The Computable Differential Equation Lecture ... - Bruce E. Shapiro
The Computable Differential Equation Lecture ... - Bruce E. Shapiro The Computable Differential Equation Lecture ... - Bruce E. Shapiro
40 CHAPTER 2. SUCCESSIVE APPROXIMATIONS Hence by equation 2.82, |φ(t) − φ n (t)| = ∣ ≤ ≤ ≤ M K ∞∑ i=n+1 ∞∑ i=n+1 ∞∑ i=n+1 (φ i (t) − φ i−1 (t)) ∣ (2.90) |φ i (t) − φ i−1 (t)| (2.91) K i−1 M |t − t 0 | i (2.92) i! ∞∑ i=n+1 |K(t − t 0 )| i Therefore by comparison with a Taylor series for e K(b−a) , i! (2.93) ‖φ(t) − φ n (t)‖ ≤ M K ≤ M K ∞∑ |K(b − a)| i i! i=n+1 ( ) n∑ e K(b−a) |K(b − a)| i − i! i=0 (2.94) (2.95) ≤ M K sup R n (t) (2.96) 0≤t≤KL where R n (t) is the Taylor Series Remainder for e t after n terms, sup R n (t) ≤ 0≤t≤KL sup 0≤{c,t}≤KL t n+1 e c (n + 1)! ≤ (KL)n+1 e KL (n + 1)! (2.97) for some unknown c between a and b.Hence proving the theorem. ‖φ(t) − φ n (t)‖ ≤ M K (KL) n+1 e KL (n + 1)! (2.98) The following example shows that this bounds is not very useful in practice. Example 2.7. Estimate the number of iterations required to obtain an solution to y ′ = t, y(0) = 1 on [0, 10] with a precision of no more that 10 −7 . Solution. Since f(t, y) = t we have f y = 0 and hence a Lipshitz constant is K = 1 (or any positive number), and we can use M = 10 on [0, 10]. The precision in the error is bounded by ɛ ≤ M(KL)n+1 e KL K(n + 1)! ≤ 10(10)n+1 e 10 (n + 1)! (2.99) We can determine the minimum value of n by using Mathematica. The following will print a list of values of 2.99 for n ranging from 1 to 50. Math 582B, Spring 2007 California State University Northridge c○2007, B.E.Shapiro Last revised: May 23, 2007
CHAPTER 2. SUCCESSIVE APPROXIMATIONS 41 errs = Table[{n, 10 (10) ˆ (n + 1) (E ˆ 10.)/(n + 1)!}, {n, 1, 50}] The output is a list of number pairs, which can be plotted with ListPlot or
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CHAPTER 2. SUCCESSIVE APPROXIMATIONS 41<br />
errs = Table[{n, 10 (10) ˆ (n + 1) (E ˆ 10.)/(n + 1)!}, {n, 1,<br />
50}]<br />
<strong>The</strong> output is a list of number pairs, which can be plotted with ListPlot or<br />
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