article

Linearization techniques for singular initial-value problems of ordinary differential equations

Author:

J. I. RamosAuthors Info & Claims

Applied Mathematics and Computation, Volume 161, Issue 2

Pages 525 - 542

https://doi.org/10.1016/j.amc.2003.12.047

Published: 01 February 2005 Publication History

Abstract

Linearization methods for singular initial-value problems in second-order ordinary differential equations are presented. These methods result in linear constant-coefficients ordinary differential equations which can be integrated analytically, thus yielding piecewise analytical solutions and globally smooth solutions. The accuracy of these methods is assessed by comparisons with exact and asymptotic solutions of homogeneous and non-homogeneous, linear and nonlinear Lane-Emden equations. It is shown that linearization methods provide accurate solutions even near the singularity or the zeros of the solution. In fact, it is shown that linearization methods provide more accurate solutions than methods based on perturbation methods. It is also shown that the accuracy of these techniques depends on the nonlinearity of the ordinary differential equations and may not be a monotonic function of the step size.

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Digital Library

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Koch, O., Kofler, P. and Weinmüller, E.B., Initial value problems for systems of ordinary first and second order differential equations with a singularity of the first kind. Analysis. v21. 373-389.

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Ramos, J.I. and Garcı¿a-López, C.M., Linearized ¿-methods I. Ordinary differential equations. Comput. Methods Appl. Mech. Engrg. v129. 255-269.

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Cited By

Omidi MArab BRasanan ARad JParand K(2022)Learning nonlinear dynamics with behavior ordinary/partial/system of the differential equations: looking through the lens of orthogonal neural networksEngineering with Computers10.1007/s00366-021-01297-838:Suppl 2(1635-1654)Online publication date: 1-Jun-2022
https://dl.acm.org/doi/10.1007/s00366-021-01297-8
Mall SChakraverty SMitra STrajković LBedi PMcIntosh SRajasree M(2015)Multi Layer Versus Functional Link Single Layer Neural Network for Solving Nonlinear Singular Initial Value ProblemsProceedings of the Third International Symposium on Women in Computing and Informatics10.1145/2791405.2791542(678-683)Online publication date: 10-Aug-2015
https://dl.acm.org/doi/10.1145/2791405.2791542
Mall SChakraverty S(2015)Numerical solution of nonlinear singular initial value problems of Emden-Fowler type using Chebyshev Neural Network methodNeurocomputing10.1016/j.neucom.2014.07.036149:PB(975-982)Online publication date: 3-Feb-2015
https://dl.acm.org/doi/10.1016/j.neucom.2014.07.036
Show More Cited By

Linearization techniques for singular initial-value problems of ordinary differential equations
1. Mathematics of computing
  1. Mathematical analysis
    1. Differential equations
    2. Numerical analysis
2. Theory of computation

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Published In

cover image Applied Mathematics and Computation

Applied Mathematics and Computation Volume 161, Issue 2

February, 2005

341 pages

ISSN:0096-3003

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Copyright © Elsevier Inc. © 2004.

Publisher

Elsevier Science Inc.

United States

Publication History

Published: 01 February 2005

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Cited By

Omidi MArab BRasanan ARad JParand K(2022)Learning nonlinear dynamics with behavior ordinary/partial/system of the differential equations: looking through the lens of orthogonal neural networksEngineering with Computers10.1007/s00366-021-01297-838:Suppl 2(1635-1654)Online publication date: 1-Jun-2022
https://dl.acm.org/doi/10.1007/s00366-021-01297-8
Mall SChakraverty SMitra STrajković LBedi PMcIntosh SRajasree M(2015)Multi Layer Versus Functional Link Single Layer Neural Network for Solving Nonlinear Singular Initial Value ProblemsProceedings of the Third International Symposium on Women in Computing and Informatics10.1145/2791405.2791542(678-683)Online publication date: 10-Aug-2015
https://dl.acm.org/doi/10.1145/2791405.2791542
Mall SChakraverty S(2015)Numerical solution of nonlinear singular initial value problems of Emden-Fowler type using Chebyshev Neural Network methodNeurocomputing10.1016/j.neucom.2014.07.036149:PB(975-982)Online publication date: 3-Feb-2015
https://dl.acm.org/doi/10.1016/j.neucom.2014.07.036
Razzaghi M(2012)Hybrid Functions for Nonlinear Differential Equations with Applications to Physical ProblemsRevised Selected Papers of the 5th International Conference on Numerical Analysis and Its Applications - Volume 823610.1007/978-3-642-41515-9_8(86-94)Online publication date: 15-Jun-2012
https://dl.acm.org/doi/10.1007/978-3-642-41515-9_8
Heo BShin NHong KGantenbein RKuo T(2011)A study on a system for measuring curvature and slope through edge information extraction of an electric pole partially hidden by an obstacleProceedings of the 2011 ACM Symposium on Research in Applied Computation10.1145/2103380.2103427(238-240)Online publication date: 2-Nov-2011
https://dl.acm.org/doi/10.1145/2103380.2103427

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