臺灣博碩士論文加值系統

我們研究了遞迴關係 q0 = α, qn+1 = qn + τ qmn ，其中 n = 0, 1, · · · ，並具有參數m > 1，α > 0，和τ > 0，這可以看作是初值問題q′(t) = qm(t), q(0) = α的前向尤拉近似。眾所周知，在這個常微分方程問題的解在有限時間T = 1(m−1)αm−1 內變得無界，並且當t 接近 T 時，數值解與精確解之間的誤差也變得無界。在本文中，我們將更詳細地研究這一誤差，並探討數值解在t接近T 時的行為。

We study the recursion relation q0 = α, qn+1 = qn + τ qmn , where n = 0, 1, · · · , with parameters m > 1, α > 0, and τ > 0, which can be regarded as the forward Euler approximation of the initial value problem q′(t) = qm(t), q(0) = α. It is known that the solution of the ODE problem becomes unbounded in a finite time T = 1
(m−1)αm−1 and the error between the numerical solution and the exact solution also becomes unbounded as t close to T . In this paper, we investigate the error in more detail and explore the behavior
of the numerical solution as t close to T .

Contents
論文審定書 i
摘要 ii
Abstract iii
1 Introduction 1
2 Asymptotic Estimations 3
3 Bounds 7
4 Conclusion 18

References
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[2] C.-H. Cho, A numerical algorithm for blow-up problems revisited, Numer. Algor., 75 (2017), 675–697.
[3] E. Isaacson and H.B. Keller, Analysis of numerical methods, Wiley, New York, 1966.
[4] J.M. Sanz-Serna and J.G. Verwer, A study of the recursion yn+1 = yn + τ ymn , J. Math. Anal. Appl., 116 (1986), 456–464.