Fast method and theoretical study of magnetohydrodynamic flow and heat transfer for fractional Maxwell fluids

This study investigates the magnetohydrodynamic flow and heat transfer in inclined pipes filled with fractional Maxwell fluids. The proposed model incorporates the momentum equation derived from the fractional constitutive relation and the fractional heat equation based on Fourier's law. Temporal and spatial discretization are implemented using the second-order fractional backward difference method and the finite element method, respectively. The stability of the fully discrete scheme is analyzed, ensuring numerical accuracy of $O({\tau }^2+h^r)$, where τ is time step size, h is space step size and r is the order of accuracy of the spatial discretization. To enhance computational efficiency, a fast numerical method is introduced. A numerical example validates the effectiveness of the proposed approach and supports the theoretical framework. Additionally, simulations are conducted to examine the effects of pipe inclination and thermal radiation on velocity and temperature distributions.