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An efficient technique based on Green’s function for solving two-point boundary value problems and its convergence analysis

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  • Tomar, Saurabh
  • Dhama, Soniya
  • Ramos, Higinio
  • Singh, Mehakpreet

Abstract

This study proposes an accurate approximation to the solution of second-order nonlinear two-point boundary value problems, including the well-known Bratu problem, using an iterative technique based on Green’s function. The approach relies on constructing an equivalent integral representation of the problem incorporating Green’s function. The proposed methodology provides a reliable approximate solution and takes just a few iterations to achieve good accuracy. The mathematical formulation is further supported by discussing in detail the convergence analysis of this approach. Different numerical examples are used to check the robustness and effectiveness of the scheme. The numerical testing for nonlinear problems with nonlinear boundary conditions demonstrates that the proposed method outperforms other existing methods, including the finite element method, the finite volume method, the finite difference method, the B-spline method, and the Adomain’s decomposition method.

Suggested Citation

  • Tomar, Saurabh & Dhama, Soniya & Ramos, Higinio & Singh, Mehakpreet, 2023. "An efficient technique based on Green’s function for solving two-point boundary value problems and its convergence analysis," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 210(C), pages 408-423.
    • Handle: RePEc:eee:matcom:v:210:y:2023:i:c:p:408-423
    DOI: 10.1016/j.matcom.2023.03.015
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    References listed on IDEAS

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    1. Singh, Randhir & Guleria, Vandana & Singh, Mehakpreet, 2020. "Haar wavelet quasilinearization method for numerical solution of Emden–Fowler type equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 174(C), pages 123-133.
    2. Ramos, Higinio & Singh, Gurjinder, 2022. "Solving second order two-point boundary value problems accurately by a third derivative hybrid block integrator," Applied Mathematics and Computation, Elsevier, vol. 421(C).
    3. Ramos, Higinio & Rufai, M.A., 2019. "A third-derivative two-step block Falkner-type method for solving general second-order boundary-value systems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 165(C), pages 139-155.
    4. Attili, Basem S. & Syam, Muhammed I., 2008. "Efficient shooting method for solving two point boundary value problems," Chaos, Solitons & Fractals, Elsevier, vol. 35(5), pages 895-903.
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