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

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

82 CHAPTER 4. IMPROVING ON EULER’S METHOD We could implement this in Mathematica as follows: =⇒ =⇒ =⇒ =⇒ =⇒ BackwardEulerFP[f , {t0 , y0 }, h , tmax , tol , imax ] := Module[{ time, yval, yvaln, yvalp, delta, eps, r}, r = {{t0, y0}}; yval = y0; For[time = t0, time < tmax, time += h, (***** Do next Backward Euler Step *****) yvaln = yval + f[time+h, yval]; (***** Do Fixed Point Iteration on y *****) For[i = 1, i ≤ imax, i++, yvalp = yvaln; yvaln = yval + f[time+h, yvalp]; delta = yvaln - yvalp; If[Abs[delta] < tol, Break[]]; ]; yval = yvaln; AppendTo[r, {time + h, yval}]; ]; Return[r]; ] We know that equation 4.103 is only guaranteed to have a fixed point, and iteration is only guaranteed to converge, if it is a contraction mapping. This requires ‖hf y (t, y)‖ < 1. Convergence becomes rapid when ‖hf y (t, y)‖ y0, t -> t0; If[Abs[JV] >= 1, Print["This problem is ill-conditioned for fixed point iteration."]; Return[$Failed]; ]; This calculates the scalar Jacobian analytically, and places the formula in the variable J , and then calculates the numerical value of the Jacobian in the variable JV . If the magnitude of the Jacobian is larger than or equal to one, then fixed point iteration is not guaranteed to work, so the program aborts. Later in the algorithm, after the next step is calculated, one also needs to test to see if the problem has entered an ill-conditioned region. To do this one can recalculate the value of the Jacobian, using the analytic formula already determined, merely substituting the current values of y and t. JV = J/.y-> ynext, t-> time If[Abs[JV]>= 1 ... 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 83 ✬ Algorithm 4.1. Backward Euler Method Using Fixed Point Iteration. To solve the initial value problem y ′ = f(t, y), y(t 0 ) = y 0 on an interval [t 0 , t max ] with a fixed step size h. 1. input: f(t, y), t 0 , y 0 , h, t max , tol 2. output: (t 0 , y 0 ) 3. let t = t 0 , y = y 0 4. while t < t max (a) let i = 0 (b) let y (0) next = y + hf(t + h, y) ⇐= (c) Repeat: y (i+1) next i = i + 1 = y + hf(t + h, y (i) next ) ⇐= ✩ ✫ until |y (i+1) next (d) let y = y (i+1) next (e) let t = t + h (f) let t n = t, y n = y − y (i) next | < tol (g) output: (t 0 , y 0 ), . . . , (t n , y n ) ✪ Theorem 4.6 (Newton’s Method). To find the root r of the scalar equation g(t) = 0, iterate on r (i+1) = r (i) − g(r(i) ) g ′ (r (i) ) To find the root r of the vector equation g(y) = 0, iterate on (4.106) ( ) ∣ ∂g(y) −1∣∣∣∣y=r r (i+1) = r (i) − g(r (i) ) ⇐= (4.107) ∂y (i) For the scalar IVP, we can use Newton’s method to find the root of g(y n ) = y n − y n−1 − hf(t, y n ) = 0 (4.108) 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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CHAPTER 2. SUCCESSIVE APPROXIMATION
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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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