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

Ilati, Mohammad; Dehghan, Mehdi

doi:10.1007/s00366-016-0458-x

Cited by 35 publications

(8 citation statements)

References 59 publications

(59 reference statements)

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“…Meshless methods can be broadly separated into two groups, namely, meshless methods based on strong form such as Kansa's-radial basis function (RBF) and collocation methods (Dehghan and Shokri, 2007;Hu et al, 2005;Jankowska et al, 2018;Kansa, 1990;Hussain and Haq, 2020;Lin et al, 2017;Singh et al, 2019;Xiong et al, 2018), smoothed particle hydrodynamics method (Liu et al, 2004) and finite point collocation method (Onate et al, 1996) and meshless methods based on weak form such as element-free Galerkin method (Belytschko et al, 1994), direct meshless local Petrov-Galerkin method (Ilati and Dehghan, 2017) and local radial point interpolation (MLRPI) method (Shivanian and Jafarabadi, 2018;Shivanian, 2015;Shivanian, 2016). In the methods of first group, by using collocation approach, governing equations and boundary conditions are discretized at the set of scattered nodes to get an algebraic system of equations.…”

Section: A Brief Review Of Numerical Methodsmentioning

confidence: 99%

Application of a collocation method based on linear barycentric interpolation for solving 2D and 3D Klein-Gordon-Schrödinger (KGS) equations numerically

Oruç¹

2020

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Purpose The purpose of this paper is to obtain accurate numerical solutions of two-dimensional (2-D) and 3-dimensional (3-D) Klein–Gordon–Schrödinger (KGS) equations. Design/methodology/approach The use of linear barycentric interpolation differentiation matrices facilitates the computation of numerical solutions both in 2-D and 3-D space within reasonable central processing unit times. Findings Numerical simulations corroborate the efficiency and accuracy of the proposed method. Originality/value Linear barycentric interpolation method is applied to 2-D and 3-D KGS equations for the first time, and good results are obtained.

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Section: A Brief Review Of Numerical Methodsmentioning

confidence: 99%

Application of a collocation method based on linear barycentric interpolation for solving 2D and 3D Klein-Gordon-Schrödinger (KGS) equations numerically

Oruç¹

2020

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“…Fu et al [331] proposed a domain-type meshless collocation method, called method of approximate particular solutions (MAPS), for numerical investigation on the effect of tumor on the thermal behavior inside the skin tissue. The Galerkin-based meshfree method has also been employed by the authors in [332,333] for numerical simulation of reaction-diffusion systems in developmental biology, which is one of the emerging areas of interest in computational biomechanics.…”

Section: Other Applicationsmentioning

confidence: 99%

Meshfree and Particle Methods in Biomechanics: Prospects and Challenges

Zhang

Ademiloye

Liew

2018

Arch Computat Methods Eng

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The use of meshfree and particle methods in the field of bioengineering and biomechanics has significantly increased. This may be attributed to their unique abilities to overcome most of the inherent limitations of mesh-based methods in dealing with problems involving large deformation and complex geometry that are common in bioengineering and computational biomechanics in particular. This review article is intended to identify, highlight and summarize research works on topics that are of substantial interest in the field of computational biomechanics in which meshfree or particle methods have been employed for analysis, simulation or/and modeling of biological systems such as soft matters, cells, biological soft and hard tissues and organs. We also anticipate that this review will serve as a useful resource and guide to researchers who intend to extend their work into these research areas. This review article includes 333 references. Highlights  We present an up to date review of meshfree and particle methods, including their advantages and limitations.  We comprehensively discuss past and recent applications of meshfree and particle methods in bioengineering and biomechanics.  We identify research areas, directions, and opportunities for the application of meshfree and particle methods in bioengineering and biomechanics.

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“…Meshless methods can be generally divided into two groups, one group is based on strong form such as Kansa's radial basis function (RBF) and collocation methods [21, 22, 28, 29, 31, 36, 38, 40–42, 53, 54, 63, 73], smoothing particle hydrodynmaics method [43], finite point collocation method [52], and so on and the other one is based on weak form such as element‐free Galerkin method [7], direct meshless local Petrov–Galerkin (DMLPG) method [33], local radial point interpolation (MLRPI) method [34, 60–62]. There are also meshless methods which use both weak and strong forms such as [18, 20].…”

Section: Introductionmentioning

confidence: 99%

Numerical simulation of two‐dimensional and three‐dimensional generalizedKlein–Gordon–Zakharovequations with power law nonlinearity via a meshless collocation method based on barycentric rational interpolation

Oruç

2022

Numerical Methods Partial

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This study presents numerical simulations of generalized two‐dimensional (2D) and three‐dimensional (3D) Klein–Gordon–Zakharov (KGZ) equations with power law nonlinearity, which are coupled nonlinear partial differential equations. A meshless collocation method based on barycentric rational interpolation is developed for space variable of the KGZ equations. For time discretization, an explicit low storage fourth order Runge Kutta method is proposed after transforming KGZ equations to system of ordinary differential equations by introducing auxiliary variables. L∞ and L2 error norms for some test problems are computed. Obtained numerical results and comparisons with finite element methods indicate that barycentric rational interpolation method is an efficient method for solving multidimensional generalized KGZ system numerically.

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

Cited by 35 publications

References 59 publications

Application of a collocation method based on linear barycentric interpolation for solving 2D and 3D Klein-Gordon-Schrödinger (KGS) equations numerically

Application of a collocation method based on linear barycentric interpolation for solving 2D and 3D Klein-Gordon-Schrödinger (KGS) equations numerically

Meshfree and Particle Methods in Biomechanics: Prospects and Challenges

Numerical simulation of two‐dimensional and three‐dimensional generalizedKlein–Gordon–Zakharovequations with power law nonlinearity via a meshless collocation method based on barycentric rational interpolation

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