Math 2.0

Ordinary differential equations Existence, uniqueness and dependence on data Concept

Picard iteration

Builds on Picard operatorIntegral equation form of an initial value problemAn iterate of the Picard operator is a contraction

Summary

Picard iteration is the construction of a solution of x′=f(t,x)x' = f(t, x), x(t0)=x0x(t_0) = x_0 as a limit of successive approximations. One starts with the constant curve φ0(t)=x0\varphi_0(t) = x_0 and sets φk+1=Pφk\varphi_{k+1} = P \varphi_k, where PP is the Picard operator: φk+1(t)=x0+∫t0tf(s,φk(s)) ds\varphi_{k+1}(t) = x_0 + \int_{t_0}^{t} f(s, \varphi_k(s))\, ds. Under the Picard–Lindelöf hypotheses, the curves φk\varphi_k converge uniformly on the time interval to the unique solution.

Remarks

Example: for x′=xx' = x, x(0)=1x(0) = 1, the iterates are φk(t)=∑j=0ktj/j!\varphi_k(t) = \sum_{j=0}^{k} t^j / j!. They converge to ete^t. Let KK be the Lipschitz constant and TT the length of the time interval. The error after kk steps is at most C(KT)k/k!C (K T)^{k} / k! for a constant CC. See an iterate of the Picard operator is a contraction.

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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 conditiondefinitionSolution of an ordinary differential equation, definition, no evidence levelSolution of an ordina…definitionInitial value problem, definition, no evidence levelInitial value problemdefinitionPicard–Lindelöf hypotheses, definition, no evidence levelPicard–Lindelöf hypot…definitionPicard operator, definition, no evidence levelPicard operatordefinitionIntegral equation form of an initial value problem, statement, no evidence levelIntegral equation for…statementAn iterate of the Picard operator is a contraction, statement, no evidence levelAn iterate of the Pic…statementPicard iteration, concept, no evidence levelPicard iterationconcept
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  1. Differential equation concept
  2. Function concept
  3. Interval concept
  4. Ordinary differential equation definition
  5. Lipschitz condition definition
  6. Solution of an ordinary differential equation definition
  7. Initial value problem definition
  8. Picard–Lindelöf hypotheses definition
  9. Picard operator definition
  10. Integral equation form of an initial value problem statement
  11. An iterate of the Picard operator is a contraction statement
  12. Picard iteration concept

Further reading

  1. Web https://en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem, Proof sketch
  2. Web https://en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem, Example of Picard iteration
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UsesPicard operator, Integral equation form of an initial value problem, An iterate of the Picard operator is a contraction

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