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論文名稱 Title |
常微分方程之雙指數收斂算法 Double-geometric Convergent Methods for ODEs |
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系所名稱 Department |
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畢業學年期 Year, semester |
語文別 Language |
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學位類別 Degree |
頁數 Number of pages |
102 |
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研究生 Author |
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指導教授 Advisor |
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召集委員 Convenor |
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口試委員 Advisory Committee |
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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 |
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統計 Statistics |
本論文已被瀏覽 5785 次,被下載 651 次 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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