探討Bessel方程解與特殊函數之關係

Studies on relationships between the solutions of Bessel equation and special functions

凌維志(Ling-Wei Chich)

指導教授 ： 林賜德

中原大學/理學院/應用數學研究所/碩士(2004年)

https://doi.org/10.6840/cycu200400748

摘要

摘要
當我們處理穩定熱流方程式 (即拉氏方程式)時 ，會導出兩個有名的二階微分方程Legendre微分方程與Bessel微分方程，傳統上數學學者都是用Frobenius的方法得到一個級數解已超過一百年了，但一直無法直接用微分或積分得到封閉式解，最近 Legendre微分方程之解由林賜德教授與日本西本勝之教授利用分數微積分的方法找出的特解，而相關的Bessel微分方程的特解由許肇謙等人求得到。在本篇論文中，將利用分數微積分來探討 Bessel微分方程的各種特解與特殊函數之關係。

關鍵字

Gamma函數 ； 分數微積分 ； 特殊函數 ； Bessel微分方程

並列摘要

Introduction
The two equations would be derived which are known as Bessel and Legendre equation while we treat that steady state heat equation (that is , Laplace equation) . Traditionally , mathematicians use the method of Frobenius to get a series solution for more than one hundred years. Traditionally , we always could not obtain a closed form solution by differential or calculus directly, new finding of particular solutions for a generalized associated Legendre equation by profession Shy-Der Lin and K. Nishimoto , and particular solutions of Bessel equation were obtained by Jaw-Chian Shyu etc. In this paper, we will discuss the relationship between the solutions of Bessel equation and special equations.

並列關鍵字

Gamma function ； Bessel differential equations ； Special functions ； Fractional calculus

參考文獻

Srivastava, Explicit Solutions of Some General Families of Ordinary and Partial Differential Equations Associated with the Bessel Equation by Means of Fractional Calculus, Journal of Fractional Calculus, Vol. 25, May (2004), 33-45.

Descartes Press, Koriyama ,1984, 1987,1989,and 1991.

【2】K. Nishimoto , An Essence of Nishimoto’s Fractional Calculus (Calculus of the 21st Century ): Integrations and Differentiations

【3】I. Podlubny , Fractional Differential Equations, Mathematics

【4】H.M. Srivastava, S. Owa, and K. Nishimoto, Some fractional differintegral equations, J. Math Anal. Appl. 106(1985),360-366.