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Application of direct meshless local Petrov–Galerkin (DMLPG) method for some Turing-type models

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Abstract

Mathematical modeling of pattern formation in developmental biology leads to non-linear reaction–diffusion systems which are usually highly stiff in both diffusion and reaction terms. In this paper, the Direct Meshless Local Petrov–Galerkin (DMLPG) procedure is applied to find the numerical solution of some non-linear time-dependent reaction–diffusion systems such as Schnakenberg model, Gierer–Meinhardt model, FitzHugh–Nagumo model and Gray–Scott model. As far as we know, it is the first time that DMLPG method is applied for solving non-linear partial differential equations (PDEs) and systems of PDEs. Computational efficiency is the most significant advantage of the DMLPG method in comparison with the classic Meshless Local Petrov–Galerkin (MLPG) method. This is due to the fact that DMLPG shifts the numerical integrations over low-degree polynomials instead of over complicated moving least squares (MLS) shape functions and this reduces the computational costs, significantly. The main aim of this paper is to show that the DMLPG method is also suitable for solving the non-linear time-dependent systems, especially reaction–diffusion systems. Numerical results support the good efficiency of the proposed method for solving non-linear reaction–diffusion systems. Also it is shown that DMLPG provides considerable savings in computational time in comparison with the classical MLPG method.

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Acknowledgments

The authors thank both reviewers for their useful comments and suggestions that improved the paper.

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  1. Department of Applied Mathematics, Faculty of Mathematics and Computer Sciences, Amirkabir University of Technology, No. 424, Hafez Ave., Tehran, 15914, Iran

    Mohammad Ilati & Mehdi Dehghan

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  1. Mohammad Ilati
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Correspondence to Mehdi Dehghan.

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Ilati, M., Dehghan, M. Application of direct meshless local Petrov–Galerkin (DMLPG) method for some Turing-type models. Engineering with Computers 33, 107–124 (2017). https://doi.org/10.1007/s00366-016-0458-x

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