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題 名 | 廣義正交多項式在邊界值問題之應用 |
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作 者 | 張義鋒; 梁頌佶; | 書刊名 | 中原學報 |
卷 期 | 19 1990.12[民79.12] |
頁 次 | 頁139-155 |
分類號 | 313.17 |
關鍵詞 | 正交多項式; 邊界值; |
語 文 | 中文(Chinese) |
中文摘要 | 本文乃是應用正交多項式之特性(1)係數遞迴關係式(2)微分遞迴關係式,再配合係數轉換矩陣和微分運算矩陣,以求解較複雜之邊界條件,諸如平滑之邊界曲線,Dirichlet條件,Neumann條件,以及混合式條件所構成之邊界值問題。其主要之構想為利用曲線匹配之概念直接將邊界函數和統御方程式配置於零點或極點上,而後獲得一組以展開係數為未知數之聯立方程式,最後再將所求得之係數轉換成所求得之解。此種方法可以推至邊界為環形域或者三維封閉曲面所構成之邊界值問題。本文中舉例說明此種方法之可行性,並且直接以正交多項式展開解析解再和所求得之解作一比較,可得到相同之精確度,因此,可知本文所使用之方法可得近似最小平方誤差之精確度。 |
英文摘要 | In this paper, we apply the coefficients recurrence and differential recurrence properties of orthogonal polynomials to derive the coeffcient transformation matrix and the operational matrix of differentiation and then to solve the boundary value problems which possess more complicated boundary conditions such as (1)Dirichlet type, (2) Neumann type, (3) Mixed type with smooth boundary curve. The main idea of our approach to use curve fitting to treat boundary functions and governing equations at locations of zeros of orthogonal polynomisls. Then a set of simultanuous algebraic equations of unknown expansion coeffcients will be obtained. Solving these coeffcients first, then multiplying the orthogonal polynomials, the solution of the problem can be obtained. This method can be extended to annular boundary condition problems or three dimensional closed surface boundary value problems. Serveral examples have been illustrated. The comparision of the results obtained by this method and the exact solutions indicate that the problems can be solved with the accuracy just like the least squares error approximation. |
本系統中英文摘要資訊取自各篇刊載內容。