A meshless method for solving nonlinear two-dimensional integral equations of the second kind on non-rectangular domains using radial basis functions with error analysis

Assari, Pouria; Adibi, Hojatollah; Dehghan, Mehdi

doi:10.1016/j.cam.2012.09.010

Cited by 79 publications

(27 citation statements)

References 41 publications

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“…These sets of scattered nodes are usually called field nodes (or collocation points). There are three types of meshless techniques: meshless techniques based on weak forms such as the element free Galerkin method , meshless techniques based on collocation techniques (strong forms) such as the meshless collocation technique based on radial basis functions (RBFs) , and meshless techniques based on the combination of weak forms and collocation technique. Because of the ill‐conditioning of the resultant linear systems in RBF‐collocation method, various approaches are proposed to overcome this difficulty; References being among them.…”

Section: Preliminariesmentioning

confidence: 99%

On the convergence analysis, stability, and implementation of meshless local radial point interpolation on a class of three‐dimensional wave equations

Shivanian

2015

Numerical Meth Engineering

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SUMMARYIn this article, the meshless local radial point interpolation method is applied to analyze three space dimensional wave equations of the form u t t C˛u t Cˇu D u xx C u yy C u´´C f .x; y;´; t/; x T D .x; y;´/ 2 Â R d ; .d D 2; 3/; t > 0 subject to given initial and Dirichlet boundary conditions. The main difficulty of the great number of methods in full 3-D problems is the large computational costs. In meshless local radial point interpolation method, it does not require any background integration cells, so that all integrations are carried out locally over small quadrature domains of regular shapes such as circles or squares in two dimensions and spheres or cubes in three dimensions. The point interpolation method with the help of radial basis functions is proposed to construct shape functions that have Kronecker delta function property. A weak formulation with the Heaviside step function converts the set of governing equations into local integral equations on local subdomains. A two-step time discretization method is employed to evaluate the time derivatives. This suggests Crank-Nicolson technique to be applied on the right hand side of the equation. The convergence analysis and stability of the method are fully discussed. Three illustrative examples are presented, and satisfactory agreements are achieved. It is shown theoretically that the proposed method is unconditionally stable for the second example whereas it is not for the first and third ones.

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Section: Preliminariesmentioning

confidence: 99%

On the convergence analysis, stability, and implementation of meshless local radial point interpolation on a class of three‐dimensional wave equations

Shivanian

2015

Numerical Meth Engineering

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“…There are three kinds of meshless methods: meshless methods based on weak forms such as the element free Galerkin (EFG) method (and its developments like for example improved EFG (IEFG), complex variable (EFG) (CVEFG) and improved complex variable (EFG) (ICVEFG)), meshless local Petrov-Galerkin (MLPG) method [4][5][6][7][8][9][10] and MLPG based on the particular solutions (MLPG-PS), meshless techniques based on collocation techniques (strong forms) such as the meshless collocation technique based on radial basis functions (RBFs) [11][12][13][14][15][16][17][18][19] and the method of approximate particular solutions (MAPS) [20] and finally meshless techniques based on the combination of weak forms and collocation technique [21][22][23][24][25][26]. Due to the ill-conditioning of the resultant linear systems in RBF-collocation method, various approaches such as some regularization methods are proposed to overcome this difficulty, Refs.…”

Section: Introductionmentioning

confidence: 99%

A new spectral meshless radial point interpolation (SMRPI) method: A well-behaved alternative to the meshless weak forms

Shivanian

2015

Engineering Analysis with Boundary Elements

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“…The books edited by Golberg [15], Delves and Mohamed [16], Atkinson [17,18], and Baker [19] contain these methods. More information for other methods, see [20][21][22][23][24][25].…”

Section: Introductionmentioning

confidence: 99%

Numerical Treatment of the Nonlocal Solution of Hammerstein –Volterra Integral Equation with Continuous Kernels

Kojok¹

2017

IJARCSSE

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A meshless method for solving nonlinear two-dimensional integral equations of the second kind on non-rectangular domains using radial basis functions with error analysis

Cited by 79 publications

References 41 publications

On the convergence analysis, stability, and implementation of meshless local radial point interpolation on a class of three‐dimensional wave equations

On the convergence analysis, stability, and implementation of meshless local radial point interpolation on a class of three‐dimensional wave equations

A new spectral meshless radial point interpolation (SMRPI) method: A well-behaved alternative to the meshless weak forms

Numerical Treatment of the Nonlocal Solution of Hammerstein –Volterra Integral Equation with Continuous Kernels

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