An Innovative Natural Transform Decomposition Method to Investigate Fractional Rosenau–Hyman Equations With Nonsingular Kernel Derivatives

Many models in fractional calculus are still being developed, investigated, and applied in the real world in numerous disciplines of engineering and science because nonlocality plays a significant role. There are still numerous nonlocal phenomena that have not been researched and are merely waiting to be discovered, despite the fact that many remarkable innovations have previously been documented by researchers in significant books and review articles. As a result, we keep discovering new uses for and facets of fractional models. In this article, the Caputo–Fabrizio (CF) and Atangana–Baleanu–Caputo (ABC) fractional operators are used to examine the two-dimensional time-fractional Rosenau–Hyman equations. We investigate the characteristics and applicability of the natural transform (NT) of the CF and ABC fractional derivatives on the CF and ABC fractional operators. In this investigation, two powerful approaches are used. The Adomian decomposition technique and the natural transformation scheme enable us to reach a unique result. A few illustrative cases are used to test the suggested technique’s accuracy, applicability, and effectiveness. The procedure is then applied to a few numerical cases, and MAPLE is used to compare the numerical solutions to the exact solutions. The exact solutions and the obtained results are compared numerically and graphically. These techniques result in a convergent series of components that are simple to compute as the solution. The effectiveness and simplicity of the current procedures are excellent. Due to the algorithm’s simple computation, high accuracy, and short processing time, it can be seen that it is extremely useful and effective.