An Approximate Analytical Series Solution for Multi-Dimensional Fractional Navier-Stokes Equations: A Hybrid Iterative Approach

The fractional Navier-Stokes (FNS) equations, as a generalization of the classical model, offer a powerful approach for describing complex fluid flow phenomena governed by fractional dynamics. However, the nonlinear structure and fractional operators in these equations pose substantial challenges in obtaining analytical or approximate solutions. This study introduces a novel hybrid analytical iterative scheme, termed the Natural Transform Iterative Algorithm (NTIA), developed for efficiently solving multidimensional time-fractional Navier–Stokes equations defined in the Caputo sense. The proposed method integrates the Natural transform, a Laplace-type integral transform adept at handling fractional derivatives with the new algorithm of the Daftardar-Gejji and Jafari method (DGJM) to construct rapidly convergent series solutions. Unlike conventional methods, NTIA avoids linearization, discretization, and perturbation assumptions, thereby reducing computational complexity while maintaining analytical accuracy. The algorithm's performance is demonstrated through several benchmark examples involving one-, two-, and three-dimensional FNS equations. Graphical and numerical comparisons with known closed-form solutions confirm the high accuracy, stability, and convergence rate of the NTIA across varying fractional orders. The method effectively captures steady-state and transient fluid behaviors while smoothly approaching classical results as the fractional order tends to unity. The findings highlight the reliability and robustness of NTIA as an analytical tool for nonlinear fractional PDEs. Its computational efficiency and adaptability make it a promising approach for solving a broad spectrum of problems in fluid mechanics, diffusion phenomena, and applied fractional dynamics.