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Math 519, Picard Iteration

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In calculus we learn a number of tricks that allow us to solve differential equations with more complicated right hand sides. These tricks all involve some manipulation of the differential equation that turns it into the simplest case (1) (1) , so we can solve it by integrating. The most important cases are separable differential equations and first order linear differential equations. Click here for a review. The existence and uniqueness theorems deal with the equations that we cannot solve using calculus. Not knowing any solution to the ODE, we begin with a first guess, namely x 0 (t)= x 0 𝑥 0 ( 𝑡 ) = 𝑥 0 . We try to improve on this guess by solving d x 1 dt =f(t, x 0 ), x 1 ( t 0 )= x 0 , 𝑑 𝑥 1 𝑑 𝑡 = 𝑓 ( 𝑡 , 𝑥 0 ) , 𝑥 1 ( 𝑡 0 ) = 𝑥 0 , which gives us a new function x 1 (t) 𝑥 1 ( 𝑡 ) . The right hand side in this differential equation only contains x 0 𝑥 0 , so it is known and we can find x 1 (t) 𝑥 1 ( 𝑡 ) by integrating. The function x 1 (t) 𝑥 1 ( 𝑡 ) tha

Math 519, Picard Iteration Existence of solutions to Differential Equations Contents The simplest case The general case The existence theorem. Proof by Picard iteration of the Existence Theorem Details of Picard's proof Choice of $\epsilon$ The $x_k(t)$ are well-defined The difference $y_k$ between two successive approximations An estimate for $m_1$ Estimating $m_k$ for $k>1$ The series $\sum y_k(t)$ converges; the sequence $x_k(t)$ converges How fast does the sequence converge? Proving that the limit is a solution Sticky points The simplest case If the right hand side of a differential equati

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