Solving Fourth Order Linear Initial and Boundary Value Problems Using an Implicit Block Method

  • Oluwaseun AdeyeyeEmail author
  • Zurni Omar
Conference paper


The introduction of new approaches to numerically approximate higher order ordinary differential equations (ODEs) is vastly being explored in recent literature. The reason for adopting these numerical approaches is because some of these higher order ODEs fail to have an approximate solution or the current numerical approach being adopted has less accuracy. The application of an implicit block method for solving fourth order ordinary differential equations (ODEs) is considered in this article. The solution encompasses both initial and boundary value problems of fourth order ODEs. The implicit block method is developed for a set of six equidistant points using a new linear block approach (LBA). The LBA produces the required family of six-step schemes to simultaneously evaluate the solution of the fourth order ODEs at individual grid points in a self-starting mode. The basic properties of the implicit block method are investigated, and the block method is seen to satisfy the property of convergence which is displayed in the numerical results obtained. Furthermore, in comparison to works of past authors the implicit block method gives more impressive results.


Implicit block method Six-step Linear block approach Fourth order Ordinary differential equations 


  1. 1.
    Fatunla, S.O.: Block methods for second order ODEs. Int. J. Comput. Math. 41(1–2), 55–63 (1991). Scholar
  2. 2.
    Bulirsch, R., Stoer, J.: Numerical treatment of ordinary differential equations by extrapolation methods. Numer. Math. 8(1), 1–13 (1966). Scholar
  3. 3.
    Fang, Y., You, X., Ming, Q.: Exponentially fitted two-derivative runge-kutta methods for the schrodinger equation. Int. J. Mod. Phys. 24(10), 1350073 (2013). Scholar
  4. 4.
    Kalogiratou, Z., Monovasilis, T., Psihoyios, G., Simos, T.E.: Runge-kutta type methods with special properties for the numerical integration of ordinary differential equations. Phys. Rep. 536(3), 75–146 (2014). Scholar
  5. 5.
    Hussain, K., Ismail, F., Senu, N.: Two embedded pairs of runge-kutta type methods for direct solution of special fourth-order ordinary differential equations. Math. Prob. Eng. 2015(196595), 1–12 (2015). Scholar
  6. 6.
    Awoyemi, D.O.: Algorithmic collocation approach for direct solution of fourth-order initial-value problems of ordinary differential equations. Int. J. Comput. Math. 82(3), 321–329 (2005). Scholar
  7. 7.
    Adeyeye, O., Kayode, S.J.: Two-step two-point hybrid methods for general second order differential equations. Afr. J. Math. Comput. Sci. Res. 6(10), 191–196 (2013). Scholar
  8. 8.
    Jator, S.N., Lee, L.: Implementing a seventh-order linear multistep method in a predictor-corrector mode or block mode: which is more efficient for the general second order initial value problem. SpringerPlus 3(1), 1–8 (2014). Scholar
  9. 9.
    Jator, S.N.: Solving second order initial value problems by a hybrid multistep method without predictors. Appl. Math. Comput. 217(8), 4036–4046 (2010). Scholar
  10. 10.
    Lambert, J.D.: Computational methods in ordinary differential equations. London (1973)Google Scholar
  11. 11.
    Hull, T.E., Enright, W.H., Fellen, B.M., Sedgwick, A.E.: Comparing numerical methods for ordinary differential equations. SIAM J. Numer. Anal. 9(4), 603–637 (2006). Scholar
  12. 12.
    Omar, Z., Kuboye, J.O.: New seven-step numerical method for direct solution of fourth order ordinary differential equations. J. Math. Fundam. Sci. 48(2):94–105 (2016).
  13. 13.
    Kayode, S.J., Duromola, M.K., Bolaji, B.: Direct solution of initial value problems of fourth order ordinary differential equations using modified implicit hybrid block method. J. Sci. Res. Rep. 3, 2790–2798 (2014). Scholar
  14. 14.
    Omar, Z., Abdelrahim, R.: Direct solution of fourth order ordinary differential equations using a one step hybrid block method of order five. Int. J. Pure Appl. Math. 109(4), 763–777 (2016). Scholar
  15. 15.
    Papakostas, S.N., Tsitmidelis, S., Tsitouras, C.: Evolutionary generation of 7th order runge–kutta–nyström type methods for solving y(4) = f(x,y). In: International Conference of Computational Methods in Sciences and Engineering, Athens, March 2015. AIP Conference Proceedings, vol. 1702, no. 1, pp. 190018-1–190018-4. AIP Publishing, New York (2015).
  16. 16.
    Jator, S.N.: Numerical integrators for fourth order initial and boundary value problems. Int. J. Pure Appl. Math. 47(4), 563–576 (2008). Scholar

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© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.School of Quantitative SciencesUniversiti Utara MalaysiaChanglunMalaysia

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