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Approximation of continuous surface differential operators with the generalized moving least-squares (GMLS) method for solving reaction–diffusion equation

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Abstract

In this paper, a meshless approximation based on generalized moving least squares is applied to solve the reaction–diffusion equations on the sphere and red-blood cell surfaces. The proposed method is based on the projected gradient of the shape functions, and it approximates the Laplace operator defined on the surfaces that is called Laplace–Beltrami. This technique only requires nodes at locations on the surface and the corresponding normal vectors to the surface. To discretize the time variable, an explicit time technique based on the fourth-order Runge–Kutta is used. Some numerical results on Turing and Fitzhugh–Nagumo partial differential equations are given for showing patterns which are appeared in biological phenomena.

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Fig. 1

Taken from Kondo and Miura (2010)

Fig. 2

Taken from Cherry and Fenton (2008)

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Acknowledgements

The authors are very grateful to reviewer for carefully reading this paper and for his (her) comments and suggestions which have improved the paper.

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Authors and Affiliations

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

    Mehdi Dehghan & Niusha Narimani

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  1. Mehdi Dehghan
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  2. Niusha Narimani
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Correspondence to Mehdi Dehghan.

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Communicated by Abimael Loula.

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Dehghan, M., Narimani, N. Approximation of continuous surface differential operators with the generalized moving least-squares (GMLS) method for solving reaction–diffusion equation. Comp. Appl. Math. 37, 6955–6971 (2018). https://doi.org/10.1007/s40314-018-0716-1

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