In mathematics, Picarditeration is the fixed point iteration discovered by Charles Émile Picard for the approximate solution of ordinary differential equations , which is also used in the proof of the local version of Picard-Lindelöf's theorem .
definition
Look through that
given initial value problem , where is a continuous and in the second argument Lipschitz continuous mapping and from a real time interval.
The Picarditeration is then given by
![x ^ {[0]} (t) = x_0](https://wikimedia.org/api/rest_v1/media/math/render/svg/31c87d42c115f08be0366d74df8f43657f9a0110)
![x ^ {[\ ell + 1]} (t) = x_0 + \ int_ {t_0} ^ tf (s, x ^ {[\ ell]} (s)) \, ds, \ quad t \ in [t_0, t_0 + \ varepsilon].](https://wikimedia.org/api/rest_v1/media/math/render/svg/adcfc488da7716cc33cd86531fba3b238daf0336)
The sequence of functions thus generated converges evenly towards the solution for sufficiently small ones .
example
Animation for the development of the sequence of functions generated by Picarditeration.
An ordinary differential equation is given by
with the starting value:
Two steps of Picard iteration are:
![x ^ {[0]} (t) = 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/73c350d8eb6b02b3cdfd147aa1c3b1679284af6e)
![x ^ {[1]} (t) = 1 + \ int_0 ^ t (\ sin (s) - 1) \, ds = 2 - \ cos (t) - t](https://wikimedia.org/api/rest_v1/media/math/render/svg/0cd1418c204ec4bce75f63925dfc9adf8c5a718d)
![x ^ {[2]} (t) = 1 + \ int_0 ^ t (\ sin (s) - (2 - \ cos (s) - s)) \, ds = 2 - \ cos (t) - 2t + \ sin (t) + \ frac {t ^ 2} {2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96adc903fe004015cc9a60a5357fd6fdb35df744)
Web links
