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Algorithm 982: Explicit Solutions of Triangular Systems of First-Order Linear Initial-Value Ordinary Differential Equations with Constant Coefficients

Published:16 August 2017Publication History
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Abstract

A method to compute explicit solutions of homogeneous triangular systems of first-order linear initial-value ordinary differential equations with constant coefficients is described. It is suitable for the limited case of well separated eigenvalues, or for multiple zero eigenvalues provided the entire column corresponding to a zero eigenvalue is zero. The solution for the case of constant inhomogeneity is described. The method requires only the computation of a constant matrix using a simple recurrence. Computing the solutions of the system from that matrix, for values of the independent variable, requires one to exponentiate only the diagonal of a matrix. It is not necessary to compute the exponential of a general triangular matrix. Although this work was motivated by a study of nuclear decay without fission or neutron absorption, which is used throughout as an example, it has wider applicability.

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References

  1. Harry Bateman. 1910. Solution of a system of differential equations occurring in the theory of radioactive transformation. Proceedings of the Cambridge Philsophical Society 15 (1910), 423--427Google ScholarGoogle Scholar
  2. Jerzy Cetnar. 2006. General solution of Bateman equations for nuclear transmutations. Annals of Nuclear Energy 33 (2006), 640--645.Google ScholarGoogle ScholarCross RefCross Ref
  3. A. G. Groff. 1983. ORIGEN2: A versatile computer code for calculating the nuclide compositions and characteristics of nuclear materials. Nuclear Technology 62 (September 1983), 335--352.Google ScholarGoogle Scholar
  4. Maple. 2013. Maple 17 User Manual. Maplesoft division of Waterloo Maple Inc., Waterloo, ON Canada.Google ScholarGoogle Scholar
  5. Cleve Moler and Charles Van Loan. 2003. Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Rev. 45, 1 (2003), 3--49.Google ScholarGoogle ScholarDigital LibraryDigital Library

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  1. Algorithm 982: Explicit Solutions of Triangular Systems of First-Order Linear Initial-Value Ordinary Differential Equations with Constant Coefficients

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        cover image ACM Transactions on Mathematical Software
        ACM Transactions on Mathematical Software  Volume 44, Issue 2
        June 2018
        242 pages
        ISSN:0098-3500
        EISSN:1557-7295
        DOI:10.1145/3132683
        Issue’s Table of Contents

        Copyright © 2017 ACM

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        Association for Computing Machinery

        New York, NY, United States

        Publication History

        • Published: 16 August 2017
        • Accepted: 1 May 2017
        • Revised: 1 April 2017
        • Received: 1 February 2017
        Published in toms Volume 44, Issue 2

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