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

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14 CHAPTER 1. CLASSIFYING THE PROBLEM delay equations might also be called initial function problems or just initial problems 5 . Let 0 ≤ τ 1 < τ 2 < · · · < tau m (1.81) be a set of m positive numbers representing the delays, the largest delay being denoted by τ m . If y appears explicitly without any delay then τ 1 = 0. Then we may define the scalar initial function problem (IFP) then as y ′ (t) = f(t, y(t − τ 1 , t − τ 2 , . . . , t − τ m )) (1.82) y(t) = g(t), t 0 − τ m ≤ t ≤ t 0 (1.83) Then there is a general existence and uniqueness theorem, which we state here without proof. Theorem 1.7. Let D ∈ [t 0 , t 0 + β) × R m be convex; suppose that f : D ↦→ R is continuous on D; and that g(t) : [t 0 − tau m ] ↦→ D. Then if r 1 ≠ 0 the DDE (equations 1.82, 1.83) has a unique solution on [t 0 , t 0 + α), where 0 < alpha ≤ β. Furthermore, if α < β then y(t) approaches the boundary of D as t → α. If τ 1 = 0 then the DDE has at most one solution. 1.7 Numerical Solutions of Differential Equations Roughly speaking, a numerical solution is a set of points on the curve that defines the solution to the differential equation (we will formalize this statement more precisely in a later section). The t-values are called a mesh or grid, and the interval between successive points is called the step size or mesh interval. By connecting the dots, using some appropriate interpolation scheme, we can recover the values of the solution at any point in its domain. We will call a numerical method consistent if it satisfies the differential equation and all associated constraints We will say that a method converges if we can obtain any desired degree of accuracy by picking the step size sufficiently small. A problem will be called well posed if there for any small change in the problem causes a bounded change in the actual solution. A numerical solution will be called stable if any small change in the problem causes a bounded change in the numerical solution. The bulk of this class will focus on finding and analyzing various techniques for solving a differential equation numerically. 5 this terminology follows R. D. Driver (1978) Introduction to Ordinary Differential Equations, Harper and Row Math 582B, Spring 2007 California State University Northridge c○2007, B.E.Shapiro Last revised: May 23, 2007
CHAPTER 1. CLASSIFYING THE PROBLEM 15 Figure 1.7: Illustration of a numerical solution. The actual solution is illustrated by the solid curve, the numerical by a set of points. The reconstruction of the solution at intermediate values is accomplished via an interpolation scheme (dashed curve). 1.8 Computer Assisted Analysis The bulk of this course will focus on finding numerical solutions, and computer programs will aide us in implemented and studying these algorithms. Additionally, computers allow us to solve and study differential equations analytically, and to examine the qualitative format of the solutions. In this section we will give some examples in Mathematica. Analytic Solution of ODEs with Mathematica The principal function here is DSolve.For example, to find the one parameter family of solutions to y ′ = 3ty (1.84) one would enter: DSolve[y’[t]==3 t y[t], y[t], t] (1.85) Recall that in Mathematica, the multiplication operator * is optional, and if there is a space between operators, then multiplication is implied. The expressions and 3 t y[t] 3*t*y[t] are equivalent. The response to input 1.85 is 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
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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 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 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 9 Differential Algebraic Eq
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Bibliography [1] Ascher, Uri M., Ma
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