臺灣博碩士論文加值系統

摘 要
在這篇論文中, 我們探討 Runge-Kutta 方法, 其利用樹狀結構處
理 1 階微分方程的數值解. 首先, 利用 Frechet derivatives 將泰
勒展開式
y(x_{n+1})=y(x_n)+hy^{(1)}(x_n)+(h^2/2!)y^{(2)}(x_n)+...+
(h^q/q!)y^{(q)}(x_n)+...
中之表示成所有q 階 elementary differentials 的線性組合, 而這
些 elementary differentials 皆有其相對應的樹狀結構. 若想求得
p 階的 Runge-Kutta 方法, 需將滿足 p 階的樹狀結構繪出, 且
Runge-Kutta 方法中的 s-stages 參數能滿足
Ψ(t)=1/γ(t),
而其所得到的方程式的解. 在文中, 將詳加敘述 Runge-Kutta 方法
4、 5 次的精確度, 且以類似的方式稍微描述 6、7 次的精確度.

Abstract
In this paper, we discuss the Runge-Kutta method for dealing with the numerical solutions of 1 order differential equation. We use the Frechet derivatives to find all derivatives of y(x), where y and x are vectors, such that they can be written in
matrix form. y(q) can be expressed as a linear combination of all elementary differentials of order q. We make use of the tree structure of elementary differentials to determine the order conditions with respect to certain accuracy. The Runge-Kutta method has order p must satisfy the order condition Ψ(t)=1/γ(t). We state the methods of order s with s stages up to s=4, but no fifth-stage method available for order 5. A sixth-stage method of order 5 can be achieved. We can have a graph as p21. Finally, we can described the methods of order 6 and 7.

CONTENTS
Section 1. Introduction ‧‧‧‧‧‧‧‧‧‧‧‧‧ 1
Section 2. Background ‧‧‧‧‧‧‧‧‧‧‧‧‧‧ 3
Section 3. The Fourth and Fifth-order methods ‧‧ 13
Section 4. Conclusion ‧‧‧‧‧‧‧‧‧‧‧‧‧ 22
References ‧‧‧‧‧‧‧‧‧‧‧‧‧‧‧‧‧‧‧ 25
Appendix ‧‧‧‧‧‧‧‧‧‧‧‧‧‧‧‧‧‧‧‧ 26

References
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Differential Equations: Runge-Kutta and General Linear
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[3.] Hairer, E., and Wanner, G.,(1997). '' Order conditions
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[6.] Lambert, J.D. (1991), Numerical methods for ordinary
differential systems: The initial value problem, Wiley,
Chichester.