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The Finite Point Method for Reaction-Diffusion Systems in Developmental Biology

Mehdi Tatari1, Maryam Kamranian2, Mehdi Dehghan2
Department of Mathematical Sciences Isfahan University of Technology, Isfahan, 84156-83111, Iran. E-mail address: mtatari@cc.iut.ac.ir (M. Tatari).
Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, No. 424, Hafez Ave., 15914, Tehran, Iran.
E-mail addresses: m.kamranian@math.iut.ac.ir (M. Kamranian), mdehghan@aut.ac.ir;mdehghan.aut@gmail.com (M. Dehghan).

Computer Modeling in Engineering & Sciences 2011, 82(1), 1-28. https://doi.org/10.32604/cmes.2011.082.001

Abstract

In this paper, the finite point method (FPM) is presented for solving nonlinear reaction-diffusion systems which are often employed in mathematical modeling in developmental biology. In order to avoid directly solving a coupled nonlinear system, a predicator-corrector scheme is applied. The finite point method is a truly meshfree technique based on the combination of the moving least squares approximation on a cloud of points with the point collocation method to discretize the governing equations. The lack of dependence on a mesh or integration procedure is an important feature, which makes the FPM simple, efficient and applicable to solve nonlinear problems in complex domains. We present numerical solutions for two cases of interest, namely the well-studied Schnakenberg model and the Gierer-Meinhardt model, in various regions to demonstrate the effects of various domain geometries on the resulting biological patterns.

Keywords

Reaction-diffusion systems, Meshfree methods, Finite point method, MLS approximation, Developmental biology, Meshless techniques.

Cite This Article

Tatari, M., Kamranian, M., Dehghan, M. (2011). The Finite Point Method for Reaction-Diffusion Systems in Developmental Biology. CMES-Computer Modeling in Engineering & Sciences, 82(1), 1–28.

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This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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