An interpolating meshless method for the numerical simulation of the time-fractional diffusion equations with error estimates

Wang, Jufeng; Sun, Fengxin

doi:10.1108/ec-03-2019-0117

Cited by 8 publications

(5 citation statements)

References 42 publications

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“…The element-free Galerkin (EFG) method, which couples the Galerkin weak form and the MLS method, is a widely used mesh-free method [29]. In order to directly apply essential boundary conditions, the interpolating element-free Galerkin (IEFG) method is proposed by coupling the IMLS method [28,[30][31][32][33]. The IEFG method not only has the advantage of directly applying boundary conditions but also has the advantage of having a smaller radius of influence compared to EFG under the same basic functions.…”

Section: Introductionmentioning

confidence: 99%

A Dimension-Splitting Variational Multiscale Element-Free Galerkin Method for Three-Dimensional Singularly Perturbed Convection-Diffusion Problems

Wang¹,

Wu²,

Xu³

et al. 2023

Computer Modeling in Engineering &Amp; Sciences

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By introducing the dimensional splitting (DS) method into the multiscale interpolating element-free Galerkin (VMIEFG) method, a dimension-splitting multiscale interpolating element-free Galerkin (DS-VMIEFG) method is proposed for three-dimensional (3D) singular perturbed convection-diffusion (SPCD) problems. In the DS-VMIEFG method, the 3D problem is decomposed into a series of 2D problems by the DS method, and the discrete equations on the 2D splitting surface are obtained by the VMIEFG method. The improved interpolation-type moving least squares (IIMLS) method is used to construct shape functions in the weak form and to combine 2D discrete equations into a global system of discrete equations for the three-dimensional SPCD problems. The solved numerical example verifies the effectiveness of the method in this paper for the 3D SPCD problems. The numerical solution will gradually converge to the analytical solution with the increase in the number of nodes. For extremely small singular diffusion coefficients, the numerical solution will avoid numerical oscillation and has high computational stability. KEYWORDSDimension-splitting multiscale interpolating element-free Galerkin (DS-VMIEFG) method; interpolating variational multiscale element-free Galerkin (VMIEFG) method; dimension splitting method; singularly perturbed convection-diffusion problems

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Section: Introductionmentioning

confidence: 99%

A Dimension-Splitting Variational Multiscale Element-Free Galerkin Method for Three-Dimensional Singularly Perturbed Convection-Diffusion Problems

Wang¹,

Wu²,

Xu³

et al. 2023

Computer Modeling in Engineering &Amp; Sciences

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“…The differential quadrature method (DQM) is developed in Saeed and Umair (2019) for solving time and space fractional nonlinear partial differential equations on a semi-infinite domain. The main aim of Wang and Sun (2019) is to present an interpolating element-free Galerkin (IEFG) method for the numerical study of the time-fractional diffusion equation. A meshless analysis of two-dimensional two-sided space-fractional wave equations is proposed in Cheng et al (2018a) based on the improved moving least-squares (IMLS) approximation.…”

Section: Introductionmentioning

confidence: 99%

A reduced-order Jacobi spectral collocation method for solving the space-fractional FitzHugh–Nagumo models with application in myocardium

Abbaszadeh,

Bagheri Salec,

Kamel Abd Al-Khafaji

2023

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PurposeThe space fractional PDEs (SFPDEs) play an important role in the fractional calculus field. Proposing a high-order, stable and flexible numerical procedure for solving SFPDEs is the main aim of most researchers. This paper devotes to developing a novel spectral algorithm to solve the FitzHugh–Nagumo models with space fractional derivatives.Design/methodology/approachThe fractional derivative is defined based upon the Riesz derivative. First, a second-order finite difference formulation is used to approximate the time derivative. Then, the Jacobi spectral collocation method is employed to discrete the spatial variables. On the other hand, authors assume that the approximate solution is a linear combination of special polynomials which are obtained from the Jacobi polynomials, and also there exists Riesz fractional derivative based on the Jacobi polynomials. Also, a reduced order plan, such as proper orthogonal decomposition (POD) method, has been utilized.FindingsA fast high-order numerical method to decrease the elapsed CPU time has been constructed for solving systems of space fractional PDEs.Originality/valueThe spectral collocation method is combined with the POD idea to solve the system of space-fractional PDEs. The numerical results are acceptable and efficient for the main mathematical model.

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“…Wang et al studied the error convergence of the IMLS method [46,47]. The meshless method based on the IMLS method has a good calculation effect [48][49][50][51]. However, the singularity of the weight function in the IMLS method is not conducive to numerical calculation.…”

Section: Introductionmentioning

confidence: 99%

A Dimension Splitting-Interpolating Moving Least Squares (DS-IMLS) Method with Nonsingular Weight Functions

2021

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By introducing the dimension splitting method (DSM) into the improved interpolating moving least-squares (IMLS) method with nonsingular weight function, a dimension splitting–interpolating moving least squares (DS–IMLS) method is first proposed. Since the DSM can decompose the problem into a series of lower-dimensional problems, the DS–IMLS method can reduce the matrix dimension in calculating the shape function and reduce the computational complexity of the derivatives of the approximation function. The approximation function of the DS–IMLS method and its derivatives have high approximation accuracy. Then an improved interpolating element-free Galerkin (IEFG) method for the two-dimensional potential problems is established based on the DS–IMLS method. In the improved IEFG method, the DS–IMLS method and Galerkin weak form are used to obtain the discrete equations of the problem. Numerical examples show that the DS–IMLS and the improved IEFG methods have high accuracy.

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An interpolating meshless method for the numerical simulation of the time-fractional diffusion equations with error estimates

Cited by 8 publications

References 42 publications

A Dimension-Splitting Variational Multiscale Element-Free Galerkin Method for Three-Dimensional Singularly Perturbed Convection-Diffusion Problems

A Dimension-Splitting Variational Multiscale Element-Free Galerkin Method for Three-Dimensional Singularly Perturbed Convection-Diffusion Problems

A reduced-order Jacobi spectral collocation method for solving the space-fractional FitzHugh–Nagumo models with application in myocardium

A Dimension Splitting-Interpolating Moving Least Squares (DS-IMLS) Method with Nonsingular Weight Functions

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