The Bernstein Collocation Method as a Powerful Numerical Technique for Solving Higher-Order BVPs and Modeling Unsteady MHD Flow

This paper presents the Bernstein Collocation Method (BCM) as a robust and efficient numerical technique for solving a wide range of higher-order linear and nonlinear boundary value problems. The method effectively transforms complex higher-order differential equations into an algebraic system of equations, enhancing the computational process. To demonstrate its versatility, we apply the BCM to model real-world phenomena, encompassing the complex unsteady MHD flow of a viscous fluid confined by moving parallel plates. Comparative analyses with established methods, including Haar wavelets, nonpolynomial spline, and quintic B-spline collocation, consistently demonstrate the superior accuracy and computational efficiency of the BCM. Five illustrative test problems further validate the efficacy, robustness, and adaptability of the BCM across diverse applications, with maximum absolute error evaluations ranging from 10 −9 to 10 −17 as the number of collocation points increases. Collectively, these results affirm the BCM as a powerful tool for solving complex boundary value problems, offering valuable insights into advanced computational methods, particularly in uid dynamics. Future work will generalize BCM to multi-dimensional and timedependent boundary value problems and and apply the method to stochastic and data-driven models for complex MHD and multiphysics flows.