A unified comparative framework for solving nonlinear Atangana–Baleanu–Caputo fractional reaction–diffusion equations using semi-analytical methods

In porous media, heterogeneous materials, biological tissues and complex physical systems, anomalous diffusion phenomena emerge that are not well described by classical integer-order diffusion models, which are not able to take into account memory and nonlocal interactions. The main objective of this study is to develop, analyze and compare, within one rigorous framework, two semi-analytical solvers for a nonlinear Burgers–Fisher-type time-fractional reaction–diffusion–convection equation governed by the Atangana–Baleanu–Caputo derivative, and to quantify how the fractional order changes the solution relative to the classical model. In this regard, a nonlinear time fractional reaction diffusion equation using the Atangana–Baleanu–Caputo (ABC) fractional derivative with its nonsingular Mittag–Leffler kernel is investigated which offers a proper model for hereditary effects without any singular behavior. Two efficient semi-analytical solution techniques, compared to each other, are developed, namely the Atangana–Baleanu Elzaki Adomian decomposition method (ABEADM) and the Atangana–Baleanu fractional reduced differential transform method (ABFRDTM). In the ABEADM, the Elzaki integral transform is applied to solve the linear terms in the model, while the polynomial decomposition technique is used to solve the nonlinear terms; On the other hand, in the ABFRDTM a recursive solution procedure is developed without any polynomial decomposition in order to reduce the computational complexity. Systematic recursive solution formulas are obtained for both the methods and convergence is shown by employing the Banach fixed point theorem along with an a priori truncation error estimate. The proposed methodologies are used on a nonlinear ABC time-fractional reaction–diffusion model, a closed-form Mittag–Leffler solution of which is derived and verified by direct substitution. The numerical results for fractional orders \(α =0.6,0.8,1.0\) show that both approaches converge quickly, and are in good agreement with the exact solution, with the classical integer-order behaviour recovered as \(α → 1\) . Moreover, the results illustrate the characteristic sub-diffusive behaviour for fractional memory and error analysis show uniform small approximation errors in the computational domain. The accuracy and computational efficiency of the proposed schemes are also competitive as evidenced by comparison with the Formable Transform Adomian Decomposition Method (FTADM). The novelty of the work lies in (i) proving that, for this benchmark, the two schemes generate identical iterates, so that the Adomian-polynomial step of ABEADM can be bypassed by the cheaper ABFRDTM recursion without loss of accuracy; (ii) a Banach contraction proof with an explicit a priori error bound for the ABC setting; and (iii) an explicit Mittag–Leffler representation of the solution. A dedicated discussion of the modelling assumptions and limitations (one space dimension, exponential initial data, constant coefficients) is also provided.