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

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156 CHAPTER 8. BOUNDARY VALUE PROBLEMS Figure 8.1: Three estimates to the solution using shooting, top to bottom curves. 8.2 Basic Theory of Boundary Value Problems In vector form the basic two point boundary value problem is given by y ′ = f(t, y) (8.14) g(y(a), y(b)) = 0 (8.15) for some function (generally nonlinear) g(u, v). When the boundary condition is linear then for some given data d B a y(a) + B b y(b) = d (8.16) for some square matrices B a and B b . In general, however, we will define B a = ∂g ∂u B b = ∂g ∂v for both linear and nonlinear boundary conditions. Example 8.1. For the boundary value problem considered in section 8.1 ( ) ( ) d u v = y ′ = f(t, y) = u dt v t 2 − v t we have linear boundary conditions ( ) 1 0 B a = ; B 0 0 b = because ( ) ( ) 1 0 u(a) 0 0 v(a) + ( ) ( 0 0 1 ; d = 1 0 1) ( ) ( ) 0 0 u(b) 1 0 v(b) ( 1 = 1) (8.17) (8.18) (8.19) (8.20) (8.21) Math 582B, Spring 2007 California State University Northridge c○2007, B.E.Shapiro Last revised: May 23, 2007
CHAPTER 8. BOUNDARY VALUE PROBLEMS 157 We assume that f is bounded and Lipshitz; hence there is a unique solution to the initial value problem y ′ (t) = f(t, y) (8.22) y(a) = c (8.23) for some unknown vector c, which we will denote by y(t, y(t); c). Substituting this back into equation 8.15 gives g(c, y(b, y(b); c)) = 0 (8.24) This is a nonlinear equation in n unknowns (n is the dimension of y), which, in general, may have any number (or no) solutions and hence is a very difficult problem to solve. We will limit our study to linear differential equations, which have the form y ′ (t) = A(t)y(t) + q(t) (8.25) for some square matrix A and some vector q. For example, the problem studied in the previous section can be written ( ) ( ( ) d u 0 1 u = dt v 1/t −1/t) 2 (8.26) v Note that the matrix does not have be constant nor does it have to be linear in t for the system to be linear. It is only linearity in the dependent variables (e.g., u and v) that matters for the problem to be considered linear. A fundamental matrix Y (t) of the differential equation is the square matrix that satisfies Y ′ (t) = A(t)y(t) (8.27) The columns of the fundamental matrix are the linearly independent solutions to the homogeneous equation. In terms of the fundamental matrix, the general solution of the differential equation 8.25 is y(t) = Y (t) [ c + ∫ t where c is determined by the boundary conditions. a ] Y −1 (s)q(s)ds (8.28) Example 8.2. Find the fundamental matrix and general solution of the problem studied in the previous section. Solution. The fundamental matrix is ⎛ t 2 + 1 ⎜ Y (t) = ⎝ 2t t 2 − 1 2t t 2 − 1 t 2 + 1 2t 2 2t 2 ⎞ ⎟ ⎠ (8.29) 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 5. RUNGE-KUTTA METHODS 101
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