Macías-Díaz, Jorge E. An efficient and fully explicit model to simulate delayed activator-inhibitor systems with anomalous diffusion. (English) Zbl 1421.92042 J. Math. Chem. 57, No. 8, 1902-1923 (2019). Summary: Departing from a two-dimensional hyperbolic system that describes the interaction between some activator and inhibitor substances in chemical reactions, we investigate a general form of that model using a finite-difference approach. The model under investigation is a nonlinear system consisting of two coupled partial differential equations with generalized reaction terms. The presence of two-dimensional diffusive terms consisting of fractional operators of the Riesz type is considered here, using spatial differentiation orders in the set \((0 , 1) \cup (1 , 2]\). We impose initial conditions on a closed and bounded rectangle, and a four-step fully explicit finite-difference methodology based on the use of fractional-order centered differences is proposed. Among the most important results of this work, we establish analytically the second-order consistency of our scheme. Moreover, a discrete form of the energy method is employed to prove the stability, the boundedness and the quadratic convergence of the technique. Some numerical simulations obtained through our method show the appearance of Turing patterns and wave instabilities, in agreement with some reports found in the literature on superdiffusive hyperbolic activator-inhibitor systems. Cited in 3 Documents MSC: 92E20 Classical flows, reactions, etc. in chemistry 35Q92 PDEs in connection with biology, chemistry and other natural sciences 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 65Q10 Numerical methods for difference equations Keywords:activator-inhibitor system; anomalously diffusive hyperbolic system; fully explicit finite difference method; convergence; stability; pattern formation in molecular dynamics; discrete energy method PDFBibTeX XMLCite \textit{J. E. Macías-Díaz}, J. Math. Chem. 57, No. 8, 1902--1923 (2019; Zbl 1421.92042) Full Text: DOI References: [1] B.K. Agarwalla, S. Galhotra, J. Bhattacharjee, Diffusion driven instability to a drift driven one: turing patterns in the presence of an electric field. J. Math. Chem. 52(1), 188-197 (2014) · Zbl 1311.92210 [2] M. Al-Ghoul, B.C. 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