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

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174 CHAPTER 9. DIFFERENTIAL ALGEBRAIC EQUATIONS 3. The local index of the pencil is ≥ 2 The following theorem effectively says that there are three broad classes of DAE’s: implicit ODE’s (index 0 DAEs); index 1 DAEs; and index 2 DAEs. Theorem 9.7. Let A(t)y ′ (t) + B(t)y(t) = f(t) be a solvable linear time-varying DAE with matrix pencil λA(t) + B(t) with index > 2 on some interval I. Then it is possible to find an analytically equivalent transformation of the DAE on some open subinterval of I with local index of 2. Theorem 9.8. Let A(t)y ′ (t) + B(t)y(t) = f(t) be a linear time-varying DAE where A(t) and B(t) are real and analytic. Then the DAE is solvable if and only if it is analytically equivalent to a system in standard canonical form using a real analytic coordinate change. Theorem 9.9. Let A(t)u ′ (t) + B(t)u(t) = f(t) be a solvable linear time-varying DAE. Then it is analytically equivalent to where N is nilpotent and C is a matrix. x ′ + Cy ′ = f(t) (9.76) Ny ′ + y = g(t) (9.77) 9.4 Hessenberg Forms for Linear and Nonlinear DAEs Recall that a matrix A is said to be a Hessenberg Matrix if A i,j = 0 for i > j + 1, i.e., only the upper triangular, diagonal, and sub-diagonal elements are non-zero: ⎛ ⎞ a 1,1 a 1,2 a 1,3 · · · a 1,n−1 a 1,n a 2,1 a 2,2 a 2,3 a 2,n−1 a 2,n 0 a3, 2 a3, 3 . . . ⎜ 0 0 .. . .. (9.78) . . ⎟ ⎝ 0 0 0 a n−1,n−2 a n−1,n−1 a n−1,n ⎠ 0 0 0 0 a n,n−1 a n,n Hessenberg forms arise commonly in physical problems due to the natural symmetry of the problems. Thus a great deal of current research is based on solving problems of this form. Definition 9.10. Let A(t)y ′ (t) + B(t)y(t) = f(t) be a linear time varying DAE. Then it is in Hessenberg Size n Form if it can be written in block form as ⎛ ⎞ ⎛ ⎞ I 0 · · · 0 x ′ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ B 11 B 12 · · · B 1,n−1 B 1,n x f(t) . 0 I .. . . B 21 B 22 · · · B 2,n−1 0 . ⎜ . .. . .. . . ⎟ . + 0 B 32 · · · B 3,n−1 . ⎝0 I 0⎠ ⎜ ⎟ ⎜ ⎝ . ⎠ ⎝ . 0 0 .. ⎟ . = . . ⎠ ⎜ ⎟ ⎜ ⎟ ⎝ . ⎠ ⎝ . ⎠ 0 · · · · · · 0 y ′ 0 · · · 0 B n,n−1 0 y g(t) (9.79) Math 582B, Spring 2007 California State University Northridge c○2007, B.E.Shapiro Last revised: May 23, 2007
CHAPTER 9. DIFFERENTIAL ALGEBRAIC EQUATIONS 175 where B n,n−1 B n−1,n−2 · · · B 21 is nonsingular Example 9.2. Write the Hessenberg forms of size 2 and 3 in block form. Solution. The size 2 form: The size 3 form: x ′ + B 11 x + B 12 y = f(t) (9.80) B 21 x = g(t) (9.81) x ′ + B 11 x + B 12 y + B 13 z = f(t) (9.82) y ′ + B 21 x + B 22 y = g(t) (9.83) B 32 y = h(t) (9.84) Theorem 9.10. A linear differential algebraic equation in Hessenberg form of size n is solvable and has local index n. For the general, nonlinear DAE F(t, y, y ′ ) = 0 (9.85) we say the DAE is in Hessenberg form of size n if it can be written explicitly as y ′ 1 = F 1 (t, y 1 , y 2 , . . . , y n ) (9.86) y ′ 2 = F 2 (t, y 1 , y 2 , . . . , y n−1 ) (9.87) . (9.88) y ′ i = F i (t, y i−1 , y i , . . . , y n−1 , t) (9.89) . (9.90) 0 = 0F n (t, y n−1 ) (9.91) and the product is nonsingular. ∂F n ∂x n−1 ∂F n−1 ∂x n−2 · · · ∂F 2 ∂x 1 ∂F 1 ∂x n (9.92) The Hessenberg form of size 1 is defined as y ′ = f(t, y, x) (9.93) 0 = g(t, y, x) (9.94) where the Jacobian ∂g/∂x is nonsingular. In principle, one could solve the constraint for x 4 and reduce the DAE to an ordinary differential equation in y. However, it will turn out that uniquenss of the solution is not guaranteed. 4 This is a consequence of the implicit function theorem and depends on the nonsingularity of the Jacobian. 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 3 Approximate Solutions 3.1
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CHAPTER 3. APPROXIMATE SOLUTIONS 45
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Chapter 5 Runge-Kutta Methods 5.1 T
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Chapter 6 Linear Multistep Methods
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CHAPTER 6. LINEAR MULTISTEP METHODS
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