Fractional Calculus of Distributions and Integral Equations

分数阶微积分是数学分析中发展最迅速的领域之一，因为它的应用范围越来越广。用于分数次微分和积分的算子已被用于各种领域，如大坝的水力学、势场以及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。它是描述长记忆过程和材料的最佳工具之一，例如异常扩散、长程相互作用、长期行为、幂定律和标度定律。另一方面，分布(或广义函数)是数学分析中推广经典函数概念的对象。分布使区分其导数不存在于经典意义上的函数成为可能，例如阶跃函数。特别地，任何局部可积函数都有分布导数。分布在偏微分方程组理论中得到了广泛的应用，证明分布(弱)解的存在性可能比经典解更容易建立。分布地研究分数阶微分方程组和积分方程组，包括第一类和第二类Abel积分方程组，一直是一个严重的挑战。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分布的分数微积分，这在分布理论中是新的。这将需要开发新的方法来计算Schwartz空间或子空间上广义函数的积分。随后，我计划利用卷积的逆算子来研究分数阶微分方程组，特别是Abel积分方程解的分布意义，并研究Mittag-Leffler函数、近似解、级数的收敛以及在拓扑结构下的Big-O表示的逼近阶。此外，我打算通过增量序列和复数分析方法来赋予所遇到的奇异分布和乘积以增量序列和复分析方法的意义，因为它们在物理学、量子场论和涉及分布的微分方程中也是广泛需要的。这项工作的意义如下：许多来自物理、工程和化学过程的应用问题导致了积分方程组，乍一看，它与阿贝尔积分方程组没有任何共同之处，由于这种认识，人们采取了更多的努力来发展求解这些方程的解析或数值方法。然而，通过应用我的卷积方法，它们转换为Abel积分方程的形式将加快求解过程，或者更重要的是，在不存在经典解的情况下导致分布解。

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.