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

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90 CHAPTER 5. RUNGE-KUTTA METHODS error of h p . We also observe that O(h) Taylor method is equivalent to Euler’s method. The second order Taylor method has y n = y n−1 + hf + h2 2 (f t + f y f) (5.11) For the test equation y ′ = λy we have f = λy, f t = 0, f y = λ, and hence y n = y n−1 + hλy n + h2 2 λ2 y n (5.12) ) = (1 + hλ + (hλ)2 y n−1 (5.13) 2 Absolute stability occurs when ∣ y n ∣∣∣ (hλ)2 ∣ = y n−1 ∣1 + hλ + 2 ∣ ≤ 1 (5.14) To find the shape of this region, set z = hλ = re iθ , and square both sides of equation 5.14 using the identity |z| 2 = zz∗ to get (1 + re iθ + r2 e 2iθ ) (1 + re −iθ + r2 e −2iθ ) ≤ 1 (5.15) 2 2 1 + re −iθ + r2 e −2iθ + re iθ + r 2 + r 3 e −iθ r2 e 2iθ 2 2 ( e −iθ + e iθ ) 2r + r 2 e−2iθ + e 2iθ + r 3 e−iθ + e iθ 2 2 2 + r 3 e iθ + r4 4 ≤ 1 (5.16) + r 2 + r4 4 ≤ 0 (5.17) 2r cos θ + r 2 cos 2θ + r 3 cos θ + r 2 + r4 4 ≤ 0 (5.18) To plot this equation pick some small interval δθ and do the following: • Set θ = 0 • Repeat 1. Set θ = θ + δθ 2. Solve 5.18 for r, accepting only real solutions 3. Plot the point (r, θ) until θ = 2π In Mathematica we can use Solve to find the solution of 5.18. For example, to find r(π), f[r , theta ] := 2r * Cos[theta] + r^2 * Cos[2*theta] + r^3 Cos[theta] + 0.25* r^4 + r^2; Solve[f[r,Pi]==0] Math 582B, Spring 2007 California State University Northridge c○2007, B.E.Shapiro Last revised: May 23, 2007
CHAPTER 5. RUNGE-KUTTA METHODS 91 which returns a list of rules for r , {{r->0.},{r->1.-1.73205i}, {r->1.+1.73205i}, {r->2.}} To convert this to a list of numbers, replace the Solve with r/.Solve[f[r,Pi]==0] This returns {0.,1.-1.73205i, 1.+1.73205i, 2.} To get only the real roots, Select[r/.Solve[f[r,Pi]==0], And[Im[#] == 0, Re[#] >= 0]&] which now returns the list {0., 2.} Figure 5.1: Region of absolute stability for second-order Taylor series methods. Numerical solutions with hλ that fall inside this region are absolutely stable Putting the whole plotting algorithm together, the following will plot 100 points on the solution. points = {}; Cartesian[{r , theta }] := {r*Cos[theta], r*Sin[theta]}; For[theta = 0, theta = 0]&]; 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 2 Successive Approximations
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CHAPTER 2. SUCCESSIVE APPROXIMATION
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Page 123 and 124: Chapter 6 Linear Multistep Methods
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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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