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

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2 CHAPTER 1. CLASSIFYING THE PROBLEM that F be sufficiently smooth 2 and that the matrix of partial derivatives ∂F/∂(y ′ ) (Jacobian matrix) be nonsingular 3 (∂F/∂(y ′ ) for a scalar equation). Then by the implicit function theorem we can solve for y ′ locally. An equation of the form 1.1 for which the Jacobian is nonsingular is thus called an ordinary differential equation, and we will focus on equations of this form in the first several chapters of these notes. It turns out that an equation for which the Jacobian is singular actually has hidden constraints: it is really a combination of differential equations and algebraic constraints, and is called a differential algebraic equation. Theorem 1.1. Implicit Function Theorem on R. 4 Let F (t, y) have continuous derivatives ∂F/∂t and ∂F/∂y in the neighborhood of a point (x 0 , y 0 ), where F (t 0 , y 0 ) = 0, Then there are intervals I and J where ∂F (t 0 , y 0 ) ∂y ≠ 0 (1.4) I = [t 0 − a, t 0 + a] (1.5) J = [y 0 − b, y 0 + b] (1.6) and a R rectangle R = I × J, such that the equation F (t, y) has precise one solution y = f(t) lying in the rectangle R, such that for all t ∈ I. Example 1.1. Solve y ′ = y. F (t, y(t)) = 0 (1.7) y(t) ∈ J (1.8) F y (t, f(t)) ≠ 0 (1.9) Solution. Writing y ′ = dy/dt and integrating we find ∫ 1 y dy = ∫ dt (1.10) ln|y| = t + C (1.11) y = Ke t (1.12) where K = ±e C . There is no restriction on the values of either C or K. 2 Throughout these notes we will assume that F is sufficiently smooth without explicitly stating so. By “sufficiently” smooth we mean that F is continuously differentiable enough times to give us the results we want. 3 Strictly speaking, nonsingularity is not really a requirement. Nonsingularity is sufficient to ensure that a solution exists, but is not required. There are examples of functions with singularities at points but for which solutions may exist. 4 For a proof, see Richard Courant and Fritz John, Introduction to Calculus and Analysis, Volume II/1, Springer Classics in Mathematics, 1998, page 225 Math 582B, Spring 2007 California State University Northridge c○2007, B.E.Shapiro Last revised: May 23, 2007
CHAPTER 1. CLASSIFYING THE PROBLEM 3 Thus we see that equation 1.1 (or 1.3) will often admit to an infinite number of solutions owing to arbitrary constants of integration that arise during its solution. For example 1.1 this is illustrated if figure 1.1, which shows the one parameter family of solutions to the example. A particular physical problem may only correspond to one member of this family. To fix down this constant, the problem must be further constrained. Such a constraint can take various forms. The nature of the constraint can have an enormous impact on our ability to solve the equation. Figure 1.1: One parameter family of solutions to y ′ = y, showing the solutions for various values of the constant of integration. c○2007, B.E.Shapiro Last revised: May 23, 2007 Math 582B, Spring 2007 California State University Northridge
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CHAPTER 6. LINEAR MULTISTEP METHODS
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Bibliography [1] Ascher, Uri M., Ma
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