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This article addresses the two analytical methods, i.e., the new iterative transform method (NITM) and the homotopy perturbation transform method (HPTM), along with an Aboodh transform (AT), to approximate the nonlinear system of two-dimensional (2D) time-fractional Navier-Stokes (TFNS) equations. We take the time-fractional derivative in the form of Atangana-Baleanu (AB). The article's suggested examples examine the accuracy and efficacy of the proposed methods, while the graphs demonstrate their potential and effectiveness. The article also provides demonstrations of uniqueness and convergence. The aforementioned techniques are straightforward and support a high rate of convergence, which helps in understanding the dynamics of fractional nonlinear systems.

In the present investigation, a new integral transform method (NITM) and the projected differential transform method (PDTM) are used to give an analytical solution to the time-fractional Klein-Gordon (TFKG) equation. The time-fractional derivative is used in the Caputo sense. The huge advantage of the suggested approach is the ease with which the nonlinear term can be effortlessly treated by projected differential transform without using Adomian's and He's polynomials. The solution of fractional partial differential equations using the aforementioned method is very simple and straightforward. The efficiency and accuracy of the proposed method are demonstrated by three examples, and the effects of various fractional Brownian motions are demonstrated graphically.