臺灣博碩士論文加值系統

摘 要
在最近幾年，許多學者利用西本勝之教授在1989年所創的分數微積分的理論，導出一些常見的二階常微分方程式的特殊解。至今，此理論為探討特殊常微分方程式特解的最有效工具。如 Gauss、Legendre、Jacobi、Chebyshev、Coulomb、Whittaker、Euler、Hermite
，和 Weber 等方程式的解。另外，可以利用分數微積分的理論，推廣上述二階常微分方程式的推廣式並求其特殊解。
這篇文章的主要目的，是在探討如何適當的應用一個更一般性的廣義常微分積分方程式（參考文獻 [6]），而得到過去所得到的某些常微分方程式的推廣式及解。本篇文章是針對在物理學及量子力學上常出現的 Weber 方程式，說明 Weber 方程式的推廣式及其解（參考文獻 [5]），即為此廣義常微分積分方程式的特例之一。
本篇論文內容共分成三章。第一章先簡述西本勝之教授所創的「分數微積分」的基本定義，及根據此定義所延伸出來的一些常用的Lemma和Property。再介紹1999年所得到的廣義 Weber方程式及其特殊解（定理A和定理B）。第二章介紹一個更一般性的線性分數微分積分方程式及其特殊解（定理1），此方程式是整合許多二階常微分方程式的推廣式而得到的。另外，為了方便應用定理1，我們找了定理2為定理1的特例。第三章則是說明過去所獲得的 Weber 方程式的推廣式及其解，就是此廣義常微分積分方程式的特例之一。

Abstract
In recent years, several workers demonstrated the usefulness of
fractional calculus in the derivation of particular solutions of a number
of familiar second-order differential equations associated (for example) with Gauss, Legendre , Jacobi, Chebyshev, Coulomb,Whittaker, Euler, Hermite, and Weber equations. The main object of this paper is to show how some of the most recent contributions on this subject, involving the Weber equations and their various generalized forms, can be obtained
by suitably applying a general theorem on particular solutions of a certain family of fractional differintegral equations.
Key words and Phrases : Fractional calculus, Differintegral equations, Weberequations, Generalized Leibniz Rule, Analytic functions, Integral
curves.

Content
Chapter 1 Introdduction, Defiinitions, and Preliminaries ---------------------------------- 1
Chapter 2 A General Theorem and Its Applications --------------------------------------------------- 6
Chapter 3 Further Deductions from Theorem 2
--------------------------------------------------10
References ------------------------------------------------ 10
Appendix -------------------------------------------------- 11

References
[1] K. Nishimito, Fractional Calculus, Vols. I , II, III , and IV, Descartes Press,
Koriyama, 1984, 1987, 1989, and 1991.
[2] K. Nishimito, An Essence of Nishimoto's Fractional Calculus ( Calculus of the
21st Century ) : Integrations and Differentiations of Arbitrary Order, Descartes
Press, Koriyama, 1991.
[3] I. Podlubny, Fractional Differential Equations, Mathematics in Science and
Engineering, Vol. 198, Academic Press, New York, London, Tokyo, and
Toronto, 1999.
[4] H.M. Srivastava, S. Owa, and K. Nishimoto, Some fractional differintegral
equations, J. Math. Anal. Appl. 106 (1985), 360-366.
[5] S.-T. Tu, Yu-Tan Huang, and I-C Chen, Generalized Weber Equations via
fractional calculus, Chung Yuan J. Vol.28, No 1, (2000), 1-5.
[6] S.-T. Tu, D.-K. Chyan, and H.M. Srivastava, Some families of ordinary and
partial fractional differintegral equations, Integral Transform. Spec. Funct. 9
(2000).