Fractional Calculus of Distributions and Integral Equations

分布和积分方程的分数阶微积分

• 批准号: RGPIN-2019-03907
• 负责人: Li, Chenkuan
• 金额: $ 1.09万
• 依托单位: Brandon University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758726
• 关键词: Fractional; Calculus; Distributions; Integral; Equations

Fractional calculus is one of the most intensively developing areas of mathematical analysis as a result of its increasing range of applications. Operators for fractional differentiation and integration have been used in various fields, such as hydraulics of dams, potential fields, and waves in liquids and gases. Fractional calculus is found in almost every realm of science and engineering. It is one of the best tools to characterize long-memory processes and materials, anomalous diffusion, long-range interactions, long-term behaviors, power laws and scaling laws, for example. On the other hand, distributions (or generalized functions) are objects that generalize the classical notion of functions in mathematical analysis.  Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense, such as the step function. In particular, any locally integrable function has a distributional derivative. Distributions are widely used in the theory of partial differential equations, where it may be easier to establish the existence of distributional (weak) solutions than classical ones. Studying fractional differential and integral equations distributionally, including Abel's integral equations of the first and second kind, has been a serious challenge. The long term goal of my research proposal is to define fractional calculus of distributions in general by convolution, normalization, and approximation, which is novel in distribution theory. This will require the development of new methods of computing integrals of generalized functions over the Schwartz space or a subspace. Subsequently, I plan to investigate fractional differential and integral equations, particularly Abel's integral equations, in the distributional sense by inverse operators in terms of convolution, and study Mittag-Leffler functions, approximate solutions, their convergence of series, and order of approximation in Big-O notation under topological structures. Additionally, I intend to give meaning to singular distributions and products encountered, while solving fractional differential and integral equations, by delta sequences and complex analysis approaches, as they are also widely needed in physics, quantum field theory and differential equations involving distributions. The significance of the work is the following: Many applied problems from physical, engineering and chemical processes lead to integral equations, which at first glance have nothing in common with Abel's integral equations, and due to this perception additional efforts are undertaken for the development of analytical or numerical procedure for solving these equations. However, through the application of my convolution approach, their transformations to the form of Abel's integral equations will speed up the solution process, or, more significantly, lead to distributional solutions in cases where classical ones do not exist.

分数阶微积分是数学分析领域中发展最活跃的领域之一，其应用范围日益扩大。分数阶微分和积分算子已被应用于各种领域，如水坝水力学、势场、液体和气体中的波动。分数阶微积分几乎存在于科学和工程的每一个领域。它是表征长记忆过程和材料、异常扩散、长程相互作用、长期行为、幂律和标度律等的最佳工具之一。另一方面，分布（或广义函数）是数学分析中函数的经典概念的推广对象。分布使得在经典意义下导数不存在的函数（如阶跃函数）的微分成为可能。特别地，任何局部可积函数都有分布导数。分布广泛应用于偏微分方程理论中，在那里建立分布（弱）解的存在性可能比经典解更容易。对分数阶微分和积分方程，包括第一类和第二类Abel积分方程的分布研究一直是一个严峻的挑战。我的研究计划的长期目标是通过卷积，归一化和近似来定义一般的分数阶分布演算，这在分布理论中是新颖的。这将需要发展新的方法来计算广义函数在施瓦茨空间或子空间上的积分。随后，我计划研究分数阶微分和积分方程，特别是阿贝尔积分方程，在分布意义上的逆算子的卷积，并研究Mittag-Leffler函数，近似解，级数的收敛性，以及在拓扑结构下的Big-O符号的近似阶。此外，我打算给奇异分布和产品的意义，而解决分数微分和积分方程，通过δ序列和复杂的分析方法，因为它们也广泛需要在物理学，量子场论和微分方程涉及分布。这项工作的意义如下：许多应用问题的物理，工程和化学过程导致积分方程，这乍一看没有什么共同点与阿贝尔的积分方程，并由于这种看法额外的努力进行发展的分析或数值程序来解决这些方程。然而，通过应用我的卷积方法，它们转换为阿贝尔积分方程的形式将加快求解过程，或者更重要的是，在经典解不存在的情况下，导致分布解。