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An efficient and fully explicit model to simulate delayed activator–inhibitor systems with anomalous diffusion

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Abstract

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.

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Acknowledgements

The author wishes to thank the anonymous reviewers and the editor in charge of handling this manuscript, for all their invaluable comments and suggestions. Their criticisms profoundly helped in improving the quality of this work. Finally, the author wishes to acknowledge the financial support from the National Council for Science and Technology of Mexico (CONACYT) through the Grant A1-S-45928.

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Correspondence to Jorge E. Macías-Díaz.

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Macías-Díaz, J.E. An efficient and fully explicit model to simulate delayed activator–inhibitor systems with anomalous diffusion. J Math Chem 57, 1902–1923 (2019). https://doi.org/10.1007/s10910-019-01046-9

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