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非整数階時間微分を含む非線形発展方程式

##article.authors##

  • 赤木, 剛朗 東北大学大学院理学研究科

DOI:

https://doi.org/10.51094/jxiv.7008

キーワード:

発展方程式、 非整数階微分、 非整数階非線形拡散方程式

抄録

非整数階微分の概念自体はライプニッツがロピタルに宛てた手紙 (1695) に登場し, その後, リーマン, リュービル, リースをはじめ多くの数学者によって研究されている古典的なテーマである. また, 非整数階解析学は, 物理学や機械工学などの応用分野との相互作用を通じて発展してきた. 例えば, 拡散の古典論から逸脱するいわゆる異常拡散 (現象) を記述するために, 分数冪ラプラシアンと並んで時間に関する非整数階微分が用いられている. 数学的な見地に立つと, 非整数階微分を含む線形微分方程式に対してはフーリエ解析やラプラス変換などの手法が引き続き適用できるが, 非整数階微分を含む非線形微分方程式に対しては有効な手法が確立していない. 特に非整数階微分に対する関数解析的手法はあまり普及していない状況で, 例えば変分法の適用にはさまざまな困難が生じる. ここでは非線形放物型方程式の研究にしばしば用いられる勾配流 (ある汎関数の値が最も急速に減少する方向に沿って状態が時間発展する系) の理論に着目し, その非整数階微分版, 言うなれば「非整数階勾配流の理論」の構築とその偏微分方程式への応用について, (執筆した時点で) 最新の話題にも触れながら解説する.

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公開日時: 2026-11-06 04:10:02 UTC
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