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

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ii Your fearless leader. Above: Typical view during a class lecture. Below: Typical view during an exam. The pictures were drawn by former students. The consumpution of cookies and caffeinated beverages during class is optional but is strongly encouraged. advised of the possibility of such damage. Math 582B, Spring 2007 California State University Northridge c○2007, B.E.Shapiro Last revised: May 23, 2007
Contents 1 Classifying The Problem 1 1.1 Defining ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Initial Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 The Fundamental Theorem . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . 10 1.5 Differential-Algebraic Equations . . . . . . . . . . . . . . . . . . . . . 12 1.6 Delay Differential Equations . . . . . . . . . . . . . . . . . . . . . . . 13 1.7 Numerical Solutions of Differential Equations . . . . . . . . . . . . . 14 1.8 Computer Assisted Analysis . . . . . . . . . . . . . . . . . . . . . . . 15 2 Successive Approximations 25 2.1 Picard Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2 Fixed Point Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.3 Fixed Point Iteration in a Normed Vector Space . . . . . . . . . . . 34 2.4 Convergence of Successive Approximations . . . . . . . . . . . . . . . 38 3 Approximate Solutions 43 3.1 The Forward Euler Method . . . . . . . . . . . . . . . . . . . . . . . 43 3.2 Epsilon-Approximate Solutions . . . . . . . . . . . . . . . . . . . . . 48 3.3 The Fundamental Inequality . . . . . . . . . . . . . . . . . . . . . . . 51 3.4 Cauchy-Euler Existence Theory . . . . . . . . . . . . . . . . . . . . . 53 3.5 Euler’s Method for Systems . . . . . . . . . . . . . . . . . . . . . . . 56 3.6 Polynomial Approximation . . . . . . . . . . . . . . . . . . . . . . . 60 3.7 Peano Existence Theorem . . . . . . . . . . . . . . . . . . . . . . . . 64 3.8 Dependence Upon a Parameter . . . . . . . . . . . . . . . . . . . . . 66 4 Improving on Euler’s Method 69 4.1 The Test Equation and Problem Stability . . . . . . . . . . . . . . . 69 4.2 Convergence, Consistency and Stability . . . . . . . . . . . . . . . . 73 4.3 Stiffness and the Backward Euler Method . . . . . . . . . . . . . . . 79 4.4 Other Euler Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5 Runge-Kutta Methods 89 5.1 Taylor Series Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.2 Numerical Quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . 92 iii
Page 1: The Computable Differential Equatio
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 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
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CHAPTER 3. APPROXIMATE SOLUTIONS 47
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Chapter 4 Improving on Euler’s Me
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CHAPTER 4. IMPROVING ON EULER’S M
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Chapter 5 Runge-Kutta Methods 5.1 T
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CHAPTER 5. RUNGE-KUTTA METHODS 91 w
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CHAPTER 5. RUNGE-KUTTA METHODS 95 5
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CHAPTER 5. RUNGE-KUTTA METHODS 97 F
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CHAPTER 5. RUNGE-KUTTA METHODS 99 T
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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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Bibliography [1] Ascher, Uri M., Ma
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