Approximation of multivariate functions and evaluation of particular solutions using Chebyshev polynomial and trigonometric basis functions

In this article, we introduce a type of basis functions to approximate a set of scattered data. Each of the basis functions is in the form of a truncated series over some orthogonal system of eigenfunctions. In particular, the trigonometric eigenfunctions are used. We test our basis functions on recovering the well-known Franke's and Peaks functions given by scattered data, and on the extension of a singular function from an irregular domain onto a square. These basis functions are further used in Kansa's method for solving Helmholtz-type equations on arbitrary domains. Proper one level and two level approximation techniques are discussed. A comparison of numerical with analytic solutions is given. The numerical results show that our approach is accurate and efficient.

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“…10. The data placed in the last row of Table III are the results by Chebyshev's approximation techniques described in [22].…”

Section: Example 5 Letmentioning

confidence: 99%

A basis function for approximation and the solutions of partial differential equations

Tian

Reutskiy²,

Chen

2007

Numerical Methods Partial

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“…(20) are real, we can begin with the following expression for the zero order modified Bessel function of second kink defined by Tikhonov and Samarskii [25]:…”

Section: Appendix II the Real Kernelsmentioning

confidence: 99%

“…The analytical particular solutions for an individual Helmholtz-type operator are referred to the works of Cheng [18] and Golberg et al [19] for splines and monomials respectively. It should be noted that the analytical particular solutions of monomials are also applicable when the Chebyshev method [20] is applied. Also, the explicit formula for the lateral displacement, slope, normal moment, and effective shear force are addressed for the kernels of MFS and the analytical particular solutions of DRM.…”

Section: Introductionmentioning

confidence: 99%

The Method of Fundamental Solutions with Dual Reciprocity for thin Plates on Winkler Foundations with Arbitrary Loadings

Tsai

2008

J. mech.

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This paper describes the combination of the method of fundamental solutions (MFS) and the dual reciprocity method (DRM) as a meshless numerical method to solve problems of thin plates resting on Winkler foundations under arbitrary loadings, where the DRM is based on the augmented polyharmonic splines constructed by splines and monomials. In the solution procedure, the arbitrary distributed loading is first approximated by the augmented polyharmonic splines (APS) and thus the desired particular solution can be represented by the corresponding analytical particular solutions of the APS. Thereafter, the complementary solution is solved formally by the MFS. In the mathematical derivations, the real coefficient operator in the governing equation is decomposed into two complex coefficient operators. In other words, the solutions obtained by the MFS-DRM are first treated in terms of these complex coefficient operators and then converted to real numbers in suitable ways. Furthermore, the boundary conditions of lateral displacement, slope, normal moment, and effective shear force are all given explicitly for the particular solutions of APS as well as the kernels of MFS. Finally, numerical experiments are carried out to validate these analytical formulas.

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“…This solution procedure may have the potential in obtaining analytical particular solutions of higher order PDEs constructed by products of Helmholtz-type operators. Also, the derived analytical particular solutions of monomials are applicable for the Chebyshev method [12].…”

Section: Introductionmentioning

confidence: 99%

The particular solutions for thin plates resting on Pasternak foundations under arbitrary loadings

Tsai¹

2009

Numerical Methods Partial

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Analytical particular solutions of splines and monomials are obtained for problems of thin plate resting on Pasternak foundation under arbitrary loadings, which are governed by a fourth-order partial differential equation (PDEs). These analytical particular solutions are valuable when the arbitrary loadings are approximated by augmented polyharmonic splines (APS) constructed by splines and monomials. In our derivations, the real coefficient operator in the governing equation is decomposed into two complex coefficient operators whose particular solutions are known in literature. Then, we use the difference trick to recover the analytical particular solutions of the original operator. In addition, we show that the derived particular solution of spline with its first few directional derivatives are bounded as r → 0. This solution procedure may have the potential in obtaining analytical particular solutions of higher order PDEs constructed by products of Helmholtz-type operators. Furthermore, we demonstrate the usages of these analytical particular solutions by few numerical cases in which the homogeneous solutions are complementarily solved by the method of fundamental solutions (MFS).

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Approximation of multivariate functions and evaluation of particular solutions using Chebyshev polynomial and trigonometric basis functions

Cited by 25 publications

References 19 publications

A basis function for approximation and the solutions of partial differential equations

A basis function for approximation and the solutions of partial differential equations

The Method of Fundamental Solutions with Dual Reciprocity for thin Plates on Winkler Foundations with Arbitrary Loadings

The particular solutions for thin plates resting on Pasternak foundations under arbitrary loadings

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