A local integral equation formulation based on moving Kriging interpolation for solving coupled nonlinear reaction-diffusion equations.

*(English)*Zbl 1302.65227Summary: The meshless local Pretrov-Galerkin method (MLPG) with the test function in view of the Heaviside step function is introduced to solve the system of coupled nonlinear reaction-diffusion equations in two-dimensional spaces subjected to Dirichlet and Neumann boundary conditions on a square domain. Two-field velocities are approximated by moving Kriging (MK) interpolation method for constructing nodal shape function which holds the Kronecker delta property, thereby enhancing the arrangement nodal shape construction accuracy, while the Crank-Nicolson method is chosen for temporal discretization. The nonlinear terms are treated iteratively within each time step. The developed formulation is verified in two numerical examples with investigating the convergence and the accuracy of numerical results. The numerical experiments revealing the solutions by the developed formulation are stable and more precise.

##### MSC:

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

35K57 | Reaction-diffusion equations |

35K40 | Second-order parabolic systems |

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\textit{K. Yimnak} and \textit{A. Luadsong}, Adv. Math. Phys. 2014, Article ID 196041, 7 p. (2014; Zbl 1302.65227)

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##### References:

[1] | A. Turing, “The chemical basis of morphogenesis,” Philosophical Transactions of the Royal Society B, vol. 237, pp. 37-72, 1952. · Zbl 1403.92034 |

[2] | T. LeppĂ¤nen, Computational studies of pattern formation in turing systems [Ph.D. thesis], Helsinki University of Technology, Espoo, Finland, 2004. |

[3] | I. Prigogine, “Etude Thermodynamics des Phenomenes Irreversibles (Study of the thermodymamics of irreversible phenomenon),” in Presented to the Science Faculty at the Free University of Brussels (1945), Dunod, Paris, France, 1947. |

[4] | A. Shirzadi, V. Sladek, and J. Sladek, “A local integral equation formulation to solve coupled nonlinear reaction-diffusion equations by using moving least square approximation,” Engineering Analysis with Boundary Elements, vol. 37, no. 1, pp. 8-14, 2013. · Zbl 1291.65296 |

[5] | L. Gu, “Moving kriging interpolation and element-free Galerkin method,” International Journal for Numerical Methods in Engineering, vol. 56, no. 1, pp. 1-11, 2003. · Zbl 1062.74652 |

[6] | L. Chen and K. M. Liew, “A local Petrov-Galerkin approach with moving Kriging interpolation for solving transient heat conduction problems,” Computational Mechanics, vol. 47, no. 4, pp. 455-467, 2011. · Zbl 1241.80005 |

[7] | E. H. Twizell, A. B. Gumel, and Q. Cao, “A second-order scheme for the “Brusselator” reaction-diffusion system,” Journal of Mathematical Chemistry, vol. 26, no. 4, pp. 297-316, 1999. · Zbl 1016.92049 |

[8] | Siraj-ul-Islam, A. Ali, and S. Haq, “A computational modeling of the behavior of the two-dimensional reaction-diffusion Brusselator system,” Applied Mathematical Modelling, vol. 34, no. 12, pp. 3896-3909, 2010. · Zbl 1201.65185 |

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