用分數微積分方法研究某特定類微分與偏微分方程之解

Study on the Solutions of Some Certain Families of Ordinary and Partial Differential Equations by Means of Fractional Calculus

洪鎮暉(Jen-Huei Hung)

指導教授 ： 林賜德

中原大學/理學院/應用數學系/碩士(2008年)

https://doi.org/10.6840/CYCU.2008.00375

摘要

當我們在處理二階線性變係數(或常係數)常微分方程時, $P(z)phi^{'}(z)+Q(z)phi^{'}(z)+R(z)phi(z)=f(z)$, 常用的方法以Frobenius的方法求級數解較多,
即是如此, 但卻無法將ㄧ般的級數解化成以微分或積分的封閉型態解. 近幾年來, 由日本的西本勝之(Katsuyaki Nishimoto)教授, 杜詩統(Shih-Tong Tu)教授, 林賜德(Shy-Der Lin)教授 $cdots$ 等, 利用分數微積分之方法去找出許多微分方程式的特解, 如
Legendre方程, Bessel方程, Gauss方程, Jacobi方程, $cdots$ 等. 由於超幾何函數在數學上的利用很廣泛, 所以在本篇論文中,
除了將Gauss方程, Jacobi方程之解形式寫成封閉形式外, 還將之化成超幾何函數形式, 並且探討彼此之間有什麼樣的關係存在.
並且介紹分數微積分的基本定義, 藉由基本定義衍生出來的引理, 對Gauss, Jacobi, 以及
$z(1-z)frac{partial^{2}phi}{partial z^{2}}+[(
ho-2lambda)z+lambda+sigma]frac{partialphi}{partial z}+lambda(
ho-lambda+1)phi=Mfrac{partial^{2}phi}{partial t^{2}}+Nfrac{partial phi}{partial t}$
偏微分方程求其特解.

關鍵字

分數微積分 ； 二階線性偏微分方程 ； Jacobi微分方程 ； Gamma函數 ； Gauss微分方程 ； 特殊方程

並列摘要

When we deal some of the linear second-order differential equations with variable coefficients (or constant coefficients),
$P(z)phi^{'}(z)+Q(z)phi^{'}(z)+R(z)phi(z)=f(z)$, the method of using regularly requests by the method of
Frobenius. However, the transformation of the solutions of series cannot be solved by the closed form of the differentiation or the integration. Recently, from Professor Katsuyaki Nishimoto
in Japan, Professor Shih-Tong Tu and Professor Shy-Der Lin in Taiwan, and so on, it drinks a lot of special differential equation types and is searched
out by using the method of fractional calculus. Such as, Legendre equation, Bessel equation, Gauss equation, Jacobi equation, and so on.
To exceed a very wide thing to use hypergeometric function on mathematics, so in this paper, making the above-mentioned functions will exceed the form of hypergeometric function.
Ahead of this, it introduces the basic definitions and results of fractional calculus, the particular solutions of the Gauss, Jacobi and to discuss and compare
$z(1-z)frac{partial^{2}phi}{partial z^{2}}+[(
ho-2lambda)z+lambda+sigma]frac{partialphi}{partial z}+lambda(
ho-lambda+1)phi=Mfrac{partial^{2}phi}{partial t^{2}}+Nfrac{partial phi}{partial t}$
.

並列關鍵字

Gamma functions ； Fractional Calculus ； Special functions ； Linear Second-Order Differential Equations ； Gauss differential equation ； Jacobi differential equation

參考文獻

Shin-Tong Tu, Ding-Kuo Chyan and H.M.Srivastava, Some Families of Ordinary and Partial Fractional Differintegral Equations, Integral Transforms and Special Functions, Vol.11, No.3, pp291-302, 2001.

H.M.Srivastava and H.L.Manocha, a treatise on Generating Functions, 1983.

W.E. Byoce, R.C.DiPrima, Elentary, Differential Equations, Wiley, 2004.

Shy-Der Lin, Wei-Chich Ling, Katsuyuki Nishimoto and H.M. Srivastava, A simple fractional-calculus approach to the solutions of the Bessel differential equation of general order and some of its applications, Computers $&$ Mathematics with Applications, Volume 49, Issues 9-10, Pages 1487-1498, May 2005.

K.Nishimoto, Fractional Calculus, Vol. I,II,III,and IV, Descartes Press, Koriyama, 1984,1987,1989,and 1991.