The numerical solution of Cahn–Hilliard (CH) equation in one, two and three-dimensions via globally radial basis functions (GRBFs) and RBFs-differential quadrature (RBFs-DQ) methods

Dehghan, Mehdi; Mohammadi, V.

doi:10.1016/j.enganabound.2014.10.008

Cited by 64 publications

(14 citation statements)

References 92 publications

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“…Ilati et al [72] used RBFs collocation and RBF-QR methods to seek the solution of one and two-dimension time-dependent coupled sine-Gorden equations and suggested that meshless techniques based on collocation methods are applicable for resolving the system of coupled nonlinear equations such as sineGorden equation. In the following of Dehghan's preceding research works, he [73] implemented two different meshless methods based on RBFs to solve Cahn-Hilliard (CH) equation in one, two and threedimensions.…”

Section: Radial Basis Function Neural Network (Rbf-nn) Backgroundmentioning

confidence: 99%

Application of a radial basis function neural network to estimate pressure gradient in water–oil pipelines

Halali

Azari

Arabloo

et al. 2016

Journal of the Taiwan Institute of Chemical Engineers

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Section: Radial Basis Function Neural Network (Rbf-nn) Backgroundmentioning

confidence: 99%

Application of a radial basis function neural network to estimate pressure gradient in water–oil pipelines

Halali

Azari

Arabloo

et al. 2016

Journal of the Taiwan Institute of Chemical Engineers

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“…Various meshless techniques have been developed in the literature [59][60][61][62] and a class of well-known meshless methods is the EFG method. The EFG method is a global method.…”

Section: A Brief Review For Element Free Galerkin (Efg)mentioning

confidence: 99%

Numerical study of three-dimensional Turing patterns using a meshless method based on moving Kriging element free Galerkin (EFG) approach

Dehghan

Abbaszadeh

2016

Computers & Mathematics with Applications

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Available online xxxx Keywords: Element free Galerkin (EFG) and movingKriging interpolation Fourth-order Runge-Kutta method Mathematical biology and Turing system Gray-Scott model and Allen-Cahn equation Ginzburg-Landau model and Brusselator equation Predator-prey model with additional food supply to predator a b s t r a c tIn this paper a numerical procedure is presented for solving a class of three-dimensional Turing system. First, we discrete the spatial direction using element free Galerkin (EFG) method based on the shape functions of moving Kriging interpolation. Then, to achieve a high-order accuracy, we use the fourth-order exponential time differencing Runge-Kutta (ETDRK4) method. Using this discretization for the temporal dimension, we obtain an explicit scheme and do not need to solve nonlinear system of equations. The EFG method uses a weak form of the considered equation that is similar to the finite element method with the difference that in the EFG method test and trial functions are moving least squares approximation (MLS) shape functions. Since the shape functions of moving least squares (MLS) approximation do not have Kronecker delta property, we cannot implement the essential boundary condition, directly. Also building shape functions of MLS approximation is a time consuming procedure. Because of the mentioned reasons we employ the shape functions of moving Kriging interpolation technique which have the mentioned property and less CPU time is required for building them. For testing this method on threedimensional PDEs, we select some equations and system of PDEs such as Allen-Cahn, Gray-Scott, Ginzburg-Landau, Brusselator models, predator-prey model with additional food supply to predator. Several test problems are solved and numerical simulations are reported which confirm the efficiency of the proposed scheme.

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“…They are classified into two main categories: RBFs meshfree methods based on strong forms such as radial basis collocation methods (Kansa's method) [27][28][29][30][31][32][33][34][35][36][37][38][39] and RBFs meshfree methods based on weak forms such as radial point interpolation method [40][41][42][43][44][45][46][47]. Very recently, Dehghan et al have developed and well-used some meshless techniques for solving various types of complicated problems [48][49][50][51][52][53][54]. Several types of meshfee methods have been developed and successfully employed to investigate various modes of fractional differential problems.…”

Section: ð1:4þmentioning

confidence: 99%

A meshfree method based on the radial basis functions for solution of two-dimensional fractional evolution equation

Ghehsareh

Bateni²,

Zaghian

2015

Engineering Analysis with Boundary Elements

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The numerical solution of Cahn–Hilliard (CH) equation in one, two and three-dimensions via globally radial basis functions (GRBFs) and RBFs-differential quadrature (RBFs-DQ) methods

Cited by 64 publications

References 92 publications

Application of a radial basis function neural network to estimate pressure gradient in water–oil pipelines

Application of a radial basis function neural network to estimate pressure gradient in water–oil pipelines

Numerical study of three-dimensional Turing patterns using a meshless method based on moving Kriging element free Galerkin (EFG) approach

A meshfree method based on the radial basis functions for solution of two-dimensional fractional evolution equation

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