Question

Explain Picard’s method of successive approximation.

Answer 1

Recall from The Method of Successive Approximations page that by The Method of Successive Approximations (Picard's Iterative Method), if dydt=f(t,y) is a first order differential equation and with the initial condition y(0)=0 (if the initial condition is not y(0)=0 then we can apply a substitution to translate the differential equation so that y(0)=0 becomes the initial condition) and if both f and ∂f∂y are both continuous on some rectangle R for which −a≤t≤a and −b≤y≤b then limn→∞ϕn(t)=limn→∞∫t0f(s,ϕn−1(s))ds=ϕ(t) where y=ϕ(t) is the unique solution to this initial value problem.

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