Volume 10, Issue 1
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Approximating Solutions of Linear Ordinary Differential Equations with Periodic Coefficients by Exact Picard Iterates
2. Picard Iterates
Let and . If is a function with values in , then consider the following general initial value problem for a system of ordinary differential equations:
The Picard iterate of a function with values in related to the initial condition is a new function defined as
Taking any initial function , let , . It can be shown that if and in equation (1) are both continuous functions in a neighborhood of the initial point , then the sequence of Picard iterates converges in a neighborhood of to the solution of the initial value problem (1). This is in fact the usual proof of the existence-uniqueness theorem for differential equations . In general we choose as the initial function the constant function equal to the initial condition . This ensures that the very first iterates are good approximations to the solution for close enough to .
Although convergence to the solution of the problem is guaranteed (and exponentially fast in the number of iterates), Picard iteration is seldom used as a method for approximating solutions to differential equations. The problem with it lies in the integration contained in the very definition (2) of the Picard iterate, which may be laborious or even impossible to perform. Take as an example the pendulum equation
whose solution cannot be written in terms of elementary functions. We rewrite it as a first-order system of the form (1) by taking , . A function onepicard that performs the operation in equation (2) is defined as follows.
Both , the function on the right-hand side of the system of differential equations, and , the function that we want to iterate, should be supplied as pure functions. As we will also be interested in iterating the operator over an initial function that is a constant function equal to the initial condition , we define the function picard.
Using to simplify matters, here is the fifth Picard iterate for the pendulum equation with initial condition .
Notice that this result is an unevaluated integral! The reader may rework the last input to find that all integrals can be calculated up to the fourth Picard iterate. This shows that high-order Picard iterates may be difficult or impossible to calculate. By comparing the graphs of the fourth Picard iterate and the solution of the same problem by a traditional numerical method, such as the one implemented in NDSolve, the reader will also notice that the approximation is not accurate enough in a time interval as small as the period of the pendulum.
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