A theoretical and computational study of the fractional Brusselator system with neural network validation

This work introduces the time-space fractional Brusselator (TSFB) model, a generalization of the classical reaction-diffusion system incorporating fractional orders $$\sigma$$ (time, $$0<\sigma \le 1$$) and $$\eta$$ (space, $$1<\eta \le 2$$) to capture memory effects and non-local interactions. A complete qualitative analysis is established, including existence, uniqueness, and Ulam–Hyers (UH) stability. A forward time central space (FTCS) scheme is developed with comprehensive stability analysis. Numerical solutions are validated using a feedforward neural network (NN) (30 and 25 neurons in two hidden layers). Detailed simulations illustrate how the fractional parameters $$\sigma$$ and $$\eta$$ influence solution behavior in two and three dimensions. Quantitative analysis reveals that decreasing $$\sigma$$ from 1.0 to 0.6 intensifies pattern formation, with peak activator concentration rising from 0.411 to 1.592. The spatial order $$\eta$$ governs pattern spacing: at $$\sigma =0.6$$, reducing $$\eta$$ from 2.0 to 1.6 produces finer patterns with characteristic spacing decreasing from 1.03 to 0.74. The combination $$(\sigma =0.6,\eta =1.6)$$ produces the most intense patterns, yielding the steepest concentration gradients of 2.168 and 3.896. The principal novelty lies in the unified framework combining fractional operators, complete analysis, the FTCS scheme, and neural computing for the TSFB system, which enhances the understanding of pattern formation and spacing in relation to the parameters $$\sigma$$ and $$\eta$$.