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On Effective Convergence of Numerical Solutions for Differential Equations

Published:01 March 2014Publication History
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Abstract

This article studies the effective convergence of numerical solutions of initial value problems (IVPs) for ordinary differential equations (ODEs). A convergent sequence {Ym} of numerical solutions is said to be effectively convergent to the exact solution if there is an algorithm that computes an N ∈ ℕ, given an arbitrary n ∈ ℕ as input, such that the error between Ym and the exact solution is less than 2-n for all mN. It is proved that there are convergent numerical solutions generated from Euler’s method which are not effectively convergent. It is also shown that a theoretically-proved-computable solution using Picard’s iteration method might not be computable by classical numerical methods, which suggests that sometimes there is a gap between theoretical computability and practical numerical computations concerning solutions of ODEs. Moreover, it is noted that the main theorem (Theorem 4.1) provides an example of an IVP with a nonuniform Lipschitz function for which the numerical solutions generated by Euler’s method are still convergent.

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      • Published in

        cover image ACM Transactions on Computation Theory
        ACM Transactions on Computation Theory  Volume 6, Issue 1
        March 2014
        92 pages
        ISSN:1942-3454
        EISSN:1942-3462
        DOI:10.1145/2600088
        Issue’s Table of Contents

        Copyright © 2014 ACM

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        Association for Computing Machinery

        New York, NY, United States

        Publication History

        • Published: 1 March 2014
        • Accepted: 1 December 2013
        • Revised: 1 November 2013
        • Received: 1 May 2013
        Published in toct Volume 6, Issue 1

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