حل معادله لژاندر و تحلیل کاربردهای آن در فزیک

Abstract

In this research, we examine Legendre differential equations, which are among variable-coefficient equations in applied mathematics. These equations are particularly significant due to their extensive applications in physics and mathematics. In general, the solutions to these equations cannot be expressed in terms of elementary functions, but solutions in the form of convergent series can be derived. This study demonstrates that many important differential equations in applied mathematics include functions described by special functions instead of elementary ones. The research method employed is library-based, and the analyses indicate that Legendre equations play a key role in providing solutions to various differential equations. Additionally, Legendre polynomials and their unique properties have been explored. Due to their remarkable characteristics, these polynomials are significant in numerous physical applications, such as the analysis of vibrations and wave phenomena. The findings of this research underscore the vital position of special functions and Legendre equations in advancing theoretical physics and advanced mathematics, and a deep understanding of them can lead to the development of new practical concepts in these fields.