Fractional Calculus of Distributions and Integral Equations

分数阶微积分是数学分析中发展最为迅速的领域之一，其应用范围日益广泛。分数阶微分和积分算子已应用于各种领域，如水坝水力学、势场、液体和气体中的波。分数阶微积分几乎在科学和工程的各个领域都有应用。它是表征长记忆过程和材料、异常扩散、远程相互作用、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）是泛化数学分析中经典函数概念的对象。分布使得微分在经典意义上不存在导数的函数成为可能，例如阶跃函数。特别地，任何局部可积函数都有一个分配导数。分布在偏微分方程理论中被广泛应用，在偏微分方程中，分布（弱）解的存在性比经典解的存在性更容易建立。分式微分方程和分式积分方程的分布研究，包括阿贝尔的第一类和第二类积分方程，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分布的分数阶微积分，这在分布理论中是新颖的。这将需要开发新的计算Schwartz空间或子空间上广义函数积分的方法。随后，我计划在卷积的逆算子上研究分数阶微分和积分方程，特别是Abel的积分方程在分布意义上的研究，并研究拓扑结构下的Mittag-Leffler函数、近似解、它们的级数收敛性以及Big-O符号下的逼近顺序。此外，我打算通过delta序列和复分析方法，在求解分数阶微分方程和积分方程时，给遇到的奇异分布和乘积赋予意义，因为它们在物理学、量子场论和涉及分布的微分方程中也被广泛需要。这项工作的意义如下：许多物理、工程和化学过程中的应用问题导致了积分方程，乍一看，它与阿贝尔的积分方程没有任何共同之处，由于这种认识，人们为解决这些方程的解析或数值过程的发展做出了额外的努力。然而，通过应用我的卷积方法，将它们转换成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，得到分布解。

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.