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

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62 CHAPTER 3. APPROXIMATE SOLUTIONS Finally, since the P n (t) are a monotonic sequence with P n+1 (t) ≥ P n (t), we know that P m (t) ≥ P n (t) whenever m > n. Therefore and hence P m (t) − P m (1) ≤ P n (t) − P n (1) (3.160) P m (t) − P n (t) ≤ P m (1) − P n (1) (3.161) lim (P m(t) − P n (t)) ≤ lim (P m(1) − P n (1)) (3.162) m→∞ m→∞ P (t) − P n (t) ≤ 1 − P n (1) (3.163) 1 − √ 1 − t − P n (t) ≤ 1 − P n (1) (3.164) =⇒ Pick any ɛ > 0. Then, since P n (1) → P (1) as n → ∞, there exists some n such that 1 − P n (1) < ɛ, in which case 1 − √ 1 − t − P n (t) < ɛ (3.165) i.e., ‖f(t) − P n (t)‖ < ɛ (3.166) This proves the Weierstrass Approximation Theorem for the special case f(t) = 1 − √ 1 − t. Next, suppose that f(t) = |t − c| for some number c ∈ [0, 1]. Let u = |t − c| and pick any ɛ > 0. Since u, ɛ ≥ 0, we know that √ Now define a new variable v, u 2 + ɛ2 4 < u2 + uɛ + ɛ2 4 = ( ɛ 2 + u ) 2 (3.167) √ u 2 + ɛ2 4 < ɛ 2 + u (3.168) u 2 + ɛ2 4 − u < ɛ 2 (3.169) v = 1 − u 2 − ɛ 2 /4 (3.170) By the previous case there is some polynomial Q(v) that approximates 1 − √ 1 − v arbitrarily closely: ∣ ∣1 − √ 1 − v − Q(v) ∣ ∣ < ɛ/2 (3.171) Define a new polynomial P (v) = 1 − Q(v); then we have ∣ √ 1 − v − P (v) ∣ < ɛ/2 (3.172) √ ∣ ∣∣∣∣ u ∣ 2 + ɛ2 4 − P (v) < ɛ/2 (3.173) Math 582B, Spring 2007 California State University Northridge c○2007, B.E.Shapiro Last revised: May 23, 2007
CHAPTER 3. APPROXIMATE SOLUTIONS 63 But since P (v) is a polynomial in v, and v is a polynomial in u, then P is also a polynomial in u. But u is a polynomial in t, hence P is a polynomial in t, which we write as follows: ∣ ∣∣∣∣ √ u 2 + ɛ2 4 − P (t) ∣ ∣∣∣∣ < ɛ 2 (3.174) Now since u 2 + ɛ2 4 < u2 + ɛ2 4 + ɛ|u| = ( |u| + ɛ 2) 2 (3.175) √ u 2 + ɛ2 4 < |u| + ɛ 2 √ u 2 + ɛ2 4 − |u| < ɛ 2 (3.176) (3.177) Adding equations 3.174 and 3.177, we get √ ∣ ∣∣∣∣ √ u ∣ 2 + ɛ2 4 − P (t) + u 2 + ɛ2 4 − |u| < ɛ (3.178) Since u 2 + ɛ 2 /4 > u 2 , we have √ ∣ ∣∣∣∣ √ ∣ ∣∣∣∣ u ∣ 2 + ɛ2 4 − P (t) + u ∣ 2 + ɛ2 4 − |u| < ɛ (3.179) √ ∣ ∣∣∣∣ u ∣ 2 + ɛ2 4 − P (t) + √u ∣ |u| − 2 + ɛ2 4 ∣ < ɛ (3.180) Using the triangle inequality on the last expression gives |P (t) − u| < ɛ (3.181) which proves the Weierstrass Approximation Theorem for f(t) = u = |t − c|. Now consider any piecewise linear function N∑ f(t) = b + a k |t − c k | (3.182) By the previous case there exist N + 1 polynomials Q 0 , Q 1 , . . . , Q N such that k=0 |a k |t − c m | − Q m (t)| < ɛ N (3.183) By the triangle inequality, where c○2007, B.E.Shapiro Last revised: May 23, 2007 |f(t) − P (t)| < ɛ (3.184) P (t) = N∑ Q k (t) (3.185) k=1 Math 582B, Spring 2007 California State University Northridge
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The Computable Differential Equatio
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Contents 1 Classifying The Problem
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CONTENTS v Timeline on Computable D
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Chapter 1 Classifying The Problem 1
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CHAPTER 1. CLASSIFYING THE PROBLEM
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Page 17 and 18: CHAPTER 1. CLASSIFYING THE PROBLEM
Page 31 and 32: Chapter 2 Successive Approximations
Page 33 and 34: CHAPTER 2. SUCCESSIVE APPROXIMATION
Page 49 and 50: Chapter 3 Approximate Solutions 3.1
Page 51 and 52: CHAPTER 3. APPROXIMATE SOLUTIONS 45
Page 67: CHAPTER 3. APPROXIMATE SOLUTIONS 61
Page 75 and 76: Chapter 4 Improving on Euler’s Me
Page 77 and 78: CHAPTER 4. IMPROVING ON EULER’S M
Page 95 and 96: Chapter 5 Runge-Kutta Methods 5.1 T
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CHAPTER 5. RUNGE-KUTTA METHODS 113
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CHAPTER 5. RUNGE-KUTTA METHODS 115
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Chapter 6 Linear Multistep Methods
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CHAPTER 6. LINEAR MULTISTEP METHODS
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Chapter 7 Delay Differential Equati
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CHAPTER 7. DELAY DIFFERENTIAL EQUAT
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Chapter 8 Boundary Value Problems 8
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CHAPTER 8. BOUNDARY VALUE PROBLEMS
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Chapter 9 Differential Algebraic Eq
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CHAPTER 9. DIFFERENTIAL ALGEBRAIC E
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Chapter 10 Appendix on Analytic Met
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CHAPTER 10. APPENDIX ON ANALYTIC ME
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CHAPTER 10. APPENDIX ON ANALYTIC ME
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Bibliography [1] Ascher, Uri M., Ma
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