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

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12 CHAPTER 1. CLASSIFYING THE PROBLEM Solution. As before we obtain y = cos t + b sin t (1.64) from the first boundary condition. But now the second condition gives hence there is precisely one solution: 1 = cos 2 + b sin 2 (1.65) b = 1 − cos 2 ≈ 1.55741 sin 2 (1.66) y = cos t + 1 − cos 2 sin 2 sin t (1.67) Thus we see that the solution of a BVP is not only not necessarily unique, but there may be an infinite number of solutions. To understand why that is the case requires delving into the theory of orthogonal functions and generalized Fourier analysis, which are subjects for other classes. Boundary value problems are also closely related to partial differential equations. 1.5 Differential-Algebraic Equations Definition 1.7 (Differential-Algebraic Equations). Let Let D ∈ R 2n+1 be a set and suppose that F (t, y, y ′ ) : D ↦→ R n . Then F (t, y, y ′ ) = 0 (1.68) where the Jacobian matrix ∂ y ′F is singular ∣ det ∂(F 1 , F 2 , . . . ) ∣∣∣ ∣ ∂(y 1 ′ , y′ 2 . . . ) = 0 (1.69) is called an explicit differential-algebraic equation (DAE). If there are some functions f(t, y, z) and g(t, y, z) and a set of additional algebraic variables z 1 (t), z 2 (t), . . . such that equation 1.68 can be rewritten as then the systems is called a semi-explicit DAE. y ′ = f(t, y, z) (1.70) 0 = g(t, y, z) (1.71) It is always ( possible to rewrite equation 1.70 as a fully explicit DAE in terms of y the vector u = , where z are the algebraic constraints, specifically, z) ( ) f(t, y, z) − y ′ F (t, y, z) = g(t, y, z) (1.72) Math 582B, Spring 2007 California State University Northridge c○2007, B.E.Shapiro Last revised: May 23, 2007
CHAPTER 1. CLASSIFYING THE PROBLEM 13 Then we have F (t, y, z) = 0 (1.73) but the Jacobian matrix becomes nonsingular: ∂F (t, y, z) ∂(y, z) = ( ) I 0 0 0| (1.74) The general theory of differential algebraic equations is still being developed. A system of the form Ay ′ (t) + By(t) = f(t) (1.75) where A and B are square matrices and A is singular, is called a linear constant coefficient DAE. If the determinant det λA + B ≠ 0 (1.76) is not identically zero (considered as a function of the variable λ), then 1.75 is solvable. 1.6 Delay Differential Equations Definition 1.8. (Functional differential equation) If a differential systems can be written explicitly as a function of a function of the abscissa variable, noteably y ′ = f(t, y(t), y(u(t)) (1.77) then it is know as a functional differential equation (FDE). If the it can be written as y ′ = f(t, y(t), y(t − τ 1 , t − τ 2 , . . . )) (1.78) then it is known as a delay differential equation (DDE). Example 1.8. A typical functional differential equation is A typical DDE is given by y ′ = y(t) + y(sin(t)) (1.79) y ′ = y(t) + y(t − 3) + 7y(t − 5) (1.80) DDEs arise frequently in control theory and population growth. The theory of functional differential equations is not as well advanced as other types of DEs and there are fewer numerical methods for solving them. The best understood functional equations are delay equations. Delay equations are similar to initial value problems, in that the value of the function must be known at the starting point. They are more complicated, however, in that the value must also be known over an entire interval preceding t 0 , of duration given by the greatest delay in the equation. Thus c○2007, B.E.Shapiro Last revised: May 23, 2007 Math 582B, Spring 2007 California State University Northridge
Page 1 and 2: The Computable Differential Equatio
Page 3 and 4: Contents 1 Classifying The Problem
Page 5 and 6: CONTENTS v Timeline on Computable D
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Chapter 4 Improving on Euler’s Me
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CHAPTER 4. IMPROVING ON EULER’S M
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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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CHAPTER 8. BOUNDARY VALUE PROBLEMS
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Bibliography [1] Ascher, Uri M., Ma
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