On a new class of fractional calculus of variations and related fractional differential equations

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2021-07-08 DOI:10.57262/die035-0506-299

Xiaobing H. Feng, Mitchell Sutton

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引用次数: 1

Abstract

This paper is concerned with analyzing a class of fractional calculus of variations problems and their associated Euler-Lagrange (fractional differential) equations. Unlike the existing fractional calculus of variations which is based on the classical notion of fractional derivatives, the fractional calculus of variations considered in this paper is based on a newly developed notion of weak fractional derivatives and their associated fractional order Sobolev spaces. Since fractional derivatives are direction-dependent, using one-sided fractional derivatives and their combinations leads to new types of calculus of variations and fractional differential equations as well as nonstandard Neumann boundary operators. The primary objective of this paper is to establish the well-posedness and regularities for a class of fractional calculus of variations problems and their Euler-Lagrange (fractional differential) equations. This is achieved first for one-sided Dirichlet energy functionals which lead to one-sided fractional Laplace equations, then for more general energy functionals which give rise to more general fractional differential equations.

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关于一类新的变分微积分及相关的分数阶微分方程

本文分析了一类分数阶变分微积分问题及其相关的欧拉-拉格朗日(分数阶微分)方程。与现有的基于经典分数阶导数概念的分数阶变分学不同，本文考虑的分数阶变分学是基于一个新发展的弱分数阶导数及其相关分数阶Sobolev空间的概念。由于分数阶导数是方向相关的，使用单侧分数阶导数及其组合导致了新的类型的变分演算和分数阶微分方程以及非标准诺伊曼边界算子。本文的主要目的是建立一类分数阶变分问题及其欧拉-拉格朗日(分数阶微分)方程的适定性和规律性。首先是单侧狄利克雷能量泛函得到单侧分数阶拉普拉斯方程，然后是更一般的能量泛函得到更一般的分数阶微分方程。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.