A Proposed Adaptive Inverse Multiquadric Shape Parameter Applied with the Dual Reciprocity BEM to Nonlinear and Coupled PDE

Kaennakham, Sayan; Chuathong, Nissaya

doi:10.3923/jas.2017.491.501

Cited by 3 publications

(4 citation statements)

References 15 publications

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“…[10][11]. From tabular illustrations and numerical results, again note that, our method gives better results and produces stable solutions than those suggested in literature 17 .…”

Section: Discussionmentioning

confidence: 75%

“…In this example the computational domain has been taken as Ω ̅ = [0, 1] × [0,1], a LSGrFEM 25 -26 is constructed. This example has been considered in literature 17 . In Tabs 5-6 the numerical solutions at 𝜖= 1 500 , ℎ = 1 20 , 𝑇 = 0.…”

Section: 𝑢(𝑥 𝑦 𝑡)mentioning

confidence: 99%

“…In Tabs 5-6 the numerical solutions at 𝜖= 1 500 , ℎ = 1 20 , 𝑇 = 0. 5 , 𝑘 = 0.001 and grid size 21 × 21 are computed, in tabular illustrations, our numerical results with the exact solutions and the solutions available in literature 17 are compared for various nodes of the grid. The tabulated results show that LSGrFEM…”

Section: 𝑢(𝑥 𝑦 𝑡)mentioning

confidence: 99%

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Numerical Analysis of Least-Squares Group Finite Element Method for Coupled Burgers' Problem

Noon

2021

Baghdad Sci.J

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In this paper, a least squares group finite element method for solving coupled Burgers' problem in 2-D is presented. A fully discrete formulation of least squares finite element method is analyzed, the backward-Euler scheme for the time variable is considered, the discretization with respect to space variable is applied as biquadratic quadrangular elements with nine nodes for each element. The continuity, ellipticity, stability condition and error estimate of least squares group finite element method are proved. The theoretical results show that the error estimate of this method is . The numerical results are compared with the exact solution and other available literature when the convection-dominated case to illustrate the efficiency of the proposed method that are solved through implementation in MATLAB R2018a.

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“…[10][11]. From tabular illustrations and numerical results, again note that, our method gives better results and produces stable solutions than those suggested in literature 17 .…”

Section: Discussionmentioning

confidence: 75%

Section: 𝑢(𝑥 𝑦 𝑡)mentioning

confidence: 99%

See 1 more Smart Citation

Numerical Analysis of Least-Squares Group Finite Element Method for Coupled Burgers' Problem

Noon

2021

Baghdad Sci.J

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“…, in Figure 2. Several investigations on this issue have shown that a good parameter depends on many factors such as the numerical method being utilised and the physics of the problem at hand [9][10][11][12]. With this problem of choosing a good shape being a pain in the neck of successfully deploying RBFNs, this study, therefore, focuses on alternative structures of RBFNs which contain no shapes at all so that they can be referred to as 'shapefree'.…”

Section: Introductionmentioning

confidence: 99%

Image reconstruction by shapefree radial basis function neural networks (RBFNs)

Paewpolsong

Sriapai

Tavaen

et al. 2022

J. Phys.: Conf. Ser.

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With the growth of artificial intelligence technologies, the research on artificial neural networks (ANNs) has been paid much more attention. Radial basis function neural networks (RBFNs) are a type of ANNs that are referred to as models that replicate the role of biological neural networks. While their applications are growing in a wide range of areas, conventional forms of RBFs contain a highly problem-dependent shape parameter, making it not as convenient as one would expect. This work investigates the numerical effectiveness of RBFs containing no shapes, so they are referred to as ‘shapefree’, under the application of image reconstruction. Nine forms of shapefree RBFs have been gathered and implemented in conjunction with the RBFNs. Two popular images (known as Lena and Plane) are damaged in Salt-and-Pepper manner before being repaired by the networks using these shapefree RBFs. The overall performances are monitored based on error norm, CPU-time and storage, and condition number. This aims to provide useful information regarding choices of RBFs for future uses, to overcome the pain one faces from choosing a suitable value of shape parameter.

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On Multiquadric Shape Determining Strategies in Image Reconstruction Applications: A Comparative Study

Sriapai

Paewpolsong

Ritthison

et al. 2021

J. Phys.: Conf. Ser.

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After being introduced to approximate two-dimensional geographical surfaces in 1971, the multivariate radial basis functions (RBFs) have been receiving a great amount of attention from scientists and engineers. Over decades, RBFs have been applied to a wide variety of problems. Approximation, interpolation, classification, prediction, and neural networks are inevitable in nowadays science, engineering, and medicine. Moreover, numerically solving partial differential equations (PDEs) is also a powerful branch of RBFs under the name of the ‘Meshfree/Meshless’ method. Amongst many, the so-called ‘Generalized Multiquadric (GMQ)’ is known as one of the most used forms of RBFs. It is of (ɛ 2 + r 2) β form, where r = ║x-x Θ║2 for x, x Θ ∈ ℝ n represents the distance function. The key factor playing a very crucial role for MQ, or other forms of RBFs, is the so-called ‘shape parameter ɛ’ where selecting a good one remains an open problem until now. This paper focuses on measuring the numerical effectiveness of various choices of ɛ proposed in literature when used in image reconstruction problems. Condition number of the interpolation matrix, CPU-time and storage, and accuracy are common criteria being utilized. The results of the work shall provide useful information on selecting a ‘suitable and reliable choice of MQ-shape’ for further applications in general.

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A Proposed Adaptive Inverse Multiquadric Shape Parameter Applied with the Dual Reciprocity BEM to Nonlinear and Coupled PDE

Cited by 3 publications

References 15 publications

Numerical Analysis of Least-Squares Group Finite Element Method for Coupled Burgers' Problem

Numerical Analysis of Least-Squares Group Finite Element Method for Coupled Burgers' Problem

Image reconstruction by shapefree radial basis function neural networks (RBFNs)

On Multiquadric Shape Determining Strategies in Image Reconstruction Applications: A Comparative Study

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