Numerical simulation of Bernstein-Legendre mixed polynomials for unsteady MHD fractional order Maxwell fluid flow in rectangular ducts

This study proposes a Bernstein-Legendre mixed polynomial algorithm based on a dual-function hybrid strategy for the numerical solution and analysis of the fractional control equations and dynamic behavior of magnetohydrodynamics in a rectangular pipe driven by a variable pressure gradient. First, the control equations describing the complex mechanical behavior of magnetohydrodynamics are established by combining the fractional Maxwell model with the Caputo fractional derivative. Second, Bernstein-Legendre polynomials are used to approximate the fluid velocity function, this method aims to discretize the governing equations into a system of algebraic equations, enabling direct numerical solution in the time domain. Additionally, error analysis of dimensionless numerical examples validates the algorithm’s effectiveness; comparisons with the absolute error of the finite difference algorithm demonstrate that the proposed algorithm achieves higher accuracy. A systematic analysis was conducted to assess the impact of critical parameters, namely fractional, relaxation time, Lorentz force, pressure gradient, and time, on the velocity profile. Numerical results indicate that all these parameters significantly affect the fluid velocity field.