In this paper an iterative technique for solving initial value problems is presented. The technique is based on Picard iterations and the Chebyshev polynomials and has been referred to by this author as the Picard-Chebyshev iteration method for initial value problems, although no such name for it appears in the literature. Its basic strength lies in the fact that it ignores completely the standard distinction between the linear and nonlinear initial value problem and that it produces a Chebyshev series solution which is easily evaluated. Its basic weakness lies in the fact that one must assume a finite series before computation can begin, and the accuracy of the solution depends completely on this assumption. In the literature this problem is overcome by simply repeating the solution with some appropriate increase in the length of the series until the accuracy desired is obtained. This proves to be an extremely lengthy and time-consuming procedure for even the simplest of initial value problems. A considerable improvement can be obtained by specifying the accuracy desired as well as the length of the series and then determine the interval over which the solution can be constructed which will meet the accuracy requirements. This is not only a workable approach to the problem but also has the added advantage that for certain accuracy requirements a much larger interval than the standard Chebyshev intervals of (0, 1) and (- 1, 1) can be used.
Proceedings of the Iowa Academy of Science
© Copyright 1977 by the Iowa Academy of Science, Inc.
Steel, Dennis R.
"Error Analysis for Picard-Chebyshev Iterations,"
Proceedings of the Iowa Academy of Science: Vol. 84:
, Article 7.
Available at: http://scholarworks.uni.edu/pias/vol84/iss3/7