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

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52 CHAPTER 3. APPROXIMATE SOLUTIONS Theorem 3.4. Fundamental Inequality. Suppose that f is continuous and Lipshitz (with constant K) on a set R that includes the point (t 0 , y 0 ). Let h > 0, and suppose that y (1) , y (2) are admissible ɛ 1 - and ɛ 2 -approximations on |t−t 0 | < h. Then |y (2) (t) − y (1) (t)| ≤ e K|t−t 0| |y (2) (t 0 ) − y (1) (t 0 )| + ɛ 1 + ɛ 2 K ( ) e K|t−t0| − 1 (3.75) Proof. Since y (1) and y (2) are ɛ-approximations with ɛ 1 and ɛ 2 respectively, we have Let ∣ ∣ dy (1) dt dy (2) dt − f(t, y (1) ) ∣ ≤ ɛ 1 (3.76) − f(t, y (2) ) ∣ ≤ ɛ 2 (3.77) (3.78) ɛ = ɛ 1 + ɛ 2 (3.79) p(t) = y (2) (t) − y (1) (t) (3.80) Then dp ∣ dt ∣ ≤ ∣ ∣f(t, y (1) ) − f(t, y (2) ) ∣ + ɛ (3.81) ≤ K ∣ ∣ ∣y (1) − y (2) + ɛ (3.82) ≤ K|p| + ɛ (3.83) except for a possible finite number of locations where dp/dt is undefined. Suppose that (t) ≠ 0, t 0 < t ≤ t 0 + h. Then since its sign does not change, p(t) is either always positive or always negative. Suppose it is positive. Then p(t) > 0, and Multiply the equation by e −Kt and rearrange to give The left hand side is exactly p ′ (t) ≤ Kp(t) + ɛ (3.84) e −Kt [ p ′ (t) − Kp(t) ] ≤ ɛe −Kt (3.85) d ( p(t)e −Kt ) = e −Kt [ p ′ (t) − Kp(t) ] (3.86) dt Hence d ( p(t)e −Kt ) ≤ ɛe −Kt (3.87) dt Math 582B, Spring 2007 California State University Northridge c○2007, B.E.Shapiro Last revised: May 23, 2007
CHAPTER 3. APPROXIMATE SOLUTIONS 53 Integrate both sides from t 0 to t, ∫ t d ( p(x)e −Kx ) ∫ t dx ≤ ɛ e −Kx dx t 0 dx t 0 (3.88) p(t)e −Kt − p(t 0 )e −Kt 0 ≤ − ɛ [ e −Kt − e −Kt ] 0 K (3.89) Solving for p(t) gives the fundamental inequality. The proof for p(t) < 0 is analogous. Now suppose that p(t) = 0 identically for all t. Then the fundamental inequality is equivalent to 0 < something positive and follows trivially. 3.4 Cauchy-Euler Existence Theory In this section the quantities f, (t 0 , y 0 ), D, h are defined as before; R is the usual 2h wide rectangle centered on (t 0 , y 0 ). Theorem 3.5. Uniqueness. Suppose that f ∈ C(D), f ∈ L(y; K)(D), and let (t 0 , y 0 ) ∈ D. Then there is at most one solution to 3.1. Proof. Suppose that y(t) and ỹ(t) are exact solutions of 3.1. Then From the fundamental inequality, equation (3.75), y ′ (t) = f(t, y(t)), y(t 0 ) = y 0 (3.90) ỹ ′ (t) = f(t, ỹ(t)), ỹ(t 0 ) = y 0 (3.91) |y(t) − ỹ(t)| ≤ e K|t−t 0| |y(t 0 ) − ỹ(t 0 )| + ɛ 1 + ɛ 2 K Hence the two functions are identical to one another for all t. ( ) e K|t−t0| − 1 = 0 (3.92) Theorem 3.6. Fundamental Existence Theorem. Let f(t, y) ∈ L(y, K)(D). Then an exact solution of the initial value problem 3.1 exists on some interval |t − t 0 | < h. Proof. Let ɛ n → 0 monotonically. Then by theorem 3.3 a sequence of ɛ n -approximate solutions y n (t) exist, such that ⇐= ∣ ∣y ′ n(t) − f(t, y n (t)) ∣ ∣ ≤ ɛ n , |t − t 0 | ≤ h (3.93) except possibly at the finite set of points t (n) i , i = 1, ..., m n By the fundamental inequality, for any integers n and p, |y n (t) − y n+p (t)| ≤ e K|t−t0| |y n (t 0 ) − y n+p (t 0 )| + ɛ n + ɛ ( ) n+p e K|t−t0| − 1 (3.94) K ≤ e K|t−t0| |y 0 − y 0 | + 2ɛ ( ) n e Kh − 1 → 0 (3.95) K c○2007, B.E.Shapiro Last revised: May 23, 2007 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
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
Page 51 and 52: CHAPTER 3. APPROXIMATE SOLUTIONS 45
Page 57: CHAPTER 3. APPROXIMATE SOLUTIONS 51
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
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CHAPTER 5. RUNGE-KUTTA METHODS 103
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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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