nLab
ordinary differential equation

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Equality and Equivalence

Contents

Idea

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 only of dimension d=1d = 1. 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. (Note that a higher-order differential equation can be turned into a system of first-order equations.)

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; given (t 0,y 0)×X(t_0, y_0) \in \mathbb{R} \times X and a,r>0a, r \gt 0, put Q[t 0a,t 0+a]×B r(y 0)Q \coloneqq [t_0 - a, t_0 + a] \times B_r(y_0)m where B r(y 0)B_r(y_0) is the closed ball in XX of radius rr about y 0y_0.

Theorem

(Picard–Lindelöf) Suppose f:QXf: Q \to X is a function satisfying the following conditions:

  • (Continuity in tt): Given any xB r(y 0)x \in B_r(y_0), the function f(,x)f(-,x) (that is tf(t,x)t \mapsto f(t, x)) is continuous from [t 0a,t 0+a][t_0 - a, t_0 + a] to XX.

  • (Lipschitz continuity in yy): 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(t, x) \in Q;

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

Then for any cmin(a,r/K)c \leq \min(a, r/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 sketch

We will define an infinite sequence of approximate solutions to the problem, prove that its limit exists, prove that this limit is an exact solution, and prove that this solution is unique.

  1. The infinite sequence is given by Picard iteration: Starting with the given constant y 0y_0, recursively define

    y n+1(t)y 0+ t=t 0f(t,y n(t))dt; y_{n+1}(t) \coloneqq y_0 + \int_{t=t_0} f(t, y_n(t)) \,\mathrm{d}t ;

    that is, y n+1y_{n+1} is that indefinite integral? of f(,y n)f(-,y_n) that takes the correct initial value. (To define y 1y_1, use abuse of notation? to interpret y 0(t)y_0(t) as y 0y_0; that is, think of y 0y_0 as a constant function.) To prove that this integral exists, use the continuity conditions and an inductive proof? that each y ny_n is continuous to show that we are integrating a continuous function.

  2. Thinking of Picard iteration as an operator between Banach spaces of continuous functions, use Lipschitz continuity and boundedness to show that the Banach fixed point theorem? applies, so that the sequence (y 1,y 2,)(y_1, y_2, \ldots) uniformly converges to a limit yy.

  3. Since uniform convergence of continuous functions behaves well with integration, the limiting instance of Picard iteration holds:

    y(t)=y 0+ t=t 0f(t,y(t))dt. y(t) = y_0 + \int_{t=t_0} f(t, y(t)) \,\mathrm{d}t .

    By differentiating with respect to tt, and by evaluating at t 0t_0, we confirm that yy is a solution of the initial value problem.

  4. Given any putative solution zz, apply Grönwall's inequality? to zyz - y to prove that z=yz = y.

References

  • The English Wikipedia has a more detailed proof. (They assume that XX is \mathbb{R}, but this is not essential.)

Revised on August 1, 2017 13:24:18 by Toby Bartels (64.89.54.51)