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International Journal of Nonlinear Sciences and Numerical Simulation

Editor-in-Chief: Birnir, Björn

Editorial Board: Armbruster, Dieter / Chen, Xi / Bessaih, Hakima / Chou, Tom / Grauer, Rainer / Marzocchella, Antonio / Rangarajan, Govindan / Trivisa, Konstantina / Weikard, Rudi


IMPACT FACTOR 2017: 1.162

CiteScore 2017: 1.41

SCImago Journal Rank (SJR) 2017: 0.382
Source Normalized Impact per Paper (SNIP) 2017: 0.636

Mathematical Citation Quotient (MCQ) 2017: 0.12

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2191-0294
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Volume 19, Issue 7-8

Issues

L-stable Explicit Nonlinear Method with Constant and Variable Step-size Formulation for Solving Initial Value Problems

Sania Qureshi
  • Corresponding author
  • Basic Sciences and Related Studies, Mehran University of Engineering and Technology, Jamshoro, Pakistan
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  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Higinio RamosORCID iD: http://orcid.org/0000-0003-2791-6230
Published Online: 2018-10-23 | DOI: https://doi.org/10.1515/ijnsns-2017-0267

Abstract

In this work, we develop a nonlinear explicit method suitable for both autonomous and non-autonomous type of initial value problems in Ordinary Differential Equations (ODEs). The method is found to be third order accurate having L-stability. It is shown that if a variable step-size strategy is employed then the performance of the proposed method is further improved in comparison with other methods of same nature and order. The method is shown to be working well for initial value problems having singular solutions, singularly perturbed and stiff problems, and blow-up ODE problems, which is illustrated using a few numerical experiments.

Keywords: nonlinear explicit method; rational approximation; blow-up ODEs; singular solutions; L-Stability; variablestep-size

PACS: 65Lxx; 65L04; 65L05; 65L11; 65L12

References

  • [1]

    S. O. Fatunla, Numerical Methods for IVPS in ODEs, Academic Press Inc. USA, 1988.

  • [2]

    U. M. Ascher and L. R. Petzold, Computer methods for ordinary differential equations and differential-algebraic equations, SIAM, 1998.Google Scholar

  • [3]

    J. D. Lambert, Numerical Methods for Ordinary Differential Systems: The Initial Value Problem, John Wiley & Sons, Inc., 1991.Google Scholar

  • [4]

    L. F. Shampine and A. Witt, Control of local error stabilizes integrations, J. Comput. Appl. Math. 62 (1995), 333–351.CrossrefGoogle Scholar

  • [5]

    H. Ramos, A non-standard explicit integration scheme for initial-value problems, Appl. Math. Comput. 189 (2007), 710–718.Web of ScienceGoogle Scholar

  • [6]

    F. D. Van Niekerk, Non-linear one-step methods for initial value problems, Comput. Math. Appl. 13 (1987), 367–371.CrossrefGoogle Scholar

  • [7]

    F. D. Van Niekerk, Rational one-step methods for initial value problems, Comput. Math. Appl. 16 (1988), 1035–1039.CrossrefGoogle Scholar

  • [8]

    H. Ramos, Contributions to the development of differential systems exactly solved by multistep finite-difference schemes, Appl. Math. Comput. 217 (2010), 639–649.Web of ScienceGoogle Scholar

  • [9]

    G. Dahlquist, A special stability problem for linear multistep methods, BIT 3 (1963), 27–43.CrossrefGoogle Scholar

  • [10]

    E. Hairer and G. Wanner, Solving ordinary differential equations II: stiff and differential-algebraic problems, Springer Series in Computational Mathematics 14, 1996.Google Scholar

  • [11]

    M. K. Jain, Numerical Solution of Differential Equations, John Wiley & Sons, Inc., 1984.Google Scholar

  • [12]

    M. Calvo and M. M. Quemada, On the stability of rational Runge-Kutta methods, J. Comput. Appl. Math. 8 (1982), 289–293.CrossrefGoogle Scholar

  • [13]

    E. Hairer, Unconditionally stable explicit methods for parabolic equations, Numer. Math. 35 (1980), 57–68.CrossrefGoogle Scholar

  • [14]

    H. Ramos, G. Singh, V. Kanwar, S. Bhatia, Solving first-order initial-value problems by using an explicit non-standard A-stable one-step method in variable step-size formulation, Appl. Math. Comput. 268 (2015), 796–805.Web of Science

  • [15]

    H. Ramos, G. Singh, V. Kanwar, S. Bhatia, An embedded 3 (2)pair of nonlinear methods for solving first order initial-value ordinary differential systems, Numer. Algorithms, 75 (2017), 509–529.CrossrefWeb of Science

  • [16]

    L. F. Shampine, I. Gladwell and S. Thompson, Solving ODEs with MATLAB, Cambridge University Press, 2003.Google Scholar

  • [17]

    J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, Springer, 2002.Google Scholar

  • [18]

    H. A. Watts, Starting step size for an ODE solver, J. Comput. Appl. Math. 9 (1983), 177–191.CrossrefGoogle Scholar

  • [19]

    L. F. Shampine and M. K. Gordon, Computer solution of ordinary differential equations: the initial value problem, Freeman, San Francisco, CA, 1975.Google Scholar

  • [20]

    A. E. Sedgwick, An effective variable-order variable-step Adams method, Dept. of Computer Science. Rept. 53, University of Toronto, Toronto, Canada, 1973.Google Scholar

About the article

Received: 2017-12-07

Accepted: 2018-10-05

Published Online: 2018-10-23

Published in Print: 2018-12-19


Citation Information: International Journal of Nonlinear Sciences and Numerical Simulation, Volume 19, Issue 7-8, Pages 741–751, ISSN (Online) 2191-0294, ISSN (Print) 1565-1339, DOI: https://doi.org/10.1515/ijnsns-2017-0267.

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