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

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

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

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

134 CHAPTER 6. LINEAR MULTISTEP METHODS of y n that is on both sides of the equation should be solved for using a Newton iteration as has been discussed previously in connection with the backward Euler method. ✬ Algorithm 6.1. Prediction-Correction Method To solve the initial value problem ✫ y ′ = f(t, y), y(t 0 ) = y 0 for the function y(t), at each time step, let EM n and IM n represent predictions at time-step n using an explicit method (EM) and an implicit method (IM), respectively. P Predict y (0) n = EM n Repeat n times (n = 1 for PEC and PECE methods) E Estimate: f (0) n = f(t n , y (0) n ) C Correct: y (1) n = IM(t n , y (0) n ). PEC methods stop here after the first iteration. P(EC) n methods stop here after n iterations. E : Estimate: f n (1) = f(t n , y n (1) ) to be used in the next iteration. PECE and P(EC) n E methods stop here. ✩ ✪ Math 582B, Spring 2007 California State University Northridge c○2007, B.E.Shapiro Last revised: May 23, 2007
Chapter 7 Delay Differential Equations 7.1 Basic Theory Definition 7.1 (Delay Differential Equation). Let 0 ≤ τ 1 < τ 2 < · · · < τ m (7.1) be a set of m positive numbers representing the delays, the largest delay being denoted by τ m . If y appears explicitly without any delay then τ 1 = 0. Then we may define the scalar delay differential equation (DDE) as y ′ (t) = f(t, y(t − τ 1 ), y(t − τ 2 ), . . . , y(t − τ m )) (7.2) y(t) = g(t), t 0 − τ m ≤ t ≤ t 0 (7.3) Theorem 7.1. Let D ∈ [t 0 , t 0 + β) × R m be convex; suppose that f : D ↦→ R is continuous on D; and that g(t) : [t 0 − τ m ] ↦→ D. Then if r 1 ≠ 0 the DDE (equations 7.2, 7.3) has a unique solution on [t 0 , t 0 + α), where 0 < α ≤ β. Furthermore, if α < β then y(t) approaches the boundary of D as t → α. If τ 1 = 0 then the DDE has at most one solution. Proof. The proof is constructive. We consider two cases: Case 1: τ 1 ≥ 0: For t 0 ≤ t ≤ τ 1 + t 0 , each y(t − τ i ) = g(t − τ i ), hence we have an explicit expression for f(t), viz., y ′ (t) = f(t, g(t − τ 1 ), g(t − τ 2 ), . . . , g(t − τ m )) (7.4) y(0) = g(0) (7.5) Hence we can integrate on (0, t) where t < τ 1 , y(t) = ∫ t t 0 y ′ (t)dt = g(0) + ∫ t t 0 f(x, g(x − τ 1 ), . . . , g(x − τ m ))dx (7.6) This uniquely determines y(t), t 0 ≤ t ≤ τ 1 . So we can repeat the process on [t 0 + τ 1 , t 0 + min(β, 2τ 1 )]. We can repeat this process until we reach β. This process is called the method of steps.Thus by construction, a solution exists. 135
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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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Chapter 2 Successive Approximations
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CHAPTER 2. SUCCESSIVE APPROXIMATION
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Chapter 3 Approximate Solutions 3.1
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Bibliography [1] Ascher, Uri M., Ma
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