The closed-form particular solutions for Laplace and biharmonic operators using a Gaussian function

Lamichhane, A.R.; Chen, C.S.

doi:10.1016/j.aml.2015.02.004

Cited by 21 publications

(3 citation statements)

References 11 publications

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“…A key feature of MAPS is that the particular solutions of a differential equation with a chosen RBF need to be derived analytically. In the literature, particular solutions for various differential operators for some commonly used RBFs have been derived . A list of the particular solutions with respect to PS and MQ for the Laplacian Δ in

{boldR}^{2}

and

{boldR}^{3}

are listed in Tables and .…”

Section: Maps Using Polyharmonic Splinesmentioning

confidence: 99%

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A modified method of approximate particular solutions for solving linear and nonlinear PDEs

Yao

Chen

Zheng

2017

Numerical Methods Partial

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The method of approximate particular solutions (MAPS) was first proposed by Chen et al. in Chen, Fan, and Wen, Numer Methods Partial Differential Equations, 28 (2012), 506–522. using multiquadric (MQ) and inverse multiquadric radial basis functions (RBFs). Since then, the closed form particular solutions for many commonly used RBFs and differential operators have been derived. As a result, MAPS was extended to Matérn and Gaussian RBFs. Polyharmonic splines (PS) has rarely been used in MAPS due to its conditional positive definiteness and low accuracy. One advantage of PS is that there is no shape parameter to be taken care of. In this article, MAPS is modified so PS can be used more effectively. In the original MAPS, integrated RBFs, so called particular solutions, are used. An additional integrated polynomial basis is added when PS is used. In the modified MAPS, an additional polynomial basis is directly added to the integrated RBFs without integration. The results from the modified MAPS with PS can be improved by increasing the order of PS to a certain degree or by increasing the number of collocation points. A polynomial of degree 15 or less appeared to be working well in most of our examples. Other RBFs such as MQ can be utilized in the modified MAPS as well. The performance of the proposed method is tested on a number of examples including linear and nonlinear problems in 2D and 3D. We demonstrate that the modified MAPS with PS is, in general, more accurate than other RBFs for solving general elliptic equations.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1839–1858, 2017

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{boldR}^{2}

and

{boldR}^{3}

are listed in Tables and .…”

Section: Maps Using Polyharmonic Splinesmentioning

confidence: 99%

“…It is slightly more accurate compared to the direct RBF collocation techniques. MAPS has been extended to many RBFs including Gaussian, MQ, IMQ, and Matérn RBFs . These RBFs all contain shape parameters and are infinitely smooth.…”

Section: Introductionmentioning

confidence: 99%

A modified method of approximate particular solutions for solving linear and nonlinear PDEs

Yao

Chen

Zheng

2017

Numerical Methods Partial

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“…It should be noted that the shape parameter in the MQ plays an essential role for the numerical solution in terms of the accuracy and stability [10,11]. us, the selection of the shape parameter in MQ is a big issue and such a problem also occurs for some other radial basis functions such as the Gaussian radial basis function [12,13]. In this paper, our investigation focuses on the conical radial basis function which is a parameter-free function [14].…”

Section: Introductionmentioning

confidence: 99%

The Conical Radial Basis Function for Partial Differential Equations

Zhang

Wang

Hou

2020

Journal of Mathematics

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The performance of the parameter-free conical radial basis functions accompanied with the Chebyshev node generation is investigated for the solution of boundary value problems. In contrast to the traditional conical radial basis function method, where the collocation points are placed uniformly or quasi-uniformly in the physical domain of the boundary value problems in question, we consider three different Chebyshev-type schemes to generate the collocation points. This simple scheme improves accuracy of the method with no additional computational cost. Several numerical experiments are given to show the validity of the newly proposed method.

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Another Mechanism to Control Invasive Species and Population Explosion: “Ecological” Damping Continued

Parshad

Yao

2017

Differ Equ Dyn Syst

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The closed-form particular solutions for Laplace and biharmonic operators using a Gaussian function

Cited by 21 publications

References 11 publications

A modified method of approximate particular solutions for solving linear and nonlinear PDEs

A modified method of approximate particular solutions for solving linear and nonlinear PDEs

The Conical Radial Basis Function for Partial Differential Equations

Another Mechanism to Control Invasive Species and Population Explosion: “Ecological” Damping Continued

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