On the Higer Order Differential-Difference Recurrences via Pde Approach

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Project title

Code/計畫編號

MOST107-2115-M019-001

Translated Name/計畫中文名

偏微分方程應用在高階微分-差分型遞迴關係的分析

Project Coordinator/計畫主持人

Hua-Huai Chern

Funding Organization/主管機關

National Science and Technology Council

Department/Unit

Department of Computer Science and Engineering

Website

https://www.grb.gov.tw/search/planDetail?id=12674629

Year

2018

Start date/計畫起

01-08-2018

Expected Completion/計畫迄

31-07-2019

Bugetid/研究經費

267千元

ResearchField/研究領域

數學

筆者擬針對幾類可由下列二階或高階微分-差分型遞回關係式 Differential-Difference Recurrences (DDR) 所刻畫的實際問題作為本項計畫的主體進行研究：\[ f_n'(x) =p_n(x) f_{n-1}(x)+q_n(x)f_{n-1}'(x)+r_n(x) f_{n-1}''(x) (1)\]或更廣義的型式（目前接觸到的範例並不算多）\[ f_n(x,y)=p_n(x,y) f_{n-1}(x,y)+\sum_{i+j=1}^m q_{n,i,j}(x,y)\frac {\partial^{i+j}}{\partial x^i\partial y^j}f_{n-1}(x,y)， (2)\]其中出現在兩種型式裡的係數數列函數 $p_n, q_n, r_n, q_{n,i,j}$ 等的條件需視問題來決定，通常會是與 $n, x, y$ 等參數有關的多項式。在有具體的實際範例的對應下, 這個計畫的主體遞回關係型式 (1) 或 (2), 雖然可視為目前正在執行計畫的延續, 然而基於高階項及偏微分項次的出現, 形成了在分析上與在可歸結於一階線性 PDE 方程或方程系統的一階 DDR 截然不同的狀況及難度。於是可作為具挑戰性的主題在下一個階段的計畫裡來探究。The author aims to study some real problems that are characterized by the following types of second or higher order Differential-Difference Recurrences (DDR) : \[ f_n'(x) =p_n(x) f_{n-1}(x)+q_n(x)f_{n-1}'(x)+r_n(x) f_{n-1}''(x), (1)\]or more general version\[ f_n(x,y)=p_n(x,y) f_{n-1}(x,y)+\sum_{i+j=1}^m q_{n,i,j}(x,y)\frac {\partial^{i+j}}{\partial x^i\partial y^j}f_{n-1}(x,y), (2)\]where those sequences of coefficient functions $p_n$, $q_n$, $r_n$ and $q_{n,i,j}$ are polynomials of variables $x,y$ and $n$. With the concrete examples found in some areas, although the core recurrences (1) and (2) can be viewed as extensions of the first order DDR that are dealt with and almost solved completely in current project, the existence of higher order differential terms makes them be more difficult to be investigated and analyzed. Accordingly, these challenging models are deserved to be studied deliberately in this project.

Keyword(s)

微分─差分型遞回關係
偏微分方程
歐拉數列
階矩分析
漸進遞移
畸點分析
極限律

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On the Higer Order Differential-Difference Recurrences via Pde Approach

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