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Numerical solution of boundary value problems by using an optimized two-step block method

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Abstract

This paper aims at the application of an optimized two-step hybrid block method for solving boundary value problems with different types of boundary conditions. The proposed approach produces simultaneously approximations at all the grid points after solving an algebraic system of equations. The final approximate solution is obtained through a homotopy-type strategy which is used in order to get starting values for Newton’s method. The convergence analysis shows that the proposed method has at least fifth order of convergence. Some numerical experiments such as Bratu’s problem, singularly perturbed, and nonlinear system of BVPs are presented to illustrate the better performance of the proposed approach in comparison with other methods available in the recent literature.

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Acknowledgments

The authors are grateful to the anonymous referees for their careful reading of the manuscript and their comments which improved the final result considerably.

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Authors and Affiliations

  1. Grupo de Computación Científica, Universidad de Salamanca, Campus Viriato, Salamanca, Spain

    Higinio Ramos

  2. Escuela Politécnica Superior de Zamora, Universidad de Salamanca, Campus Viriato, 49022, Zamora, Spain

    Higinio Ramos

  3. Department of Mathematical Sciences, The Federal University of Technology at Akure, P.M.B. 704, Akure, Ondo State, Nigeria

    M. A. Rufai

Authors
  1. Higinio Ramos
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  2. M. A. Rufai
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Correspondence to Higinio Ramos.

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Ramos, H., Rufai, M.A. Numerical solution of boundary value problems by using an optimized two-step block method. Numer Algor 84, 229–251 (2020). https://doi.org/10.1007/s11075-019-00753-3

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  • DOI: https://doi.org/10.1007/s11075-019-00753-3

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