| dc.description.abstract | 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. | en_US |