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

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72 CHAPTER 4. IMPROVING ON EULER’S METHOD The error after n steps is then e n (t) = |y(t) − ŷ(t)| = δe λ(t−t 0) 1 − e −nλh ∣ e λh − 1 ∣ (4.25) For |nλh| t 0 + nh, the e λ(t−t 0) factor still dominates the fraction, and hence the error still tends towards zero. The analogous vector test equation is written as y ′ = Ay, y(0) = y 0 (4.27) where A is a square matrix of constants. The solution is given by the matrix exponential y = e At y 0 (4.28) where the matrix exponential is defined by its Taylor Series, ∞∑ e At t k A k = k! k=0 (4.29) If the matrix A is diagonalizable, or equivalently, has n linearly independent eigenvectors v 1 , . . . , v n , with corresponding eigenvalues λ 1 , . . . , λ n , then where and e At = UEU −1 (4.30) E = diag(e λ 1t , . . . , e λnt ) (4.31) U = (v 1 , . . . , v n ) (4.32) Example 4.1. Solve the initial value problem ( ) ( y ′ 0 1 1 = Ay, A = , y(0) = −2 0 0) (4.33) Solution. A is diagonalizable with eigenvalues ±i √ 2 and corresponding eigenvectors ( ) ( ) √ i 2 − √i and 2 (4.34) 1 1 We have then y = e At y 0 = = = ( − √i i 2 1 1 √ 2 ( e −i √ ) 2t +e i√ 2t 2 i ei√ 2t −e −i√ 2t √ 2 ( cos (√ 2t ) − √ 2 sin (√ 2t ) ) ) ( ) ( e i√ 2t i 0 0 e −i√ 2t √ 1 2 2 − √ i 1 2 2 ) (1 0 ) (4.35) (4.36) (4.37) Math 582B, Spring 2007 California State University Northridge c○2007, B.E.Shapiro Last revised: May 23, 2007
CHAPTER 4. IMPROVING ON EULER’S METHOD 73 More generally, the matrix A may not be diagonalizable;however, there is still a similarity transform to Jordan Canonical Form J = U −1 AU (4.38) where U is the matrix of generalized eigenvectors v k , k = 1, . . . , n, where (A − λI) k v k = 0, (4.39) The Jordan Canonical Form of a matrix is the block-diagonal matrix ⎛ ⎞ Λ 1 0 · · · 0 J = 0 Λ 2 . ⎜ ⎝ . . .. ⎟ ⎠ 0 · · · 0 Λ m (4.40) where each Jordan Block corresponds to one eigenvalue of the original matrix ⎛ ⎞ λ i 1 0 · · · 0 . 0 λ i 1 .. . Λ i = . ⎜ . .. . .. . .. 0 (4.41) ⎟ ⎝ 0 0 λ i 1 ⎠ 0 · · · 0 λ i In this case we have e At = Ue Jt U −1 (4.42) It is generally easier to calculate the solution y = m∑ ∑k i i=1 p=1 t k i−p (k i − p)! v p, ie λ it (4.43) where eigenvalue λ i is repeated k i times with generalized eigenvectors v p,i , p = 1, . . . , k i . 1 Theorem 4.1. The vector test equation is stable if all of the eigenvalues satisfy either Re(λ) < 0, or Re(λ) = 0 and λ is simple. The solution is asymptotically stable if all the eigenvalues satisfy Re(λ) < 0. 4.2 Convergence, Consistency and Stability Definition 4.4 (One Step Method). Any method that can be written in the form is called a one-step method. y n+1 = y n + hφ(t n , y n , h) (4.44) 1 For a derivation of this result see Gray, A., Mezzino,M. & Pinsky,M.A. Introduction to Ordinary Differential Equations with Mathematica, Springer (1997) 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
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Chapter 1 Classifying The Problem 1
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CHAPTER 1. CLASSIFYING THE PROBLEM
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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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Bibliography [1] Ascher, Uri M., Ma
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