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博碩士論文 etd-0611113-231801 詳細資訊
Title page for etd-0611113-231801
論文名稱
Title
常微分方程之雙指數收斂算法
Double-geometric Convergent Methods for ODEs
系所名稱
Department
畢業學年期
Year, semester
語文別
Language
學位類別
Degree
頁數
Number of pages
102
研究生
Author
指導教授
Advisor
召集委員
Convenor
口試委員
Advisory Committee
口試日期
Date of Exam
2013-06-27
繳交日期
Date of Submission
2013-07-18
關鍵字
Keywords
牛頓法、Picard迭代法、譜方法、徑向基法、常微分方程、收斂速度、雙指數收斂、超指數收斂
spectral method, Picard’s iteration, radial basis function, ordinary differential equation, speed of convergence, Newton’s method, super-geometric convergence, double-geometric convergence
統計
Statistics
本論文已被瀏覽 5785 次,被下載 651
The thesis/dissertation has been browsed 5785 times, has been downloaded 651 times.
中文摘要
我們先回顧數值方法上所有可能的收斂速度,接著我們研究譜方法、徑向基法和Picard法解常微分方程之超指數收斂行為,此類的收斂速度比指數收斂還快。最後我們發現牛頓法求解常微分方程的冪級數解能達到最快的雙指數收斂。
Abstract
We first review all possible convergent speeds of existing numerical methods.
Then we focus on super-geometric convergent behaviors, which is faster than exponential one, of spectral, Kansa’s and Picard’s methods for solving ordinary differential equations. We discover that Newton’s method on power series domain possesses the fastest double-exponential convergence.
目次 Table of Contents
致謝辭 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
摘要. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Abstract. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Speed of Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
3 Spectral and Kansa's Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.1 Spectral Collocation Method. . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.2 Kansa’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20
4 Semi-analytic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.1 Picard’s Iteration and Its Modification. . . . . . . . . . . . . . . . . . . . 30
4.2 Newton’s Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45
5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63
5.1 Initial Value Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63
5.2 Boundary Value Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .90
參考文獻 References
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