Comparative Analysis of Advection–Dispersion Equations with Atangana–Baleanu Fractional Derivative

In this study, we solve the fractional advection–dispersion equation (FADE) by applying the Laplace transform decomposition method (LTDM) and the variational iteration transform method (VITM). The Atangana–Baleanu (AB) sense is used to describe the fractional derivative. This equation is utilized to determine solute transport in groundwater and soils. The FADE is converted into a system of non-linear algebraic equations whose solution leads to the approximate solution for this equation using the techniques presented. The proposed approximate method’s convergence is examined. The suggested method’s applicability is demonstrated by testing it on several illustrative examples. The series solutions to the specified issues are obtained, and they contain components that converge more quickly to the precise solutions. The actual and estimated results are demonstrated in graphs and tables to be quite similar, demonstrating the usefulness of the proposed strategy. The innovation of the current work is in the application of an effective method that requires less calculation and achieves a greater level of accuracy. Furthermore, the proposed approaches may be implemented to prove their utility in tackling fractional-order problems in science and engineering.