Volume 10, Issue 1
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Staff and Contributors
Approximating Solutions of Linear Ordinary Differential Equations with Periodic Coefficients by Exact Picard Iterates
We explained our implementation of a method for producing very good approximations to solutions of linear differential equations with periodic coefficients. The method is based on calculating exact high-order Picard iterates by using a fast integration function and also works for equations or initial conditions depending on parameters. As a first accuracy test, we compared the numerical values for the first Mathieu characteristic numbers as predicted by our method (with 20 iterates) and the ones calculated by built-in Mathematica functions. The agreement was very good for the parameter values considered. Of course, the built-in functions are much faster, but we cannot forget that the proposed method is very general and the Mathieu equation is just an interesting example.
We also visually compared the stability chart for the vertically driven upside-down double pendulum obtained by two different methods, both based on calculating Picard iterates. The first method was to directly calculate the Picard iterates for a system of four equations and the second was to decouple the system of four equations into two Mathieu equations, for which we had already calculated the necessary Picard iterates. Again, the results with both methods were visually indistinguishable, providing an indirect check of the accuracy of the Picard iterates. We also checked directly that one of the points in that stability chart agreed numerically with the exact calculated value within the tolerance of the bisection method used.
By running the FastPicard package with Mathematica 4.1 on a 1.4 GHz Pentium 4 with 256 MB of RAM, we were able to calculate up to 50 iterates for the Mathieu equation in less than one hour. Results were reported in .
We should also note that there exist mathematically rigorous upper bounds on the difference between the exact solution of an equation and the th Picard iterate approximating it. These bounds decrease exponentially fast in . In principle, they can be used with symbolic Picard iterates to give computer-assisted proofs of properties of solutions for differential equations depending on parameters.
We used the phrase "in principle" in the previous paragraph for the following reason. The reader should have noticed from the example with the Mathieu equation that the errors committed in approximating solutions by a fixed number of Picard iterates increase with time. In our stability examples, it was necessary to approximate the solution of the differential equations up to time . This is indeed quite a long time interval for the number of iterates we are able to calculate. As a consequence, the error bounds are not very interesting. In applications where the time interval is shorter, it is possible that the error bounds could produce interesting proofs.
There also exist rigorous bounds on the errors committed by most traditional numerical methods. But these methods are difficult to implement in the case of equations depending on parameters and, even when the equations do not depend on parameters, the calculations in these methods are subject to roundoff errors that are difficult to overcome.
Therefore we propose exact Picard iteration as an alternative method for producing approximate solutions to linear differential equations with periodic coefficients depending on parameters. For problems in which the time intervals are short, we may even provide interesting upper bounds for the errors due to calculating only a finite number of iterates.
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