The Computable Differential Equation Lecture ... - Bruce E. Shapiro

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46 CHAPTER 3. APPROXIMATE SOLUTIONS y 3 = y 2 + hf(t 2 , y 2 ) (3.26) = 1.5625 + (0.25)(1.5625) = 1.953125 (3.27) t 3 = t 2 + h = 0.5 + 0.25 = 0.75 (3.28) (t 3 , y 3 ) = (0.75, 1.953125) (3.29) y 4 = y 3 + hf(t 3 , y 3 ) (3.30) = 1.953125 + (0.25)(1.953125) = 2.44140625 (3.31) t 4 = t 3 + 0.25 = 1.0 (3.32) (t 4 , y 4 ) = (1.0, 2.44140625) (3.33) Since t 4 = 1 we are done. The solutions are tabulated in table 3.1 for this and other step sizes. These solutions are also illustrated in figure 3.1. The green dots in the figure correspond to the numbers calculated here; the other colors correspond to other step sizes. The points are joined by straight lines in the figure, but the straight lines are not part of the solution, only the dots. The figure also shows the exact solution. Table 3.1: Approximate values for different step sizes. See example 3.1. t h = 1/2 h = 1/4 h = 1/8 h = 1/16 exact solution 0.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0625 1.0625 1.0645 0.1250 1.1250 1.1289 1.1331 0.1875 1.1995 1.2062 0.2500 1.2500 1.2656 1.2744 1.2840 0.3125 1.3541 1.3668 0.3750 1.4238 1.4387 1.4550 0.4375 1.5286 1.5488 0.5000 1.5000 1.5625 1.6018 1.6242 1.6487 0.5625 1.7257 1.7551 0.6250 1.8020 1.8335 1.8682 0.6875 1.9481 1.9887 0.7500 1.9531 2.0273 2.0699 2.1170 0.8125 2.1993 2.2535 0.8750 2.2807 2.3367 2.3989 0.9375 2.4828 2.5536 1.0000 2.2500 2.4414 2.5658 2.6379 2.7183 Math 582B, Spring 2007 California State University Northridge c○2007, B.E.Shapiro Last revised: May 23, 2007
CHAPTER 3. APPROXIMATE SOLUTIONS 47 Figure 3.1: Solution curves for example 3.1. The curves correspond to h = 1 (red); h = 1/2 (green); h = 1/4 (blue); h = 1/8 (purple); and the exact solution (orange). Taylor Series Theorem 3.1. Taylor’s Theorem with Remainder. Let f : J ↦→ R be C n+1 on J ⊂ R, and suppose that a ∈ interior(J). Then for any x ∈ J, there exists some number c ∈ J, where c is between a and x, inclusive, such that f(x) = P n (x) + R n (x) (3.34) where the The Taylor Polynomial of order n for f about x =a is and P n (x) = f(a) + (x − a)f ′ (a) + (x − a)2 f ′′ (a) 2! R n (x) = (x − a)n+1 f (n+1) (c) (n + 1)! Theorem 3.2. If, in addition, f ∈ C ∞ then + · · · + (x − a)n f (n) (a) ) (3.35) n! (3.36) ∞∑ f k (a)(x − a) k f(x) = k! k=0 (3.37) which is called the Taylor Series of f about x =a. c○2007, B.E.Shapiro Last revised: May 23, 2007 Math 582B, Spring 2007 California State University Northridge
Page 1 and 2: The Computable Differential Equatio
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Page 5 and 6: CONTENTS v Timeline on Computable D
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Bibliography [1] Ascher, Uri M., Ma
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