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2023 Fiscal Year Final Research Report

Numerical analysis of analytic functions based on hyperfunction theory

Research Project

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Project/Area Number 21K03366
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Review Section Basic Section 12040:Applied mathematics and statistics-related
Research InstitutionThe University of Electro-Communications

Principal Investigator

Ogata Hidenori  電気通信大学, 大学院情報理工学研究科, 教授 (50242037)

Project Period (FY) 2021-04-01 – 2024-03-31
Keywords数値解析 / 数値計算 / 超函数 / 佐藤超函数 / 解析関数 / 複素関数論 / 複素解析 / 変数変換
Outline of Final Research Achievements

Hyperfunction theory is a theory of generalized functions based on complex function theory, where generalized functions called hyperfunctions are expressed as the differences of the boundary values on the real axis of complex analytic functions called defining functions. Computations of hyperfunctions are carried out using the defining functions. Remarking it, we proposed a method of function approximation, numerical differentiation, numerical indefinite integration and a numerical solver of initial value problems of ordinary differential equations based on hyperfunction theory.
Our study is also a study of numerical analysis of analytic functions. In this point of view, we studied numerical computations using variable transformations. To be specific, we proposed a numerical indefinite integration, a numerical solution method of ordinary differential equations and so on using the IMT-type transform, which is used the IMT-type numerical integration formula.

Free Research Field

数値解析

Academic Significance and Societal Importance of the Research Achievements

科学技術計算において解析関数はよく現れ,その数値計算法は精力的に研究されている。佐藤超函数論は複素関数論に基づく一般化関数の理論であり,それを用いればデルタ関数など特異性のある関数が複素解析関数を用いて記述される。したがって,佐藤超函数論を用いれば,数値的に扱いの困難な特性を持つ関数の計算が,よく研究されている解析関数の数値計算法を応用して行うことができる。その意味で,本研究は数値計算の可能性を大きく広げたと言える。
数値計算の変数変換技法は複素関数論と関連し,その意味で本研究と関連する。IMT型変換はDE変換と比べてあまり研究されておらず,その意味で同研究分野の可能性を大きく広げたと言える。

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Published: 2025-01-30  

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