Opuscula Math. 43, no. 4 (2023), 575-601
https://doi.org/10.7494/OpMath.2023.43.4.575

Opuscula Mathematica

# Bernstein operational matrix of differentiation and collocation approach for a class of three-point singular BVPs: error estimate and convergence analysis

Nikhil Sriwastav
Amit K. Barnwal
Abdul-Majid Wazwaz
Mehakpreet Singh

Abstract. Singular boundary value problems (BVPs) have widespread applications in the field of engineering, chemical science, astrophysics and mathematical biology. Finding an approximate solution to a problem with both singularity and non-linearity is highly challenging. The goal of the current study is to establish a numerical approach for dealing with problems involving three-point boundary conditions. The Bernstein polynomials and collocation nodes of a domain are used for developing the proposed numerical approach. The straightforward mathematical formulation and easy to code, makes the proposed numerical method accessible and adaptable for the researchers working in the field of engineering and sciences. The priori error estimate and convergence analysis are carried out to affirm the viability of the proposed method. Various examples are considered and worked out in order to illustrate its applicability and effectiveness. The results demonstrate excellent accuracy and efficiency compared to the other existing methods.

Keywords: Bernstein polynomials, collocation method, three-point singular BVPs, convergence analysis, error estimate.

Mathematics Subject Classification: 34B05, 34B15, 34B16, 34B18, 34B27, 34B60.

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• Communicated by Alexander Domoshnitsky.
• Revised: 2023-03-31.
• Accepted: 2023-05-24.
• Published online: 2023-06-13.