This article introduces a new iterative transform method and homotopy perturbation transform method along with a natural transform to analyze the multi-dimensional Navier-Stokes equations. To solve the fractional-derivative, the Caputo-Fabrizio definition of the fractional derivative was employed. Four examples were considered to examine the efficacy and accuracy of the proposed methods. The efficiency and accuracy were also demonstrated by the solution comparison via graphs. The proposed methods' convergence and uniqueness are also discussed. The methods mentioned above are straightforward and support a high rate of convergence.

This study investigates the two-dimensional nonlinear complex coupled Maccari equations, which are significant in describing solitary waves concentrated in small spatial regions. These equations have applications across various fields, including hydrodynamics, nonlinear optics, and the study of sonic Langmuir solitons. Using the Bäcklund transformation, we explore a broad range of soliton solutions for this system, focusing on their spectral properties. The proposed method stands out for its simplicity and comprehensive results compared to traditional approaches. The obtained solutions are expressed in rigorous, trigonometric, and hyperbolic forms, providing deeper insights into the dynamics of the system. To enhance understanding, we present contour and three-dimensional graphical representations of the solutions. This study has potential applications in energy and industry by advancing the understanding of nonlinear wave phenomena, which are crucial in optimizing energy transfer processes and designing efficient systems in hydrodynamic and optical engineering. Additionally, the soliton solutions obtained here contribute to technologies in power transmission and high-speed optical communications, offering a foundation for innovations in sustainable energy systems and industrial applications.