@article {737,
	title = {Towards the resolution of the Gibbs phenomena},
	journal = {Journal of Computational and Applied Mathematics},
	volume = {161},
	number = {1},
	year = {2003},
	note = {ISI Document Delivery No.: 742YDTimes Cited: 27Cited Reference Count: 25},
	month = {Dec},
	pages = {41-65},
	type = {Article},
	abstract = {It is well known that the expansion of an analytic nonperiodic function on a finite interval in a Fourier series leads to spurious oscillations at the interval boundaries. This result is known as the Gibbs phenomenon. The present paper introduces a new method for the resolution of the Gibbs phenomenon which follows on the reconstruction method of Gottlieb and coworkers (SIAM Rev. 39 (1997) 644) based on Gegenbauer polynomials orthogonal with respect to weight function (1-x(2))(lambda-1/2). We refer to their approach as the direct method and to the new methodology as the inverse method. Both methods use the finite set of Fourier coefficients of some given function as input data in the re-expansion of the function in Gegenbauer polynomials or in other orthogonal basis sets. The finite partial sum of the new expansion provides a spectrally accurate approximation to the function. In the direct method, this requires that certain conditions are met concerning the parameter lambda in the weight function, the number of Fourier coefficients, N and the number of Gegenbauer polynomials, m. We show that the new inverse method can give exact results for polynomials independent of lambda and with m=N. The paper presents several numerical examples applied to a single domain or to subdomains of the main domain so as to illustrate the superiority of the inverse method in comparison with the direct method. (C) 2003 Elsevier B.V. All rights reserved.},
	keywords = {equation, Gibbs phenomenon, spectral methods},
	isbn = {0377-0427},
	url = {<Go to ISI>://000186545000003},
	author = {Shizgal, B. D. and Jung, J. H.}
}
@article {504,
	title = {Spectral methods based on nonclassical basis functions: the advection-diffusion equation},
	journal = {Computers \& Fluids},
	volume = {31},
	number = {4-7},
	year = {2002},
	note = {ISI Document Delivery No.: 533JJTimes Cited: 4Cited Reference Count: 17},
	month = {May-Sep},
	pages = {825-843},
	type = {Article},
	abstract = {The advection diffusion equation has a long history as a benchmark for numerical methods. If advection dominates over diffusion the numerical solution is difficult, especially if there are boundary layers to resolve. The eigenvalues of the approximate discretized spatial operator can be complex, and if the real part of any one of these is positive, the temporal development of the discretized equations by finite differences is unstable. The stability or the time development by finite difference methods is usually discussed in terms of the eigenvalues of the first and second derivative spatial operators. In this paper, the eigenvalues of the spatial operator in the advection-diffusion equation determined with a Galerkin method based on a new set of nonclassical basis functions are all real and negative. A collocation solution of the time dependent advection-diffusion equation is also considered and results using Chebyshev-Gauss-Lobatto and Legendre-Gauss-Lobatto quadratures are compared with results based on new basis functions. The results demonstrate that improved convergence can be obtained with new basis functions defined with respect to nonclassical weight functions. (C) 2002 Elsevier Science Ltd. All rights reserved.},
	keywords = {advection diffusion, collocation, EIGENVALUES, FOKKER-PLANCK EQUATION, OPERATOR, QUADRATURE DISCRETIZATION METHOD, spectral methods},
	isbn = {0045-7930},
	url = {<Go to ISI>://000174525900027},
	author = {Shizgal, B. D.}
}
@article {5033,
	title = {A spectral solution of the Sturm-Liouville equation: comparison of classical and nonclassical basis sets},
	journal = {Journal of Computational and Applied Mathematics},
	volume = {136},
	number = {1-2},
	year = {2001},
	note = {ISI Document Delivery No.: 475METimes Cited: 16Cited Reference Count: 28},
	month = {Nov},
	pages = {17-35},
	type = {Article},
	abstract = {This paper considers a spectral method of solution of the Stunn-Liouville equation and the associated Schroedinger equation. The main objective is to develop a collocation method based on quadrature (collocation) points generated from nonclassical polynomials. The polynomials and associated quadrature points are calculated with Gautschi{\textquoteright}s Stieltjes procedure from some specified weight function. The particular spectral method used here with nonclassical basis sets is referred to as the Quadrature Discretization Method (QDM). The QDM and a related weighted QDM are applied to several Sturm-Liouville and Schroedinger equations and the results are compared with the traditional spectral methods based on Chebyshev and Legendre quadrature points. The results are also compared with the results of other workers wherever available. The QDM was found to give the most rapid convergence relative to other methods for the problems studied. (C) 2001 Elsevier Science B.V. All rights reserved.},
	keywords = {LAGRANGE, MESHES, METHOD QDM, nonclassical polynomials, NUMERICAL-INTEGRATION, ORTHOGONAL POLYNOMIALS, QUADRATURE DISCRETIZATION METHOD, SCHRODINGER-EQUATION, Schroedinger equation, spectral methods, Sturm-Liouville equation},
	isbn = {0377-0427},
	url = {<Go to ISI>://000171167400002},
	author = {Chen, H. and Shizgal, B. D.}
}
