(untitled) (Contemporary Mathematics, 2024)
This study employs the natural transform decomposition method (NTDM) to examine analytical solutions of the nonlinear time-fractional Richards equation (TFRE). The NTDM is an innovative and attractive hybrid integral transform strategy that elegantly combines the Adomian decomposition method and the natural transform method. This solution strategy effectively generates rapidly convergent series-type solutions through an iterative process involving fewer calculations. The convergence and uniqueness of the solutions are presented. To demonstrate the efficiency of the considered solution method, two test cases of the TFRE are investigated within the framework of the Caputo-Fabrizio and Atangana-Baleanu-Caputo derivatives whose definitions incorporate nonsingular kernel functions. Numerical comparisons between the obtained approximate solutions, exact solutions, and those from existing related literature are presented to show the validity and accuracy of the technique. Graphical representations demonstrating the effect of varying non-integer, temporal, and spatial parameters on the behavior of the obtained model solutions are also presented. The results indicate that the execution of the method is straightforward and can be employed to explore complex physical systems governed by time-fractional nonlinear partial differential equations.