Anomalous diffusion shows up in places where ordinary diffusion equations fall short: porous media, heterogeneous materials, biological tissues and other complex physical systems. Classical integer-order diffusion models cannot account for memory and nonlocal interactions, which is precisely where fractional calculus enters. A new study published in Discover Applied Sciences tackles this gap by developing, analyzing and comparing, 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 by quantifying how the fractional order changes the solution relative to the classical model.
The Atangana–Baleanu–Caputo (ABC) fractional derivative carries a nonsingular Mittag–Leffler kernel, which the authors describe as offering a proper model for hereditary effects without any singular behavior. The equation under study is nonlinear, combining reaction, diffusion and convection terms in a single time-fractional model, making it a demanding benchmark for any solution technique.
Two efficient semi-analytical solution techniques are developed and compared to each other. The first is the Atangana–Baleanu Elzaki Adomian decomposition method, abbreviated ABEADM. In this approach the Elzaki integral transform is applied to solve the linear terms in the model, while the polynomial decomposition technique is used to handle the nonlinear terms. The second is the Atangana–Baleanu fractional reduced differential transform method, or ABFRDTM, in which a recursive solution procedure is developed without any polynomial decomposition in order to reduce computational complexity. Both methods operate on the same governing equation and target the same fractional order, allowing a direct head-to-head comparison.
Systematic recursive solution formulas are obtained for both methods, and convergence is established by employing the Banach fixed point theorem along with an a priori truncation error estimate.
The proposed methodologies are applied to a nonlinear ABC time-fractional reaction–diffusion model for which a closed-form Mittag–Leffler solution is derived and verified by direct substitution. The explicit Mittag–Leffler representation of the solution is listed among the novelties of the work.
Numerical results for fractional orders α = 0.6, 0.8 and 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 approaches 1. The results also illustrate the characteristic sub-diffusive behaviour associated with fractional memory, and error analysis shows uniform small approximation errors across 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). But the most striking finding is structural: the authors prove that, for this benchmark, the two schemes generate identical iterates. This means the Adomian-polynomial step of ABEADM can be bypassed by the cheaper ABFRDTM recursion without any loss of accuracy.
The novelty of the work lies in three elements: the proof that the two schemes generate identical iterates for this benchmark, the Banach contraction proof with an explicit a priori error bound for the ABC setting, and the explicit Mittag–Leffler representation of the solution.
A dedicated discussion of the modelling assumptions and limitations is also provided. The authors note that the framework is restricted to one space dimension, uses exponential initial data, and assumes constant coefficients.
Subject of Research: Semi-analytical solution methods for nonlinear time-fractional reaction–diffusion equations with the Atangana–Baleanu–Caputo derivative
Article Title: A unified comparative framework for solving nonlinear Atangana–Baleanu–Caputo fractional reaction–diffusion equations using semi-analytical methods
Article References: Alshammari, S., Al Oweidi, K. F., Sanaee, H., Alshammari, M., Alsheekhhussain, Z., & Al-sawalha, M. M. (2026). A unified comparative framework for solving nonlinear Atangana–Baleanu–Caputo fractional reaction–diffusion equations using semi-analytical methods. Discover Applied Sciences. https://doi.org/10.1007/s42452-026-09824-2
Image Credits: AI Generated
DOI: 10.1007/s42452-026-09824-2
Keywords: fractional calculus, Atangana–Baleanu–Caputo derivative, reaction–diffusion equations, semi-analytical methods, Adomian decomposition, Elzaki transform, reduced differential transform, Mittag–Leffler function, convergence analysis, sub-diffusion, porous media, anomalous diffusion
News Source: Denise Maddox. (November 9, 2026). Two semi-analytical solvers converge on fractional reaction–diffusion equations. Scienmag.
