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

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48 CHAPTER 3. APPROXIMATE SOLUTIONS 3.2 Epsilon-Approximate Solutions To prove that a Euler’s method solution can actually be constructed we need to formalize the definition of an approximate solution a little. 1 Definition 3.1. Let f(t, y) ∈ C 0 on some domain D ⊂ R 2 that contains the rectangle R = [t 0 − a, t 0 + a] × [y 0 − b, y 0 + b] (3.38) Then y(t) is an ɛ-approximate solution of the initial value problem 3.1 all of the following conditions are true: if 1. y(t) is admissable on D, i.e., f(t, y(t)) ∈ D for all t ∈ [t 0 − a, t 0 + a]; 2. y(t) is continuous on [t 0 − a, t 0 + a]; 3. y ′ (t) exists except at finitely many points t 1 , t 2 , ..., t n and is continuous on t i < t < t i+1 for all i; 4. |y ′ (t) − f(t, y(t)| ≤ ɛ for all t ∈ [t 0 − a, t 0 + a] except possibly at t 1 , t 2 , .... Theorem 3.3. Let (t 0 , y 0 ), R, D and f be defined as in definition (3.1); furthermore, suppose that f is bounded on R, namely that there exists some M > 0 such that |f(t, y)| ≤ M (3.39) for all (t, y) ∈ R; and define h = min(a, b/M) (3.40) Then it is possible to construct an ɛ-approximate solution to (3.1). Proof. We will construct the solution going to the “right”, namely, for t > t 0 . The proof for the left-side solution is analogous. Define the rectangle S ⊂ R by Since S = [t 0 − h, t 0 + h] × [y 0 − Mh, y 0 + Mh] (3.41) h = min(a, b/M) ≤ a (3.42) Mh = min(Ma, b) ≤ b (3.43) then S ⊂ R. Hence all of the properties of f that hold on R also hold on S. =⇒ Let ɛ > 0 be given. Then since f(t, y) is continuous, it is uniformly continuous on S, and therefore there exists some δ > 0 such that Press. 1 This discussion parallels W. Hurewicz (1958) Lectures on Ordinary Differential Equations, MIT Math 582B, Spring 2007 California State University Northridge c○2007, B.E.Shapiro Last revised: May 23, 2007
CHAPTER 3. APPROXIMATE SOLUTIONS 49 Figure 3.2: Construction of the approximate solution. P ✟ S ✛ h ✲ ❍ (t 2 , y 2 ) ❍ ❍❍❍❍ O ✟✥✥✥✥✥ ✟✟✟✟✟✟✟✟✟✟✟✟✟ (t 1 , y 1 ) ❍ ✻ (t 0 , y 0 ) α ❍ ❍❍❍❍❍❍❍❍❍❍❍❍❍ ❍ Mh ❄ Q |ˆt − t|, |ŷ − y| < δ =⇒ |f(ˆt, ŷ) − f(t, y)| ≤ ɛ (3.44) ∀(ˆt, ŷ), (t, y) ∈ S. Pick any meshing satisfying t 0 < t 1 < · · · < t n−1 < t n = t 0 + h (3.45) t i − t i−1 ≤ min(δ, δ/M), i = 1, 2, . . . , n (3.46) To simplify the notation we define the points O = (t 0 , y 0 ) (3.47) P = (t 0 + h, y 0 + Mh) (upper right hand corner of S) (3.48) Q = (t 0 + h, y 0 − Mh) (lower right hand corder of S) (3.49) Construct the solution by drawing a line segment from (t 0 , y 0 ) with slope f(t 0 , y 0 ) until it intersects the line t = t 1 . The end point then has coordinates (t 1 , y 1 ) where y 1 − y 0 t 1 − t 0 = f(t 0 , y 0 ) (3.50) y 1 = y 0 + (t 1 − t 0 )f(t 0 , y 0 ) (3.51) The reader should compare the construction of y 1 in equation(3.51) with equation (3.9); the step size h n in (3.9) corresponds to the time-step t 1 − t 0 in (3.51). So we are constructing the first segment of the solution using the Forward Euler Method. We then continue to construct additional points on the solution at subsequent steps, 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
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 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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Page 75 and 76: Chapter 4 Improving on Euler’s Me
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Page 95 and 96: Chapter 5 Runge-Kutta Methods 5.1 T
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CHAPTER 5. RUNGE-KUTTA METHODS 99 T
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CHAPTER 5. RUNGE-KUTTA METHODS 101
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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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Bibliography [1] Ascher, Uri M., Ma
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