The Computable Differential Equation Lecture ... - Bruce E. Shapiro
The Computable Differential Equation Lecture ... - Bruce E. Shapiro The Computable Differential Equation Lecture ... - Bruce E. Shapiro
6 CHAPTER 1. CLASSIFYING THE PROBLEM Figure 1.3: The one-parameter family of solutions for y ′ = (3 − y)/2 for different values of the constant of integration, and the solution to the initial value (heavy line) problem through (t 0 , y 0 ) = (2π, 4). The initial condition is indicated by the large gray dot. • The solution is unique; • The solution depends continuously on the data. If a problem is not well posed then there is no point in trying to solve it numerically, so we begin our study of initial value problems by looking at what it takes to make a problem well posed. We will find that a Lipshitz Condition, defined below in definition 1.5 is sufficient to ensure that the problem is well posed. 1.3 The Fundamental Theorem The importance (and usefulness) of initial value problems is enhanced by a general existence theorem and the fact that under appropriate conditions (namely, a Lipshitz Condition) the solution is unique. While we will defer the proof of this statement until later, we will present one of many different versions of the fundamental existence theorem. Definition 1.5. [Lipshitz Condition]. A function f(t, y) on D is said to be Lipshitz (or Lipshitz continuous, or satisfy a Lipshitz condition) on y if there exists some constant K > 0 if for all (x, y 1 ), (x, y 2 ) ∈ D then |f(x, y 1 ) − f(x, y 2 )| ≤ K|y 1 − y 2 | (1.24) The constant K is called a this as f ∈ L(y; K)(D). Lipshitz Constant for f. We will sometimes denote Math 582B, Spring 2007 California State University Northridge c○2007, B.E.Shapiro Last revised: May 23, 2007
CHAPTER 1. CLASSIFYING THE PROBLEM 7 Theorem 1.3. Fundamental Existence and Uniqueness Theorem Suppose that f(t, y) ∈ L(y; K)(R) for some convex domain R. Then for any point (t 0 , y 0 ) ∈ R there exists a neighborhood N of (t 0 , y 0 ) and a unique differentiable function φ(t) on N satisfying y ′ = f(t, φ(t)) (1.25) such that y ′ (t 0 ) = y 0 . The existence theorem is illustrated in figure 1.4. Given any initial value, there is some solution that passes through the point. Observe that the existence of the solution is not guaranteed globally, only within some open neighborhood of the initial condition. Figure 1.4: Illustration of the existence of a solution. Theorem 1.4 (Continuous dependence on IC). Under the same conditions, the solution depends continuously on the initial data, i.e., if ỹ is a solution satisfying the same ODE with ỹ(t 0 ) = ỹ 0 , then |y(t) − ỹ(t)| ≤ e Kt |y 0 − ỹ 0 | (1.26) Theorem 1.5 (Perturbed Equation). Under the same conditions, suppose that ỹ is a solution of the perturbed ODE, ỹ ′ = f(t, ỹ) + r(t, ỹ) (1.27) where r is bounded on D, i.e., there exists some M > 0 such that |r(t)| ≤ M on D. Then |y(t) − ỹ(t)| ≤ e Kt |y 0 − ỹ 0 | + M K (eKt − 1) (1.28) Proving that a function is Lipshitz is considerably eased by the following theorem. c○2007, B.E.Shapiro Last revised: May 23, 2007 Math 582B, Spring 2007 California State University Northridge
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CHAPTER 1. CLASSIFYING THE PROBLEM 7<br />
<strong>The</strong>orem 1.3. Fundamental Existence and Uniqueness <strong>The</strong>orem Suppose<br />
that f(t, y) ∈ L(y; K)(R) for some convex domain R. <strong>The</strong>n for any point (t 0 , y 0 ) ∈ R<br />
there exists a neighborhood N of (t 0 , y 0 ) and a unique differentiable function φ(t) on<br />
N satisfying<br />
y ′ = f(t, φ(t)) (1.25)<br />
such that y ′ (t 0 ) = y 0 .<br />
<strong>The</strong> existence theorem is illustrated in figure 1.4. Given any initial value, there<br />
is some solution that passes through the point. Observe that the existence of the<br />
solution is not guaranteed globally, only within some open neighborhood of the<br />
initial condition.<br />
Figure 1.4: Illustration of the existence of a solution.<br />
<strong>The</strong>orem 1.4 (Continuous dependence on IC). Under the same conditions, the<br />
solution depends continuously on the initial data, i.e., if ỹ is a solution satisfying<br />
the same ODE with ỹ(t 0 ) = ỹ 0 , then<br />
|y(t) − ỹ(t)| ≤ e Kt |y 0 − ỹ 0 | (1.26)<br />
<strong>The</strong>orem 1.5 (Perturbed <strong>Equation</strong>). Under the same conditions, suppose that ỹ is<br />
a solution of the perturbed ODE,<br />
ỹ ′ = f(t, ỹ) + r(t, ỹ) (1.27)<br />
where r is bounded on D, i.e., there exists some M > 0 such that |r(t)| ≤ M on D.<br />
<strong>The</strong>n<br />
|y(t) − ỹ(t)| ≤ e Kt |y 0 − ỹ 0 | + M K (eKt − 1) (1.28)<br />
Proving that a function is Lipshitz is considerably eased by the following theorem.<br />
c○2007, B.E.<strong>Shapiro</strong><br />
Last revised: May 23, 2007<br />
Math 582B, Spring 2007<br />
California State University Northridge
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