Sandwich/SqueezeTheorem,.
Picard’s Method
The followingmethodappearedin 1891by EmilePicard,a famousFrenchmathematician.It is a method
for solvinginitialvalueproblemsin differentialequationsthat producesa sequenceof functionswhichconverge
to the solution.Startwith the initialvalueproblem:
y= f (x, y) with y(x 0 ) = y 0
Iff(x,y) andfx(x,y) are both continuousthen a uniquesolutionto the initialvalueproblemexistsby Picard’s
theory. Now ify(x) is the solutionto the givenproblem,then a reformulationof the differentialequationis
possible:
Nowthe FundamentalTheoremof Calculusis utilizedto integratethe left handside of the problemand
uponisolating, the followingresultis obtained:
The equationaboveis the startingpointfor the Picarditerationbecauseit will be usedto buildthe sequence
of functionswhichwill describethe actualsolutionto the initialvalueproblem.The Picardsequenceof
functionsis calculatedas follows:
Step 1- DefineY 0 (x) =y 0
Step 2 - SubstituteY 0 (t) =y 0 fory(t) inf(t,y(t)):
Step 3 - Repeatstep 2 withY 1 (t) fory(t) :
The substitutionprocessis repeatedntimesand generatesa sequenceof functions{Yn(x)} whichconverges
to the initialvalueproblem.To summarizethis proceduremathematically,
Picard’s Method