Math 2.0

Ordinary differential equations Definition

Maximal solution

Builds on Solution of an ordinary differential equationInitial value problemUniqueness of solutions for Lipschitz vector fields

Definition

Let DD be an open subset of ℝ×ℝn\mathbb{R} \times \mathbb{R}^n and let f:D→ℝnf : D \to \mathbb{R}^n be continuous. Let II be an open interval and let γ:I→ℝn\gamma : I \to \mathbb{R}^n be a solution of x′=f(t,x)x' = f(t, x) whose graph lies in DD. The solution γ\gamma is maximal if there is no solution γ~:I~→ℝn\tilde{\gamma} : \tilde{I} \to \mathbb{R}^n on an open interval I~\tilde{I} with I⊊I~I \subsetneq \tilde{I} and γ~(t)=γ(t)\tilde{\gamma}(t) = \gamma(t) for each t∈It \in I. The interval II of a maximal solution is its maximal interval of existence.

Remarks

If ff is locally Lipschitz in xx, each initial value problem x(t0)=x0x(t_0) = x_0 with (t0,x0)∈D(t_0, x_0) \in D has exactly one maximal solution. Each other solution of the problem is a restriction of it. The maximal interval (t−,t+)(t_-, t_+) can depend on the initial value, and it can be smaller than ℝ\mathbb{R}. If t+<∞t_+ < \infty, then (t,γ(t))(t, \gamma(t)) leaves every compact subset of DD as t→t+t \to t_+. For D=ℝ×ℝnD = \mathbb{R} \times \mathbb{R}^n, this means ∥γ(t)∥→∞\|\gamma(t)\| \to \infty. See blow-up in finite time.

Reading path

To understand this page from the ground up, read these 10 pages in order. The foundations come first.

Show the path as a graph
Differential equation, concept, no evidence levelDifferential equationconceptFunction, concept, no evidence levelFunctionconceptInterval, concept, no evidence levelIntervalconceptOrdinary differential equation, definition, no evidence levelOrdinary differential…definitionLipschitz condition, definition, no evidence levelLipschitz conditiondefinitionGrönwall's inequality, statement, no evidence levelGrönwall's inequalitystatementSolution of an ordinary differential equation, definition, no evidence levelSolution of an ordina…definitionInitial value problem, definition, no evidence levelInitial value problemdefinitionUniqueness of solutions for Lipschitz vector fields, statement, no evidence levelUniqueness of solutio…statementMaximal solution, definition, no evidence levelMaximal solutiondefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Differential equation concept
  2. Function concept
  3. Interval concept
  4. Ordinary differential equation definition
  5. Lipschitz condition definition
  6. Grönwall's inequality statement
  7. Solution of an ordinary differential equation definition
  8. Initial value problem definition
  9. Uniqueness of solutions for Lipschitz vector fields statement
  10. Maximal solution definition

Where this leads

47 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Ordinary_differential_equation, Global uniqueness and maximum domain of solution
About this page
Tag019VSXSFMQ
URIm2://019VSXSFMQ
Aliasesmaximal-interval-of-existence
Kinddefinition
WritingW0 stub
MSC34A12
DefinesMaximal solution
UsesSolution of an ordinary differential equation, Initial value problem, Uniqueness of solutions for Lipschitz vector fields

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by