Study guide: Solving nonlinear ODE and PDE problems

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Picard iteration

Nonliner equation from Backward Euler scheme for logistic ODE: $$ F(u) = au^2 + bu + c = 0$$

Let $ u^{-} $ be an available approximation of the unknown $ u $.

Linearization of $ u^2 $: $ u^{-}u $ $$ F(u)\approx\hat F(u) = au^{-}u + bu + c = 0$$

But

Problem: the solution $ u $ of $ \hat F(u)=0 $ is not the exact solution of $ F(u)=0 $

Solution: set $ u^{-}=u $ and repeat the procedure