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Numerical solution of singular boundary value problems using Green’s function and improved decomposition method

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Abstract

In this work, an effective technique for solving a class of singular two point boundary value problems is proposed. This technique is based on the Adomian decomposition method (ADM) and Green’s function. The technique relies on constructing Green’s function before establishing the recursive scheme for the solution components. In contrast to the existing recursive schemes based on ADM, the proposed recursive scheme avoids solving a sequence of nonlinear algebraic or transcendental equations for the undetermined coefficients. The approximate solution is obtained in the form of series with easily calculable components. For the completeness, the convergence and error analysis of the proposed scheme is supplemented. Moreover, the numerical examples are included to demonstrate the accuracy, applicability, and generality of the proposed scheme. The results reveal that the method is very effective, straightforward, and simple.

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur, 721302, India

    Randhir Singh, Jitendra Kumar & Gnaneshwar Nelakanti

Authors
  1. Randhir Singh
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  2. Jitendra Kumar
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  3. Gnaneshwar Nelakanti
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Corresponding author

Correspondence to Randhir Singh.

Appendix

Appendix

Construction of Green’s function of following the problem

Consider the linear differential equation

Integrating the above equation twice first from x to 1 and then from 0 to x, changing the order of integration, and applying the boundary conditions, we obtain

where Green’s function is given by

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Singh, R., Kumar, J. & Nelakanti, G. Numerical solution of singular boundary value problems using Green’s function and improved decomposition method. J. Appl. Math. Comput. 43, 409–425 (2013). https://doi.org/10.1007/s12190-013-0670-4

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  • DOI: https://doi.org/10.1007/s12190-013-0670-4

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