一种新的分数阶微分学理论
A new theory of fractional differential calculus
摘要：
本文提出了一种完备的一维弱分数微分学新理论。这一新理论的关键是引入了弱分数阶导数的概念，这是整数阶弱导数的自然推广;它还有助于统一现有的多个分数阶导数定义，并刻画什么函数是分数阶可微的。对弱分数阶导数建立了各种微积分规则，包括微积分基本定理、积和链式法则、分部积分分部积分法公式。此外，还建立了弱分数阶可微函数与经典分数阶导数的关系以及弱分数阶可微函数的详细刻画。此外，还将弱分数阶导数的概念系统地推广到一般分布，而不仅仅是某些特殊分布。这一新的理论为以后的工作中系统而严格地发展分数索伯列夫空间、分数变分法和分数偏微分方程的新理论及其数值解奠定了坚实的理论基础。本文是参考文献[9]第1-4节和第6节材料的简要介绍。
英文摘要：
This paper presents a self-contained new theory of weak fractional differential calculus in one-dimension. The crux of this new theory is the introduction of a weak fractional derivative notion which is a natural generalization of integer order weak derivatives; it also helps to unify multiple existing fractional derivative definitions and characterize what functions are fractionally differentiable. Various calculus rules including a fundamental theorem calculus, product and chain rules, and integration by parts formulas are established for weak fractional derivatives. Additionally, relationships with classical fractional derivatives and detailed characterizations of weakly fractional differentiable functions are also established. Furthermore, the notion of weak fractional derivatives is also systematically extended to general distributions instead of only to some special distributions. This new theory lays down a solid theoretical foundation for systematically and rigorously developing new theories of fractional Sobolev spaces, fractional calculus of variations, and fractional PDEs as well as their numerical solutions in subsequent works. This paper is a concise presentation of the materials of Sections 1-4 and 6 of reference [9].
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
