This work developed an adaptive power expansion iterative scheme (APEIS) for numerically investigating nonlinear fractional systems involving Atangana–Baleanu–Caputo (ABC) operators. The proposed framework was applied to a time-fractional Hindmarsh–Rose (TF-HR) reaction–diffusion system to analyze memory-dependent signal propagation dynamics. Existence, uniqueness, convergence, and Hyers–Ulam stability results were established to provide the theoretical foundation of the proposed methodology. Numerical simulations were carried out for different fractional orders through approximate solution profiles, residual-error analysis, and wave-propagation investigations. A comparative study with the classical Adomian decomposition method (ADM) demonstrated close agreement between the two approaches, while APEIS consistently produced smaller residual errors and improved convergence behavior. The numerical results revealed that fractional memory effects strongly influence oscillatory dynamics and signal propagation. In particular, smaller fractional orders lead to smoother wave evolution with stronger memory effects, whereas larger fractional orders generate faster oscillatory behavior with weaker memory influence. These findings confirm the accuracy, efficiency, and reliability of the proposed APEIS framework for memory-dependent nonlinear reaction–diffusion systems.
Citation: Othman Abdullah Almatroud, Faten H. Damag, Sahar Albosaily, Marwa Ennaceur. Adaptive power expansion iterative scheme for time-fractional Hindmarsh–Rose systems: Analysis of nonlinear signal propagation and memory dynamics[J]. AIMS Mathematics, 2026, 11(8): 26897-26926. doi: 10.3934/math.20261079
This work developed an adaptive power expansion iterative scheme (APEIS) for numerically investigating nonlinear fractional systems involving Atangana–Baleanu–Caputo (ABC) operators. The proposed framework was applied to a time-fractional Hindmarsh–Rose (TF-HR) reaction–diffusion system to analyze memory-dependent signal propagation dynamics. Existence, uniqueness, convergence, and Hyers–Ulam stability results were established to provide the theoretical foundation of the proposed methodology. Numerical simulations were carried out for different fractional orders through approximate solution profiles, residual-error analysis, and wave-propagation investigations. A comparative study with the classical Adomian decomposition method (ADM) demonstrated close agreement between the two approaches, while APEIS consistently produced smaller residual errors and improved convergence behavior. The numerical results revealed that fractional memory effects strongly influence oscillatory dynamics and signal propagation. In particular, smaller fractional orders lead to smoother wave evolution with stronger memory effects, whereas larger fractional orders generate faster oscillatory behavior with weaker memory influence. These findings confirm the accuracy, efficiency, and reliability of the proposed APEIS framework for memory-dependent nonlinear reaction–diffusion systems.
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Othman Abdullah Almatroud, Faten H. Damag, Sahar Albosaily, Marwa Ennaceur. Adaptive power expansion iterative scheme for time-fractional Hindmarsh–Rose systems: Analysis of nonlinear signal propagation and memory dynamics[J]. AIMS Mathematics, 2026, 11(8): 26897-26926. doi: 10.3934/math.20261079
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