Numerical solutions of elliptic partial differential equations using Chebyshev polynomials

Ghimire, B. Khatri; Tian, Haiyan; Lamichhane, A.R.

doi:10.1016/j.camwa.2016.06.012

Cited by 27 publications

(8 citation statements)

References 19 publications

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“…φ j (x, y) is the j th basis function. The basis functions, φ(x, y), are the RBFs [8]- [9], the Chebyshev polynomials [19]- [20] and high-order polynomial [21]. In this paper, we use the trigonometric functions to be the basis functions, which will be presented in the next subsection.…”

Section: Methods Of Particular Solutions (Mps)mentioning

confidence: 99%

“…Other than the RBFs as the basis function in the MPS, there are some other choices. In [19]- [20], the Chebyshev polynomials have been considered to be the basis functions in the MPS instead of the RBFs. Karageorghis and Kyza [19] adopted the Chebyshev polynomials to approximate the inhomogeneous term and used the MPS to accurately analyzed Poisson equation, inhomogeneous Helmholtz equation and inhomogeneous biharmonic equation.…”

Section: Introductionmentioning

confidence: 99%

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The method of particular solutions using trigonometric basis functions

Tian

Fan

et al. 2018

Journal of Computational and Applied Mathematics

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In this paper, the method of particular solutions (MPS) using trigonometric functions as the basis functions is proposed to solve two-dimensional elliptic partial differential equations. The inhomogeneous term of the governing equation is approximated by Fourier series and the closed-form particular solutions of trigonometric functions are derived using the method of undetermined coefficients. Once the particular solutions for the trigonometric basis functions are derived, the standard MPS can be applied for solving partial differential equations. In comparing with the use of radial basis functions and polynomials in the MPS, our proposed approach provide another simple approach to effectively solving two-dimensional elliptic partial differential equations. Five numerical examples are provided in this paper to validate the merits of the proposed meshless method.

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Section: Methods Of Particular Solutions (Mps)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The method of particular solutions using trigonometric basis functions

Tian

Fan

et al. 2018

Journal of Computational and Applied Mathematics

View full text Add to dashboard Cite

show abstract

“…the remaining challenge is to choose the series coefficients a i so that the residual of Equation ( 2) can be minimized. Although many types of "basis functions" are available, a good choice for most of all applications is the Fourier series [39][40][41][42][43]. Another popular used "basis function" is the well-known Chebyshev series, which is just a Fourier cosine expansion with a change of variable [25,40,42,44,45].…”

Section: Introductionmentioning

confidence: 99%

“…Although many types of "basis functions" are available, a good choice for most of all applications is the Fourier series [39][40][41][42][43]. Another popular used "basis function" is the well-known Chebyshev series, which is just a Fourier cosine expansion with a change of variable [25,40,42,44,45]. Once the particular solutions have been obtained, the solution of the original problem can then be converted to a homogeneous one which can be solved by using the BEM/MFS-based methods [11,13,14,19,[46][47][48][49][50][51][52][53][54][55][56][57][58][59][60][61][62][63][64].…”

Section: Introductionmentioning

confidence: 99%

A Pseudo-Spectral Fourier Collocation Method for Inhomogeneous Elliptical Inclusions with Partial Differential Equations

Wang

Yu³

2022

Mathematics

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Inhomogeneous elliptical inclusions with partial differential equations have aroused appreciable concern in many disciplines. In this paper, a pseudo-spectral collocation method, based on Fourier basis functions, is proposed for the numerical solutions of two- (2D) and three-dimensional (3D) inhomogeneous elliptic boundary value problems. We describe how one can improve the numerical accuracy by making some extra “reconstruction techniques” before applying the traditional Fourier series approximation. After the particular solutions have been obtained, the resulting homogeneous equation can then be calculated using various boundary-type methods, such as the method of fundamental solutions (MFS). Using Fourier basis functions, one does not need to use large matrices, making accrual computations relatively fast. Three benchmark numerical examples involving Poisson, Helmholtz, and modified-Helmholtz equations are presented to illustrate the applicability and accuracy of the proposed method.

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“…There are many papers concerned with the numerical solutions of elliptic type boundary value problems by using the meshless and mesh reduction methods, such as Zhu et al, 1998Zhu et al, , 1999Zhu, 1998a, 1998b;Atluri et al, 1999;Atluri and Shen, 2002;Li et al, 2007 andGhimire et al, 2016. The collocation techniques together with the expansion of trial solutions by utilizing different basis-functions were employed to solve the elliptic type boundary value problems; see, for example, Cheng et al, 2003;Hu et al, 2005;Tian et al, 2008, and Hu and Chen, 2008.…”

Section: Introductionmentioning

confidence: 99%

Directional Method of Fundamental Solutions for Three-dimensional Laplace Equation

Liu¹,

Fu²,

Kuo³

2017

JMR

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We propose a simple extension of the two-dimensional method of fundamental solutions (MFS) to a two-dimensional like MFS for the numerical solution of the three-dimensional Laplace equation in an arbitrary interior domain. In the directional MFS (DMFS) the directors are planar orientations, which can take the geometric anisotropy of the problem domain into account, and more importantly the order of the logarithmic singularity with ln R of the new fundamental solution is reduced than that of the conventional three-dimensional fundamental solution with singularity 1/r. Some numerical examples are used to validate the performance of the DMFS.

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Numerical solutions of elliptic partial differential equations using Chebyshev polynomials

Cited by 27 publications

References 19 publications

The method of particular solutions using trigonometric basis functions

The method of particular solutions using trigonometric basis functions

A Pseudo-Spectral Fourier Collocation Method for Inhomogeneous Elliptical Inclusions with Partial Differential Equations

Directional Method of Fundamental Solutions for Three-dimensional Laplace Equation

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