论文标题
分数差分积分和分数Sobolev空间的新理论：一维情况
A new theory of fractional differential calculus and fractional Sobolev spaces: One-dimensional case
论文作者
论文摘要
本文介绍了一维弱分数微积分和分数Sobolev空间的独立新理论。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。还获得了各种微积分规则，包括微积分，产品和链条规则的基本定理，以及针对弱分数衍生物的零件公式的集成以及与经典衍生物的关系。基于弱分数导数概念，引入了新的分数SOBOLEV空间，并建立了许多重要的定理和属性，例如密度/近似定理，扩展定理，跟踪定理以及这些SOBOLEV空间中的各种嵌入定理。此外，还建立了与现有的分数Sobolev空间的一些关系。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。新理论为系统和严格地开发了变化理论的分数和分数PDE理论及其在后续作品中的数值解决方案奠定了一个坚实的理论基础。
This paper presents a self-contained new theory of weak fractional differential calculus and fractional Sobolev spaces 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 of calculus, product and chain rules, and integration by parts formulas are established for weak fractional derivatives and relationships with classical derivatives are also obtained. Based on the weak fractional derivative notion, new fractional order Sobolev spaces are introduced and many important theorems and properties, such as density/approximation theorem, extension theorems, trace theorem, and various embedding theorems in these Sobolev spaces are established. Moreover, a few relationships with existing fractional Sobolev spaces are also established. Furthermore, the notion of weak fractional derivatives is also systematically extended to general distributions instead of only to some special distributions. The new theory lays down a solid theoretical foundation for systematically and rigorously developing a fractional calculus of variations theory and a fractional PDE theory as well as their numerical solutions in subsequent works.
