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

More documents

Recommendations

Info

$NAME: Math 103L: Exercise on Functions ... - Bruce E. Shapiro$

$Introducing Latex - Bruce E. Shapiro$

$Applied Differential Equations, Math 280 at CSUN - Bruce E. Shapiro$

$Math 103 Section 1.2: Linear Equations and ... - Bruce E. Shapiro$

60 CHAPTER 3. APPROXIMATE SOLUTIONS for the first data set, and { {t 0 , y 0,2 }, {t 1 , y 1,2 }, {t 2 , y 2,2 }, . . . } for the second data set. We get these from firstDataSet = Most /@ Flatten /@ r; secondDataSet = Drop[#, {2}] & /@ Flatten /@ r; Each “@” symbol is shorthand for the Map function. Thus Most /@Flatten /@r is equivalent to Map[Flatten[r]]. The Map function applies a function to every element in a list, namely, f/@{a, b, c}is the same as {f[a], f[b], f[c]}. Putting the whole solution together in one place, here is our program: y0={1,0}; f[t , y ]:= {y[[2]], -y[[1]]}; r=ForwardEuler[f,{0, y0}, 0.1π, 2π]; firstDataSet = Most /@ Flatten /@ r; secondDataSet = Drop[#, {2}] & /@ Flatten /@ r; p1 = JoinedListPlot[FirstDataSet, Red, DisplayFunction->Identity]; p2 = JoinedListPlot[SecondDataSet, Blue, DisplayFunction->Identity]; Show[p1, p2, DisplayFunction -> $DisplayFunction, AspectRatio->0.2] 3.6 Polynomial Approximation Theorem 3.9 (Weierstrass Approximation Theorem). Any continuous function can be approximated by a polynomial to any desired degree of accuracy. More specifcally, suppose that f(t) ∈ C(R). Then for any ɛ > 0, there exists a polynomial P (t) such that ‖P − f‖ < ɛ (3.149) =⇒ =⇒ Proof. We will prove 2 the theorem on [0, 1]; the generalization to any closed interval [a, b] follows by an algebraic transformation ˆt = (b − a)t + a. We first prove the 2 The proof follows the Encyclopedia of Mathematics object number 6145, AMS classification 41A10, online edition only at http://planetmath.org. The proof can be skipped without any loss of continuity. Math 582B, Spring 2007 California State University Northridge c○2007, B.E.Shapiro Last revised: May 23, 2007
CHAPTER 3. APPROXIMATE SOLUTIONS 61 theorem for f(t) = 1 − √ 1 − t. Construct the sequence 3 of Polynomials on [0, 1]. P 0 (t) = 0 (3.150) P n+1 (t) = 1 ( Pn (t) 2 + t ) 2 (3.151) We observe first that 0 ≤ P (t) ≤ 1. This may be proven by induction. First, ⇐= P 1 = t/2 ≤ 1 on [0, 1]. Next, assume that P n < 1. Then P n+1 = 1 2 (P 2 n + t) ≤ 1 2 (12 + 1) = 1. Furthermore, each P n is monotonically increasing. Again, prove by induction. Certainly P 1 = t/2 is monotonically increasing with slope 1/2. Assume P ′ n > 0. Then P ′ n+1 = 1 2 (2P nP ′ n + 1) > 0 because the P n are all positive. Hence the function is monotonically increasing. Next we observe that the sequence of polynomials is itself increasing. To prove this, observe that P n+2 (t) − P n+1 (t) = 1 ( P 2 2 n+1 (t) + t − P n 2 (t) − t ) (3.152) = 1 ( P 2 2 n+1 (t) − P n 2 (t) ) (3.153) = 1 2 (P n+1(t) + P n (t)) (P n+1 (t) − P n (t)) (3.154) Since t/2 = P 1 (t) ≥ P 0 (x) = 0 for all t ∈ [0, 1], by applying this recursively we see that the sequence is monotonically increasing: P n+1 (t) ≥ P n t on [0, 1] . Since the sequence P 0 , P 1 , . . . is bounded and monotonically increasing it converges to some limit P (t). But lim P 1 n+1(t) = lim n→∞ n→∞ 2 (P n(t) 2 + t) (3.155) P (t) = 1 2 (P 2 (t) + t) (3.156) 0 = P 2 (t) − 2P (t) + t (3.157) P (t) = 1 ± √ 1 − t (3.158) We need to take the negative square root because of the restriction P (t) ≤ 1. Hence lim P n(t) = 1 − √ 1 − t (3.159) n→∞ 3 Use of the fixed point iteration x 0 = 0, x n+1 = 1 2 (u + x2 n) to find √ z, where z = 1 − u and x = 1 − √ z, was discovered in ancient Babylonia. 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
Page 3 and 4:
Contents 1 Classifying The Problem
Page 5 and 6:
CONTENTS v Timeline on Computable D
Page 7 and 8:
Chapter 1 Classifying The Problem 1
Page 9 and 10:
CHAPTER 1. CLASSIFYING THE PROBLEM
Page 11 and 12:
Page 13 and 14:
Page 15 and 16: 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 65: CHAPTER 3. APPROXIMATE SOLUTIONS 59
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
Page 97 and 98: CHAPTER 5. RUNGE-KUTTA METHODS 91 w
Page 99 and 100: CHAPTER 5. RUNGE-KUTTA METHODS 93 w
Page 101 and 102: CHAPTER 5. RUNGE-KUTTA METHODS 95 5
Page 103 and 104: CHAPTER 5. RUNGE-KUTTA METHODS 97 F
Page 105 and 106: CHAPTER 5. RUNGE-KUTTA METHODS 99 T
Page 107 and 108: CHAPTER 5. RUNGE-KUTTA METHODS 101
Page 117 and 118:
CHAPTER 5. RUNGE-KUTTA METHODS 111
Page 119 and 120:
Page 121 and 122:
Page 123 and 124:
Chapter 6 Linear Multistep Methods
Page 125 and 126:
CHAPTER 6. LINEAR MULTISTEP METHODS
Page 127 and 128:
Page 129 and 130:
Page 131 and 132:
Page 133 and 134:
Page 135 and 136:
Page 137 and 138:
Page 139 and 140:
Page 141 and 142:
Chapter 7 Delay Differential Equati
Page 143 and 144:
CHAPTER 7. DELAY DIFFERENTIAL EQUAT
Page 145 and 146:
Page 147 and 148:
Page 149 and 150:
Page 151 and 152:
Page 153 and 154:
Page 155 and 156:
Page 157 and 158:
Page 159 and 160:
Chapter 8 Boundary Value Problems 8
Page 161 and 162:
CHAPTER 8. BOUNDARY VALUE PROBLEMS
Page 163 and 164:
Page 165 and 166:
Page 167 and 168:
Page 169 and 170:
Page 171 and 172:
Page 173 and 174:
Chapter 9 Differential Algebraic Eq
Page 175 and 176:
CHAPTER 9. DIFFERENTIAL ALGEBRAIC E
Page 177 and 178:
Page 179 and 180:
Page 181 and 182:
Page 183 and 184:
Page 185 and 186:
Page 187 and 188:
Page 189 and 190:
Page 191 and 192:
Page 193 and 194:
Page 195 and 196:
Page 197 and 198:
Page 199 and 200:
Chapter 10 Appendix on Analytic Met
Page 201 and 202:
CHAPTER 10. APPENDIX ON ANALYTIC ME
Page 203 and 204:
CHAPTER 10. APPENDIX ON ANALYTIC ME
Page 205 and 206:
Bibliography [1] Ascher, Uri M., Ma
show all

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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?