Numerical Analysis of the Caputo Fractional Richards Equation Using Yang Transform-Based Hybrid Methods

The Richards equation is the renowned equation for studying the characteristics of infiltration in unsaturated soil areas, such as porous media. This paper’s primary goal is to demonstrate how the water transport problem behaves in unsaturated soil. In this work, we investigated approximate solutions of the non-linear time-fractional Richards equation (TFRE) with the help of the Yang transform iterative method (YTIM) and the homotopy perturbation transform method (HPTM). The methods used are novel and attractive, successfully combining the Yang transform method, the new iterative method, and the homotopy perturbation method. Both techniques use an iterative process with fewer computations to efficiently provide rapidly convergent series-type solutions. Two cases of the TFRE are examined through the Caputo derivative in order to illustrate the effectiveness of the applied methods. Graphs for various fractional orders are drawn to illustrate the physical behavior of the obtained solutions. Numerical comparisons among the derived approximate solutions and the precise solution are provided in order to demonstrate the efficacy of the YTIM and HPTM. The absolute error obtained using the suggested approaches was contrasted with that obtained using the q-homotopy analysis transform method (q-HATM). The impact of changing non-integer, spatial, and temporal parameters on the behavior of the resulting solution is also illustrated graphically. Mathematica software is used to obtain the approximate series solution and to create graphical displays of several fractional orders. The findings show that the proposed techniques are easy to implement and may be used to investigate sophisticated physical systems controlled by time-fractional non-linear partial differential equations.