Spectral solutions of nonlinear fractional ADR equations with ABC derivative for groundwater contaminant transport

In this study, we introduce a two-dimensional shifted Legendre spectral method to address nonlinear fractional advection–dispersion–reaction (ADR) equations governed by the generalized Atangana–Baleanu–Caputo (ABC) fractional derivative with a Mittag-Leffler kernel. The significance of this work lies in its ability to tackle the inherent challenges posed by the nonlinear and fractional characteristics of such models, which frequently arise in complex physical and engineering systems. By employing the proposed spectral scheme, we ensure robust numerical stability and high accuracy, even in the presence of strong nonlinearity. To validate the effectiveness of the method, two illustrative examples are solved, with results presented through detailed figures and tables in Sec. 5 . These results consistently demonstrate excellent accuracy, reflected in remarkably small absolute errors. Consequently, the proposed approach establishes itself as a reliable and powerful computational tool for solving nonlinear partial fractional differential equations, offering valuable insights and practical utility in diverse scientific and engineering applications. The proposed method yields very small absolute errors: for Example 5.1 the maximum absolute error is of order [Formula: see text] (and still less than [Formula: see text] with other fractional orders), while for Example 5.2 the absolute error still within [Formula: see text] for all presented cases, confirming the high accuracy and stability of the proposed scheme.