, 准教授 (70375393) ; KUBO Takayuki 筑波大学, 数理物質系, 講師 (90424811)
• Co-Investigator(Renkei-kenkyūsha): ASHINO Ryuichi 大阪教育大学, 教育学部, 教授 (80249490)
• Project Period (FY): 2012-04-01 – 2016-03-31
• Keywords: 関数方程式論 / ウェーブレット / 数値解析

Outline of Final Research Achievements

As for the wavelet theory, we could find some practical wavelets and construct an interesting counter example concerned with the unconditional convergence of wavelet expansions. As for the partial differential equations theory, we succeeded to get the very useful representation formula of the solutions to the Cauchy problem for the second order wave type equations. These research results are sufficiently satisfactory as an each result in the wavelet theory and the partial differential equations theory. Combining the partial differential equations theory with the wavelet theory, we also obtained a result concerned with the relations between the wellposedness and the oscillations of the coefficients by visualizing with the wavelet transform. This seems to be still scope for improvement and will become the research subject in the future.

Free Research Field

解析学