Numerical simulation of Turing patterns in a fractional hyperbolic reaction-diffusion model with Grünwald differences

  • J. E. Macías-DíazEmail author
  • Ahmed S. Hendy
Regular Article


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 (1, 2] . We impose initial conditions on a closed and bounded rectangle, and a finite-difference methodology based on the use of weighted-shifted Grünwald differences is proposed. Among the most important results of this work, we establish analytically the second-order consistency of our scheme. Moreover, the discrete energy method is employed to prove the stability 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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Copyright information

© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Departamento de Matemáticas y FísicaUniversidad Autónoma de AguascalientesAguascalientesMexico
  2. 2.Department of Computational Mathematics and Computer Science, Institute of Natural Sciences and MathematicsUral Federal UniversityYekaterinburgRussia
  3. 3.Department of Mathematics, Faculty of ScienceBenha UniversityBenha CityEgypt

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