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| 題 名 | 一個使用運算矩陣之新奇分數微積分數值方法=A Novel Numerical Method for Solving Fractional Calculus via Operational Matrices |
|---|---|
| 作 者 | 吳俊霖; | 書刊名 | 統計與資訊評論 |
| 卷 期 | 9 2007.11[民96.11] |
| 頁 次 | 頁23-43 |
| 分類號 | 440.11 |
| 關鍵詞 | 分數微積分; 分數微分方程式; 運算矩陣; 正交函數; Fractional calculus; Fractional differential equation; Operational matrix; Orthogonal functions; |
| 語 文 | 中文(Chinese) |
| 中文摘要 | 分數微積分（fractional calculus）就是把一般所熟知的微分與積分運算推廣到任意非整數次方，而含有分數（fractional）微分或積分運算子，像是d□／dt□和d□／dt□，的微分方程則稱之為分數微分方程式(fractional differential equation)。它們在科學及工程上有很多的應用，例如利用分數微積分可以建立許多自然科學系統的模型，而分數微分方程在擴散模型、電磁學、控制等領域中也都有相當好的成功應用。但是通常它們只在某些限制條件下才有解析解，而且也很難以一般的數值方法來求解。 我們通常以[9066]D□(t)來表示一個任意實數α次方的微分運算，其中下標a稱為該分數微分的端點(terminal)，也就是在t = a之分數微分。運算矩陣(operational matrix)的觀念被廣泛地應用在許多領域，像是解微分方程、系統參數鑑別，以及求解線性系統的最佳化等。運算矩陣的主要特性是其可以將微分方程式轉換為代數(algebra)的形式，如此不但可以簡化問題而且可以加快計算的速度。然而傳統所用的正交函數或正交多項式之運算矩陣都是以0為起點，所以只能求解端點a = 0之分數微積分或分數微分方程。本論文推導出一類新的正交函數運算矩陣來克服此一限制，使其能計算端點不為0的分數微積分，[9066]D□(t)，之數值解及相關之分數微分方程。實驗結果顯示所提方法比其他傳統方法簡單而且具電腦計算導向(computer-oriented)。 |
| 英文摘要 | Fractional calculus is a branch of mathematics that deals with generalization of well-known operations of differential and integration to arbitrary non-integer order – which can be non-integer real or even imaginary numbers. And a differential equation involving the fractional calculus operators such as d□／dt□ and d□／dt□ is called the fractional differential equation. The idea of modeling dynamic systems by fractional differential equations can be used in many fields of science and engineering including fluid flow, electric networks, electromagnetic theory and statistics. But not only its analytical solutions exist only for a limited number of cases, but also the numerical methods are difficult to solve. We usually use the notation, [9066]D□ f(t), to denote the fractional derivative of arbitrary real order α, where α is called the terminal of fractional differentiation. The idea of the operational matrix has been widely applied to many fields such as solving the differential equations, identifying and optimizing linear systems etc. The main character of the operational method is to convert a differential equation into an algebraic one, it not only simplifies the problem but also speed up the computation. But the traditional operational approaches are able to calculate the fractional derivative only at terminal α=0. In this paper, we derive a new numerical method based on the operational matrix of orthogonal functions to solve this problem. Several examples are given to demonstrate the powerfulness of the proposed method. |
本系統中英文摘要資訊取自各篇刊載內容。