A Regularized Volterra-Fredholm Integral Approach for the Two-Dimensional Brusselator System and Turing Pattern Simulation

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In this paper, we propose a regularized integral approach for the two-dimensional Brusselator reaction–diffusion system posed on the unit disk. The nonlinear source terms are first extended and regularized by convolution with a compactly supported mollifier, which naturally leads to an enlarged computational domain Bε = B(0, 1 + ε). Using Duhamel’s principle and the heat kernel, the original reaction– diffusion problem is transformed into a nonlinear Volterra–Fredholm integral system. We establish local Lipschitz estimates for the associated nonlinear integral operators in a suitable Banach space and prove the local existence and uniqueness of the regularized solution by a fixed-point argument. For the numerical approximation, a Nystr¨om-type scheme is developed by applying quadrature rules simultaneously to the Volterra time integral and to the Fredholm spatial integral. Since the spatial domain is circular, the discretization is performed in polar coordinates. Numerical experiments are carried out with spatially heterogeneous input and output source terms, combining radial feed gradients and small stochastic perturbations. The simulations show the emergence of clear Turing-type patterns, characterized by localized activator peaks and complementary inhibitor distributions. Compared with standard finite-difference simulations of Brusselator-type models, the present integral formulation produces sharper and more clearly separated spatial structures.

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