@inbook {2380,
	title = {Compact Collision Kernels for Hard Sphere and Coulomb Cross Sections; Fokker-Planck Coefficients},
	booktitle = {Rarefied Gas Dynamics},
	series = {Aip Conference Proceedings},
	volume = {1084},
	year = {2009},
	note = {ISI Document Delivery No.: BJF97Times Cited: 0Cited Reference Count: 26Chang, Yongbin Shizgal, 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 = {140-145},
	publisher = {Amer Inst Physics},
	organization = {Amer Inst Physics},
	address = {Melville},
	abstract = {A compact collision kernel is derived for both hard sphere and Coulomb cross sections. The difference between hard sphere interaction and Coulomb interaction is characterized by a parameter eta. With this compact collision kernel, the calculation of Fokker-Planck coefficients can be done for both the Coulomb and hard sphere interactions. The results for arbitrary order Fokker-Planck coefficients are greatly simplified. An alternate form for the Coulomb logarithm is derived with concern to the temperature relaxation in a binary plasma.},
	keywords = {Boltzmann collision term, collision frequency, cross, cut-off, EIGENVALUES, Fokker-Planck coefficients, GAS, Kernel, relaxation, SCATTERING, section, temperature relaxation},
	isbn = {0094-243X978-0-7354-0615-5},
	url = {<Go to ISI>://000265564800022},
	author = {Chang, Y. B. and Shizgal, B. D.},
	editor = {Abe, T.}
}
@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 {2995,
	title = {RELAXATION DYNAMICS OF HOT PROTONS IN A THERMAL BATH OF ATOMIC-HYDROGEN},
	journal = {Physical Review E},
	volume = {49},
	number = {1},
	year = {1994},
	note = {ISI Document Delivery No.: MV514Times Cited: 10Cited Reference Count: 58},
	month = {Jan},
	pages = {347-358},
	type = {Article},
	abstract = {We present a rigorous kinetic theory formulation of the relaxation of hot protons (H+) in a bath of thermal atomic hydrogen (H). We apply the (well-known) quantum-mechanical scattering theory to (H+,H) collisions and calculate the differential elastic cross section as a function of collision energy and scattering angle. This calculation includes the effects Of both direct and charge-exchange scattering. We then solve the time-dependent Boltzmann equation numerically for the H+ distribution function with an initial delta-function distribution. We also consider two approximate models for the collision dynamics, each based on the assumption that charge exchange dominates the relaxation and that no momentum is transferred in a collision (the linear-trajectory approximation). The first model uses the Rapp-Francis [J. Chem. Phys. 37, 2631 (1962)] energy-dependent cross section in the exact kernel which defines the Boltzmann collision operator. The second model uses a hard-sphere cross section in an approximate collision kernel. We compare the relaxation behavior calculated with the approximate formulations with the exact solution. We also calculate the mobility of H+ in H and compare the exact and-approximate; formulations. This study has applications to processes in astrophysics and aeronomy such as the non-thermal escape of H from planetary atmospheres.},
	keywords = {CHARGE-EXCHANGE, COLLISION KERNELS, EIGENVALUES, ENERGIES, equation, ESCAPE, EXOSPHERE, TRANSPORT, VENUS},
	isbn = {1063-651X},
	url = {<Go to ISI>://A1994MV51400048},
	author = {Clarke, A. S. and Shizgal, B.}
}
@article {7109,
	title = {COMPARISON OF WKB (WENTZEL-KRAMERS-BRILLOUIN) AND SWKB SOLUTIONS OF FOKKER-PLANCK EQUATIONS WITH EXACT RESULTS - APPLICATION TO ELECTRON THERMALIZATION},
	journal = {Canadian Journal of Physics},
	volume = {69},
	number = {6},
	year = {1991},
	note = {ISI Document Delivery No.: FZ294Times Cited: 5Cited Reference Count: 41},
	month = {Jun},
	pages = {712-719},
	type = {Article},
	abstract = {A comparison of WKB (Wentzel-Kramers-Brillouin) and SWKB eigenfunctions of the Schrodinger equation for potentials in the class encountered in supersymmetric quantum mechanics is presented. The potentials that are studied are those that result from the transformation of a Fokker-Planck eigenvalue problem to a Schrodinger equation. Linear Fokker-Planck equations of the type considered in this paper give the probability distribution function for a large number of physical situations. The time-dependent solutions can be expressed as a sum of exponential terms with each term characterized by an eigenvalue of the Fokker-Planck operator. The specific Fokker-Planck operator considered is the one that describes the thermalization of electrons in the inert gases. The WKB and SWKB semiclassical approximations are compared with exact numerical results. Although the eigenvalues can be very close to the exact values, we find significant departures for the eigenfunctions.},
	keywords = {ACTIVATED, APPROXIMATION, DISCRETE-ORDINATE METHOD, EIGENVALUES, INVARIANCE, ISOMERIZATION, NUCLEATION, RARE-GAS MODERATORS, RATE-PROCESSES, SHAPE, SOLVABLE POTENTIALS, SUPERSYMMETRIC QUANTUM-MECHANICS},
	isbn = {0008-4204},
	url = {<Go to ISI>://A1991FZ29400011},
	author = {Shizgal, B. and Demeio, L.}
}
