A meshless RBF method for computing a numerical solution of unsteady Burgers’-type equations

Bouhamidi, A.; Hached, Mustapha; Jbilou, Khalide

doi:10.1016/j.camwa.2014.05.022

Cited by 14 publications

(6 citation statements)

References 25 publications

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“…and ζ n can be computed from Equation (15). One can write Equations (5), (17), (18) and boundary conditions (2) in the matrix form as follows…”

Section: Lie-group Methodsmentioning

confidence: 99%

“…Fan et al [16] developed a mesh-free numerical method based on a combination of the multiquadric RBFs (LRBFCM) and the fictitious time integration method (FTIM) to solve the two-dimensional Burgers' equations. Bouhamidi et al [17] presented RBFs interpolation technique for spatial discretization and implicit Runge-Kutta (IRK) schemes for temporal discretization of the unsteady coupled Burgers'-type equations. Xie et al [18] approximated the solution of the Burgers' equation spatially by the multiquadric MQ-RBF and used C-N finite difference scheme as temporal discretization technique.…”

Section: Introductionmentioning

confidence: 99%

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A Meshless Method for Burgers’ Equation Using Multiquadric Radial Basis Functions With a Lie-Group Integrator

Seydaoğlu

2019

Mathematics

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An efficient technique is proposed to solve the one-dimensional Burgers’ equation based on multiquadric radial basis function (MQ-RBF) for space approximation and a Lie-Group scheme for time integration. The comparisons of the numerical results obtained for different values of kinematic viscosity are made with the exact solutions and the reported results to demonstrate the efficiency and accuracy of the algorithm. It is shown that the numerical solutions concur with existing results and the proposed algorithm is efficient and can be easily implemented.

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“…and ζ n can be computed from Equation (15). One can write Equations (5), (17), (18) and boundary conditions (2) in the matrix form as follows…”

Section: Lie-group Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Meshless Method for Burgers’ Equation Using Multiquadric Radial Basis Functions With a Lie-Group Integrator

Seydaoğlu

2019

Mathematics

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“…methods, which are based on radial basis functions (RBFs), have been developed for solving PDEs by Bouhamidi et al (2014), Dehghan and Mohammadi (2017); Mehrabi and Voosoghi (2018), Golbabai et al (2015).…”

Section: Radial Basis Functionmentioning

confidence: 99%

A local radial basis function differential quadrature semi-discretisation technique for the simulation of time-dependent reaction-diffusion problems

Jiwari

Gerisch

2021

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Purpose This paper aims to develop a meshfree algorithm based on local radial basis functions (RBFs) combined with the differential quadrature (DQ) method to provide numerical approximations of the solutions of time-dependent, nonlinear and spatially one-dimensional reaction-diffusion systems and to capture their evolving patterns. The combination of local RBFs and the DQ method is applied to discretize the system in space; implicit multistep methods are subsequently used to discretize in time. Design/methodology/approach In a method of lines setting, a meshless method for their discretization in space is proposed. This discretization is based on a DQ approach, and RBFs are used as test functions. A local approach is followed where only selected RBFs feature in the computation of a particular DQ weight. Findings The proposed method is applied on four reaction-diffusion models: Huxley’s equation, a linear reaction-diffusion system, the Gray–Scott model and the two-dimensional Brusselator model. The method captured the various patterns of the models similar to available in literature. The method shows second order of convergence in space variables and works reliably and efficiently for the problems. Originality/value The originality lies in the following facts: A meshless method is proposed for reaction-diffusion models based on local RBFs; the proposed scheme is able to capture patterns of the models for big time T; the scheme has second order of convergence in both time and space variables and Nuemann boundary conditions are easy to implement in this scheme.

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“…also proposed and implemented the method of particular solutions using inverse MQ and polyharmonic RBFs based on the finite difference scheme in time to solve one-dimensional time-dependent inhomogeneous Burgers' equation. A meshless method for the computation of a numerical solution of the unsteady Burgers-type equation based on thin-plate spline RBF was discussed in (Bouhamidi et al 2014). Fan et al (2014) proposed a meshfree approximation scheme derived from a combination of the MQ-RBF collocation method and the fictitious time integration method to approximate the solutions of the two-dimensional Burgers' equations.…”

Section: Introductionmentioning

confidence: 99%

An efficient Strang splitting technique combined with the multiquadric-radial basis function for the Burgers’ equation

2021

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In the present paper, two effective numerical schemes depending on a second-order Strang splitting technique are presented to obtain approximate solutions of the one-dimensional Burgers' equation utilizing the collocation technique and approximating directly the solution by multiquadric-radial basis function (MQ-RBF) method. To show the performance of both schemes, we have considered two examples of Burgers' equation. The obtained numerical results are compared with the available exact values and also those of other published methods. Moreover, the computed L 2 and L ∞ error norms have been given. It is found that the presented schemes produce better results as compared to those obtained almost all the schemes present in the literature.

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A meshless RBF method for computing a numerical solution of unsteady Burgers’-type equations

Cited by 14 publications

References 25 publications

A Meshless Method for Burgers’ Equation Using Multiquadric Radial Basis Functions With a Lie-Group Integrator

A Meshless Method for Burgers’ Equation Using Multiquadric Radial Basis Functions With a Lie-Group Integrator

A local radial basis function differential quadrature semi-discretisation technique for the simulation of time-dependent reaction-diffusion problems

An efficient Strang splitting technique combined with the multiquadric-radial basis function for the Burgers’ equation

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