160
Views
14
CrossRef citations to date
0
Altmetric
Research Article

An adaptive one-point second-derivative Lobatto-type hybrid method for solving efficiently differential systems

ORCID Icon & ORCID Icon
Pages 1687-1705 | Received 29 Jun 2021, Accepted 17 Oct 2021, Published online: 22 Nov 2021

References

  • P. Henrici, Discrete Variable Methods in ODEs, John Wiley, New York, 1962.
  • J.D. Lambert, Numerical Methods for Ordinary Differential Systems, John Wiley, New York, USA, 1991.
  • E. Hairer and G. Wanner, Solving Ordinary Differential Equations II, Springer Series in Computational Mathematics, Vol. 14, Springer Verlag, Berlin, 2010. Stiff and differential-algebraic problems, Second revised edition, paperback.
  • E. Hairer, S.P. Norsett and G. Wanner, Solving Ordinary Differential Equations I, Nonstiff Problems, Springer-Verlag, Berlin, Heidelberg, New York, 1993.
  • J.C. Butcher, Numerical Methods for Ordinary Differential Equations, John Wiley and Sons, 2008.
  • W.B. Gragg and H.J. Stetter, Generalized multistep predictor-corrector methods, J ACM 11(2) (1964), pp. 188-–209.
  • M.A Rufai, M.K. Duromola and A.A. Ganiyu, Derivation of one-sixth hybrid block method for solving general fist order ordinary differential equations, IOSR-JM 12(5) (2016), pp. 20–27.
  • H. Dym, Linear Algebra in Action, AMS, Providence, Rhode Island, 2007.
  • E.A. Areo and M.A. Rufai, A new uniform fourth order one-third step continuous block method for direct solutions of y″=f(x,y,y′), Br. J. Math. Comput. Sci. Sci. Domain Int. 15(4) (2016), pp. 1–12.
  • M.A. Rufai and H. Ramos, Numerical solution of second-order singular problems arising in astrophysics by combining a pair of one-step hybrid block Nystrm methods, Astrophys. Space Sci.365 (2020), pp. 96. https://doi.org/https://doi.org/10.1007/s10509-020-03811-8.
  • S.N. Jator and J. Li, A self-starting linear multi-step method for a direct solution of the general second order initial value problem, Int. J. Comput. Math. 86 (2009), pp. 827–836.
  • M.A. Rufai and H. Ramos, One-step hybrid block method containing third derivatives and improving strategies for solving Bratu's and Troesch's problems, Numer. Math. Theory Methods Appl. 13 (2020), pp. 946–972.
  • H. Ramos and M.F. Patricio, Some new implicit two-step multiderivative methods for solving special second-order IVP's, Appl. Math. Comput. 239 (2014), pp. 227–241.
  • H. Ramos, Z. Kalogiratou, T. Monovasilis and T.E. Simos, An optimized two-step hybrid block method for solving general second order initial-value problems, Numer. Algorithms 72(4) (2016), pp. 1089–1102.
  • H. Ramos and M.A. Rufai, Third derivative modification of k-step block Falkner methods for the numerical solution of second order initial-value problems, Appl. Math. Comput. 333 (2018), pp. 231–245.
  • N.S. Hoang, Collocation Runge-Kutta-Nyström methods for solving second-order initial value problems, Int. J. Comput. Math. (2021). DOI:https://doi.org/10.1080/00207160.2021.1900567.
  • M.A. Rufai and H. Ramos, A variable step-size fourth-derivative hybrid block strategy for integrating third-order IVPs, with applications, Int. J. Comput. Math. (2021). DOI:https://doi.org/10.1080/00207160.2021.1907357.
  • A. Abdi and Z. Jackiewicz, Towards a code for nonstiff differential systems based on general linear methods with inherent Runge-Kutta stability, Appl. Numer. Math. 136 (2019), pp. 103–121.
  • H. Ramos and G. Singh, A note on variable step-size formulation of a Simpson's-type second derivative block method for solving stiff systems, Appl. Math. Lett. 64 (2017), pp. 101–107.
  • H. Barucq, M. Duruflè and M. N'diaye, High-order locally a-stable implicit schemes for linear ODEs, J. Sci. Comput. 85 (2020), pp. 31.
  • P.O. Olatunji and M.N.O. Ikhile, Strongly regular general linear methods, J. Sci. Comput., 82 (2020), pp. 31. https://doi.org/https://doi.org/10.1007/s10915-019-01107-w.
  • G. Califano, G. Izzo and Z. Jackiewicz, Strong stability preserving general linear methods with Runge-Kutta stability, J. Sci. Comput. 76 (2018), pp. 943–968.
  • M.B. Suleiman, H. Musa, F. Ismail and N. Senu, A new variable step size block backward differentiation formula for solving stiff initial value problems, Int. J. Comput. Math. 90(11) (2013), pp. 2391–2408.
  • A.N. Fairuz and Z.A. Majid, Rational methods for solving first-order initial value problems, Int. J. Comput. Math. 98(2) (2021), pp. 252–270.
  • A. Abdi and G. Hojjati, Implementation of Nørdsieck second derivative methods for stiff ODEs, Appl. Numer. Math. 94 (2015), pp. 241–253.
  • S.N. Jator, A.O. Akinfenwa, S.A. Okunuga and A.B. Sofoluwe, High-order continuous third derivative formulas with block extensions for y″=f(x,y,y′), Int. J. Comput. Math. 90(9) (2013), pp. 1899–1914.
  • J. Vigo-Aguiar and H. Ramos, Variable step-size implementation of multistep methods for y″=f(x,y,y′), J. Comput. Appl. Math. 192 (2006), pp. 114–131.
  • L.F. Shampine, B.P. Sommeijer and J.G. Verwer IRKC, an IMEX solver for stiff diffusion-reaction PDEs J, Comput. Appl. Math. 196(2) (2006), pp. 485-–497.
  • X. Piao, S. Bu, D. Kim and P. Kim, An embedded formula of the Chebyshev collocation method for Stiff problems, J. Comput. Phys., 351 (2017), PP. 376–391.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.