@article {983,
	title = {Neutral particle collisional relaxation of suprathermal distributions in aeronomy},
	journal = {Planetary and Space Science},
	volume = {52},
	number = {10},
	year = {2004},
	note = {ISI Document Delivery No.: 858UKTimes Cited: 3Cited Reference Count: 46},
	month = {Aug},
	pages = {915-922},
	type = {Article},
	abstract = {There are numerous situations in laboratory experiments and in atmospheric science that are characterized by distributions of energetic neutral species with extended high-energy tails. This paper provides a detailed analysis of the relaxation of such isotropic nonequilibrium distributions of neutral species. We consider a minor constituent, referred to as the test-particle dilutely dispersed in a second component that acts as a heat bath at equilibrium. A hard sphere cross section is assumed for the collisions of the test particles and the heat bath particles, and collisions between test particles are not included in the analysis. We study the approach to equilibrium with a finite difference method of solution of the Boltzmann equation. The solution of the Boltzmann equation is also presented as an expansion in the eigenfunctions of the Boltzmann collision operator. The main objective of the paper is the calculation of the energy-dependent relaxation times for the distribution function. It is anticipated that these relaxation times will not exhibit a strong energy dependence owing to the energy-independent hard sphere cross section. This relaxation behaviour is important in the characterization of nonthermal populations of energetic atoms in aeronomy. The results are compared with a similar analysis for Coulomb collisions described in the preceeding paper. (C) 2004 Elsevier Ltd. All rights reserved.},
	keywords = {BINARY GASES, Boltzmann equation, CHARGE-EXCHANGE, distribution functions, ENERGY-TRANSFER, FAST NITROGEN-ATOMS, FOKKER-PLANCK EQUATION, HYDROGEN, KINETIC-THEORY, MARS, O(P-3) ATOMS, OXYGEN-ATOMS, relaxation times, suprathermal populations},
	isbn = {0032-0633},
	url = {<Go to ISI>://000224217300003},
	author = {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 {4226,
	title = {The Quadrature Discretization Method (QDM) in the solution of the Schrodinger equation},
	journal = {Journal of Mathematical Chemistry},
	volume = {24},
	number = {4},
	year = {1998},
	note = {ISI Document Delivery No.: 165EGTimes Cited: 12Cited Reference Count: 63},
	pages = {321-343},
	type = {Article},
	abstract = {The Quadrature Discretization Method (QDM) is employed in the solution of several one-dimensional Schrodinger equations that have received considerable attention in the literature. The QDM is based on the discretization of the wave function on a grid of points that coincide with the points of a quadrature. The quadrature is based on a set of non-classical polynomials orthogonal with respect to a weight function. For a certain class of problems with potentials of the form that occur in supersymmetric quantum mechanics, the ground state wavefunction is known. In the present paper, the weight functions that are used are related to the ground state wavefunctions if known, or some approximate form. The eigenvalues and eigenfunctions of four different potential functions discussed extensively in the literature are calculated and the results are compared with published values.},
	keywords = {ACCURATE EIGENVALUES, ANHARMONIC-OSCILLATORS, DISCRETE-ORDINATE, ENERGY-LEVELS, FOKKER-PLANCK EQUATION, HILL DETERMINANT APPROACH, INNER-PRODUCT TECHNIQUE, method, NONCLASSICAL BASIS FUNCTIONS, OPTICAL WAVE-GUIDES, QUANTUM-MECHANICAL MODELS},
	isbn = {0259-9791},
	url = {<Go to ISI>://000078506600002},
	author = {Chen, H. L. and Shizgal, B. D.}
}
@article {3811,
	title = {The quadrature discretization method (QDM) in the solution of the Schrodinger equation with nonclassical basis functions},
	journal = {Journal of Chemical Physics},
	volume = {104},
	number = {11},
	year = {1996},
	note = {ISI Document Delivery No.: TY728Times Cited: 34Cited Reference Count: 66},
	month = {Mar},
	pages = {4137-4150},
	type = {Article},
	abstract = {A discretization method referred to as the Quadrature Discretization Method (QDM) is introduced for the solution of the Schrodinger equation. The method has been used previously for the solution of Fokker-Planck equations. The Fokker-Planck equation can be transformed to a Schrodinger equation with a potential of the form that occurs in supersymmetric quantum mechanics, For this class of potentials, the groundstate wave function is known. The QDM is based on the discretization of the wave function on a grid of points that coincide with the points of a quadrature. The quadrature is based on a set of nonclassical polynomials orthogonal with respect to a weight function determined by the potential function in the Schrodinger equation. For the Fokker-Planck operator, the weight function that provides rapid convergence of the eigenvalues are the steady distributions at infinite time, that is, the ground state wave functions. In the present paper, the weight functions used in an analogous solution of the Schrodinger equation are related to the ground state wave functions if known, or some approximate form. Calculations are carried out for a model systems, the Morse potential, and for the vibrational levels of O-2 and Ar-Xe with realistic pair potentials. For O-2, the wave functions are used to calculate the vibrationally inelastic transition amplitudes for a Morse potential and compared with exact analytic results. The eigenvalues of a two-dimensional Schrodinger equation with the Henon-Heiles potential are also calculated. The rate of convergence of the eigenvalues and the eigenfunctions of the Schrodinger equation is very rapid with this approach. (C) 1996 American Institute of Physics.},
	keywords = {BOUND-STATES, COEFFICIENTS, DISCRETE-ORDINATE METHOD, EIGENVALUE PROBLEMS, FOKKER-PLANCK EQUATION, ORTHOGONAL POLYNOMIALS, POTENTIALS, QUANTUM-MECHANICS, RECURRENCE, ROVIBRATIONAL STATES},
	isbn = {0021-9606},
	url = {<Go to ISI>://A1996TY72800030},
	author = {Shizgal, B. D. and Chen, H.}
}
@article {7177,
	title = {COULOMB MILNE PROBLEM},
	journal = {Physical Review A},
	volume = {45},
	number = {6},
	year = {1992},
	note = {ISI Document Delivery No.: HK384Times Cited: 9Cited Reference Count: 62},
	month = {Mar},
	pages = {3687-3699},
	type = {Article},
	abstract = {The transport properties of an ensemble of charged test particles diffusing from infinity through a second charged species that fills the positive half-space are studied. It is assumed that there are no incoming particles at the boundary between the medium in the positive half-space and the vacuum that fills the negative half-space. This problem has been referred to as the Milne problem and has previously been considered for neutral species [M. J. Lindenfold and B. Shizgal, Phys. Rev. A 27, 1657 (1983)]. The present paper examines the rarefied-gas dynamical effects for the Coulomb interaction between the two species. The distribution function of the background gas is taken to be a Maxwellian and the test-particle distribution is determined from a solution of the Boltzmann equation with its expansion in Burnett functions, products of Laguerre polynomials, and spherical harmonics. The extrapolation length is calculated together with the density and temperature profiles and compared with the Chapman-Enskog results valid far from the boundary. A comparison with the previous results for the hard-sphere cross section describing collisions between two neutral species is also presented.},
	keywords = {ABSORBING SPHERE, ANOMALOUS, BOUNDARY-LAYER, COUPLED TEST PARTICLE, FOKKER-PLANCK EQUATION, HEAT-FLUX, KINETIC, PLASMA, SOLAR-WIND, STEADY-STATE, STRONGLY INHOMOGENEOUS SYSTEMS, TRANSPORT},
	isbn = {1050-2947},
	url = {<Go to ISI>://A1992HK38400038},
	author = {Barrett, J. and Demeio, L. and Shizgal, B.}
}
@article {7108,
	title = {A UNIFORM WKB ANALYSIS OF THE COUPLING OF ELECTRON-ATTACHMENT AND THERMALIZATION IN GASES},
	journal = {Journal of Physics B-Atomic Molecular and Optical Physics},
	volume = {24},
	number = {13},
	year = {1991},
	note = {ISI Document Delivery No.: FX594Times Cited: 11Cited Reference Count: 44},
	month = {Jul},
	pages = {2909-2923},
	type = {Article},
	abstract = {The presence of small amounts of strongly electronegative gases such as SF6 and CCl4 can greatly perturb from equilibrium the electron energy distribution function in a variety of electron devices. The effect of the addition of these electron attaching gases on the relaxation behaviour of a non-equilibrium electron distribution is studied with a linear Fokker-Planck equation. The time dependence of the electron mobility as well as other transport properties can be expressed as a sum of exponential terms with each term characterized by an eigenvalue of the Fokker-Planck operator. This eigenvalue problem can be transformed to a Schrodinger equation with a potential function parametrized by the electron-atom cross sections, the electric field strength and the gas temperature. In the present paper, the eigenvalues and eigenfunctions are calculated with a WKB semiclassical approximation and compared with exact results. The application of the WKB approximation provides a useful way to calculate these strongly perturbed distributions. The specific applications in this paper are for the relaxation of electrons in the inert gases in the presence of small amounts of SF6 or CCl4.},
	keywords = {CROSS-SECTIONS, DISCHARGES, FOKKER-PLANCK EQUATION, MIXTURES, MODERATORS, RARE, relaxation, SF6, SWARM DEVELOPMENT, TRANSPORT-COEFFICIENTS},
	isbn = {0953-4075},
	url = {<Go to ISI>://A1991FX59400007},
	author = {Shizgal, B.}
}
