利用分數微積分解某些特定類偏微分方程

Explicit Solutions of Some Certain Class of Fractional Partial Differential Equations by Means of Fractional Calculus

陸家弘(Chia-Hung Lu)

指導教授 ： 林賜德

中原大學/理學院/應用數學研究所/博士(2013年)

https://doi.org/10.6840/CYCU.2013.00096

摘要

最近幾年,分數微積分的各種算子(即任意實數或複數階的微分與積分計算方法)已實際在許多科學與工程不同領域上被研究與應用。
很多作者已經證明出分數微積分在許多二階與高階線性常微分、偏微分方程特解推理的有用性。本論文是推廣相關的柯西尤拉微分方程求其特解與應用。
對於分數微積分在線性常微分、偏微分方程在二階與高階推理的有用性,本論文亦用了分數微積分的方法再加上微分分數次數的拉氏轉換與負二項式定理以及反拉氏轉換的方法來求出特解。
目前仍然未解的波動偏微分方程之柯西問題,其初始條件為函數值時,我們亦可使用分數微積分的方法得到的結果與用傳統變數分離方法所得的結果相同,若再加上邊界條件可利用傅立葉級數方法求得特解。

關鍵字

分數微積分

並列摘要

In recent years,various operators of fractional calculus(thatis,calculus of integrals and deriva-tives of arbitrary real or complex orders)have been investigated and applied in many remarkably diverse fields of science and engineering.
Many authors have demonstrated the usefulness of fractional calculus in the derivation of particular solutions of anumber of linear ordinary and partial differential equations of the second and higher orders.
The purpose of this paper is to present a certain class of the explicit particular solutions of associated Cauchy-Euler fractional partial differential equation of arbitrary real or complex orders and their applications.And to show how this simple fractional calculus method to the solutions of some families of fractional partial differential equations would lead naturally to several interesting consequences.
The methodology presented here is based chiefly upon some general theorems on explicit particular solutions of some families of fractional differential equations with Laplace transform and the expansion coefficients of binomial series.
One open question for the wave(diffusion)equation,we can use fractional calculus and seperated variable methods to solve the classical Cauchy problem with the functional initial conditions. We also can use Fourier series to obtain the particular solution.

並列關鍵字

fractional calculus

參考文獻

References

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