| 摘要: | 學生： 賈敏翔 指導教授： 方向
國立中央大學數學系
摘要
分數階微積分是數學中的一個領域，它將微分和積分的傳統概念擴展到非整數甚至複數階。經典微積分處理整數階導數和積分，而分數階微積分則探索實階甚至复階函數積分和微分的可能性。
分數階微積分的根源可以追溯到17 世紀，幾乎早在經典微積分誕生的時候，萊布尼茨和歐拉等數學家就開始思考分數階導數的含義和性質。然而，直到19 世紀末和20世紀初，人們在理解和形式化這門數學學科方面才取得了重大進展。
分數階微積分在許多科學領域都有應用，包括物理、工程、信號處理和金融。它可以對錶現出非本地和內存依賴行為的複雜系統進行建模和分析。例如，分數階微積分已被用來描述反常擴散、粘彈性和分形幾何等現象。
此外，分數階微積分為求解分數階微分方程提供了強大的工具。通過考慮分數階導數和積分，研究人員可以解決傳統微積分難以解決的方程。這為理解複雜的動力系統和開發更準確的數學模型開闢了新途徑。
總之，分數階微積分提供了一個用於分析和操作具有非整數階微分和積分的函數的框架。它擴展了傳統微積分的範圍，使人們能夠更深入地理解複雜系統，並為各種應用提供強大的數學工具。
本文主要利用了Schur’s test 來討論分數階積分在L^p空間上的有界性。同時我們定義了新的算子並研究有界性。;A Class of Bilinear Forms via Fractional Integration Operators
Student: MIN-XIANG JIA Advisor: XIANG FANG
Submitted to Department of Mathematics
National Central University
ABSTRACT
Fractional calculus is an area in mathematics which extends the traditional concepts of differentiation and integration to non-integer, or even complex, orders. While classical calculus deals with integer-order derivatives and integrals, fractional calculus explores the possibilities of integrating and differentiating functions with real or even complex orders.
The roots of fractional calculus can be traced back to the 17th century, almost as early as the birth of the classical calculus, when mathematicians like Leibniz and Euler pondered over the meaning and properties of fractional derivatives. It wasn’t until the late 19th and early 20th centuries, however, that significant progress was made in understanding and formalizing this mathematical discipline.
Fractional calculus finds applications in many scientific fields, including physics, engineering, signal processing, and finance. It enables the modeling and analysis of complex
systems that exhibit non-local and memory-dependent behavior. For instance, fractional calculus has been used to describe phenomena such as anomalous diffusion, viscoelasticity, and fractal geometry.
Moreover, fractional calculus offers powerful tools for solving differential equations with fractional orders. By considering fractional derivatives and integrals, researchers can tackle equations that traditional calculus struggles to address. This opens up new avenues for understanding intricate dynamical systems and developing more accurate mathematical models.
In summary, fractional calculus provides a framework for analyzing and manipulating functions with non-integer orders of differentiation and integration. It expands the scope of traditional calculus, enabling a deeper understanding of complex systems and offering powerful mathematical tools for various applications.
This paper mainly uses Schur’s test to discuss the boundedness of fractional integrals on L^p space. At the same time we define new operators and study boundedness. |
