@inbook {2604,
	title = {Relaxation Behavior for the Lorentz Fokker Planck Equation: Model Cross Sections and Nonextensive Entropy},
	booktitle = {Rarefied Gas Dynamics},
	series = {Aip Conference Proceedings},
	volume = {1084},
	year = {2009},
	note = {ISI Document Delivery No.: BJF97Times Cited: 0Cited Reference Count: 19Shizgal, Bernie D.Proceedings Paper26th International Symposium on Rarefied Gas Dynmaics (RGD26)JUN 20-JUL 25, 2008Kyoto, JAPANJapan Soc Promot Sci, Japan Aerpspace Explorat Agcy, Soc Promot Space Sci, Iwantani Naoji Fdn, Inoue Fdn Sci, Casio Sci Promot Fdn, Kaijma Fdn, IHI Corp, IHI Aerospac Engn Co Ltd, Osaka Vaccuun Ltd, Nissin Inc2 HUNTINGTON QUADRANGLE, STE 1NO1, MELVILLE, NY 11747-4501 USA},
	pages = {39-44},
	publisher = {Amer Inst Physics},
	organization = {Amer Inst Physics},
	address = {Melville},
	abstract = {The relaxation behavior of a test particle in a background gas at equilibrium for which the test particle background particle mass ratio, m(t)/m(b) tends to zero, that is the Lorentz limit, is studied with the Lorentz-Fokker-Planck equation. This is the situation for electrons in inert gases. If there is also an applied electric field the stationary distribution is not the equilibrium Maxwellian. In the present paper, the evolution of the isotropic distribution function is studied with both a finite difference solution and one based on the expansion of the distribution function in the eigenfunctions of the Fokker-Planck operator. The eigenvalue spectrum depends strongly on the velocity dependence of the momentum transfer cross section in the Fokker-Planck equation. In particular, the spectrum is composed of a discrete spectrum and a continuum, and the continuum can dominate the spectrum. The role of the eigenvalue spectrum of this operator on the time evolution of the distribution function is studied. In particular, the time dependence of the entropy is determined and we show that the Kullback-Leibler entropy rationalizes the approach to a stationary distribution. The Tsallis non-extensive entropy is not required to explain the evolution of the distributions.},
	keywords = {Boltzmann equation, DEGRADATION, electron, Maxwell molecules, METHOD QDM, NONCLASSICAL BASIS FUNCTIONS, pseudospectral method, QUADRATURE DISCRETIZATION METHOD, quantum cross section, RARE-GAS MODERATORS, SUPRATHERMAL PARTICLE DISTRIBUTIONS, THERMALIZATION},
	isbn = {0094-243X978-0-7354-0615-5},
	url = {<Go to ISI>://000265564800005},
	author = {Shizgal, B. D.},
	editor = {Abe, T.}
}
@article {2165,
	title = {An efficient mapped pseudospectral method for weakly bound states: vibrational states of He-2, Ne-2, Ar-2 and Cs-2},
	journal = {Journal of Physics B-Atomic Molecular and Optical Physics},
	volume = {41},
	number = {18},
	year = {2008},
	note = {ISI Document Delivery No.: 347PITimes Cited: 2Cited Reference Count: 54Lo, Joseph Q. W. Shizgal, Bernie D.},
	month = {Sep},
	pages = {13},
	type = {Article},
	abstract = {This paper reports the application of pseudospectral methods to the calculation of the bound states of He-2, Ne-2, Ar-2 and Cs-2. The calculation of the uppermost loosely bound states, only one for He-2, presents a challenge for most numerical methods. We employ the quadrature discretization method which is a collocation based on quadrature points defined with weight functions that approximate the ground states. A mapping procedure is proposed that greatly improves the convergence of the uppermost states that are very loosely bound and characterized by very diffuse eigenfunctions. Comparisons with the results with other methods and experiment are also carried out wherever possible.},
	keywords = {BASIS FUNCTIONS, DIATOMIC-MOLECULES, DIMERS, DISCRETE, HELIUM DIMER, LAGRANGE-MESH METHOD, LONG-RANGE MOLECULES, NONCLASSICAL, QUADRATURE DISCRETIZATION METHOD, QUANTUM-MECHANICAL PROBLEMS, RARE-GAS, ULTRAVIOLET-LASER SPECTROSCOPY, VARIABLE REPRESENTATION},
	isbn = {0953-4075},
	url = {<Go to ISI>://000259153100008},
	author = {Lo, J. Q. W. and Shizgal, B. D.}
}
@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.}
}
@article {4757,
	title = {A direct spectral collocation Poisson solver in polar and cylindrical coordinates},
	journal = {Journal of Computational Physics},
	volume = {160},
	number = {2},
	year = {2000},
	note = {ISI Document Delivery No.: 315BRTimes Cited: 22Cited Reference Count: 33},
	month = {May},
	pages = {453-469},
	type = {Article},
	abstract = {In this paper, we present a direct spectral collocation method for the solution of the Poisson equation in polar and cylindrical coordinates. The solver is applied to the Poisson equations for several different domains including a part of a disk, an annulus, a unit disk, and a cylinder. Unlike other Poisson solvers for geometries such as unit disks and cylinders, no pole condition is involved for the present solver. The method is easy to implement, fast, and gives spectral accuracy. We also use the weighted interpolation technique and nonclassical collocation points to improve the convergence. (C) 2000 Academic Press.},
	keywords = {ARBITRARY ORDER ACCURACY, coordinates, cylindrical, EXPANSION, METHOD QDM, NONCLASSICAL BASIS FUNCTIONS, Poisson solver, polar coordinates, QUADRATURE DISCRETIZATION METHOD, SCHRODINGER-EQUATION, SINGULARITIES, spectral collocation, TSCHEBYSCHEFF POLYNOMIALS},
	isbn = {0021-9991},
	url = {<Go to ISI>://000087093200002},
	author = {Chen, H. L. and Su, Y. H. and Shizgal, B. D.}
}
