<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>R</mml:mi> </mml:math> omanovski-Jacobi polynomials approach for fractional mobile-immobile advection-dispersion model with Atangana-Baleanu-Caputo derivative

Abstract The study used a non-singular Caputo–Baleanu time derivative (including the Mittag-Leffler kernel) to examine the fractional mobile–immobile advection–dispersion (FMI-AD) model with variable coefficients. Using <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">R</mml:mi> </mml:math> omanovski–Jacobi ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">R</mml:mi> </mml:math> J) polynomials, we obtain for the first time an operational matrix (OM) for the Atangana–Baleanu–Caputo (ABC) fractional derivative. This enables the conversion of the FMI-AD equation into an algebraic system using a spectral collocation technique. A thorough error analysis and a stability proof for the technique are examples of theoretical contributions. Compared to traditional fractional models, the non-singular ABC kernel offers more realistic memory effects and increased numerical stability, which makes it ideal for simulating transport in heterogeneous media. Multiple numerical examples are provided to support the methodology. Accuracy is verified through comparison with prior research and exact solutions. The outcomes show stability in the presence of perturbed initial conditions, high accuracy, and fast convergence. It should be noted that the current work includes both 1D and 2D problems, and all calculations were done using Mathematica.