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A Green’s function approach for the numerical solution of a class of fractional reaction–diffusion equations

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  • Hernandez-Martinez, Eliseo
  • Valdés-Parada, Francisco
  • Alvarez-Ramirez, Jose
  • Puebla, Hector
  • Morales-Zarate, Epifanio

Abstract

Reaction–diffusion equations with spatial fractional derivatives are increasingly used in various science and engineering fields to describe spatial patterns arising from the interaction of chemical or biochemical reactions and anomalous diffusive transport mechanisms. Most numerical schemes to solve fractional reaction–diffusion equations use finite difference schemes based on the Grünwald–Letnikov formula. This work introduces a new systematic approach based on Green’s function formulations to obtain numerical schemes for fractional reaction–diffusion equations. The idea is to pose an integral formulation of the equation in terms of the underlying Green’s function of the fractional operator to subsequently use numerical quadrature to obtain a set of ordinary differential equations. To illustrate the numerical accuracy of the method, dynamic and steady-state situations are considered and compared with analytical and numerical solutions via Grünwald finite differences schemes. Numerical simulations show that the scheme proposed improves the performance and convergence of traditional finite differences schemes based on Grünwald formula.

Suggested Citation

  • Hernandez-Martinez, Eliseo & Valdés-Parada, Francisco & Alvarez-Ramirez, Jose & Puebla, Hector & Morales-Zarate, Epifanio, 2016. "A Green’s function approach for the numerical solution of a class of fractional reaction–diffusion equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 121(C), pages 133-145.
    • Handle: RePEc:eee:matcom:v:121:y:2016:i:c:p:133-145
    DOI: 10.1016/j.matcom.2015.09.004
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    References listed on IDEAS

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    1. Paradisi, Paolo & Cesari, Rita & Mainardi, Francesco & Tampieri, Francesco, 2001. "The fractional Fick's law for non-local transport processes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 293(1), pages 130-142.
    2. Gafiychuk, V.V. & Datsko, B.Yo., 2006. "Pattern formation in a fractional reaction–diffusion system," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 365(2), pages 300-306.
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