论文标题
分数差分微积分的新理论
A new theory of fractional differential calculus
论文作者
论文摘要
本文介绍了一维弱差分微积分的独立新理论。这种新理论的症结在于引入弱的分数衍生物概念，这是整数阶弱衍生物的自然概括。它还有助于统一多个现有的分数定义，并表征哪些函数在分数上可区分。为弱分数衍生物建立了各种微积分规则，包括基本定理微积分，产品和链条规则以及零件公式的集成。另外，还建立了与经典分数衍生物的关系以及弱分数可区分函数的详细特征。此外，弱分数衍生物的概念也被系统地扩展到一般分布，而不仅仅是某些特殊分布。这一新理论奠定了一个坚实的理论基础，用于系统，严格地开发分数Sobolev空间的新理论，变化的分数计算以及分数PDE及其在后续作品中的数值解决方案。本文是参考文献第1-4节和第6节的材料的简洁介绍。
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].
