Bernstein Operational Matrix for Solving Boundary Value Problems

This paper outlines a numerical method called the Bernstein operational matrix of derivative (BOMD) of order two and order three with the approach of the Chebyshev collocation technique to solve boundary value problems (BVP). BOMD with suitable collocation points is implemented to solve the BVP using the linear combination of Bernstein polynomials with unknown coefficients to approximate the solutions. The derivatives featured in the problem sets will be approximated by utilizing the matrix. The subsequent examination involves a mathematical analysis of the proposed method, including evaluating its order, absolute error metrics and comparative assessments with alternative methodologies. Four problems involving linear and non-linear equations and systems, along with practical real-world problems, are addressed to assess the reliability of the proposed method.