nLab
ordinary differential equation

Context

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

Equality and Equivalence

An ordinary differential equation is a differential equation involving derivatives of a function with respect to one argument only, i.e. the function is on a manifold of only 11 dimension. This function can be vector valued, what is sometimes viewed as a system of possibly coupled equations; still all of them have the derivatives taken with respect to the same parameter.

Existence and uniqueness (Picard-Lindelöf theorem)

A basic theorem concerns existence and uniqueness of local solutions to initial value problems. Let XX be a Banach space; denote the ball of radius rr about xXx \in X by B r(x)B_r(x). Given (t 0,x 0)×X(t_0, x_0) \in \mathbb{R} \times X and a,r>0a, r \gt 0, put Q(a,r)[t 0a,t 0+a]×B r(x 0)Q(a, r) \coloneqq [t_0 - a, t_0 + a] \times B_r(x_0).

Theorem

(Picard-Lindelöf) Suppose f:Q(a,b)Xf: Q(a, b) \to X is a continuous function satisfying the following conditions:

  • (Lipschitz condition) There is a Lipschitz constant LL such that

    f(t,x)f(t,x)Lxx\|f(t, x) - f(t, x')\| \leq L\|x - x'\|

    for all (t,x)Q(a,b)(t, x) \in Q(a, b);

  • (Boundedness) There is a constant KK such that sup (t,x)Q(a,b)f(t,x)K\sup_{(t, x) \in Q(a, b)} \|f(t, x)\| \leq K.

Then for any y 0Xy_0 \in X and cmin(a,b/K)c \coloneqq \min(a, b/K), there exists exactly one solution y:[t 0c,t 0+c]Xy: [t_0 - c, t_0 + c] \to X to the initial value problem

y(t)=f(t,y(t)),y(t 0)=y 0.y'(t) = f(t, y(t)), \qquad y(t_0) = y_0.
Proof

Revised on October 8, 2014 14:06:48 by Todd Trimble (67.81.95.215)